[{"references": ["instAddNat", "PartialOrder.toPreorder", "RelSeries.toFun", "Preorder.toLT", "Fin", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "RelSeries.length", "SemilatticeInf.toPartialOrder", "HarderNarasimhan.PayoffFunction.relSeries_step_lt", "instLTNat", "Exists", "And", "BoundedOrder", "instNeZeroNatHAdd_1", "Bot.bot", "RelSeries.last", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "RelSeries.head", "Nontrivial", "Lattice", "Top.top", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction.semistableRel", "HarderNarasimhan.PayoffFunction.Admissible", "Nat.cast", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "RelSeries", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "Nat.instNeZeroSucc", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Fin.NatCast.instNatCast", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "WellFoundedGT", "OrderBot.toBot", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Lattice.toSemilatticeInf", "instHAdd", "HarderNarasimhan.PayoffFunction.relSeries_succ_step_lt", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "LE.le", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.exists_relSeries_semistableRel", "constType": "∀ {ℒ : Type u_1} [Nontrivial ℒ] [inst : Lattice ℒ] [inst_1 : BoundedOrder ℒ] [WellFoundedGT ℒ] {S : Type u_2}\n [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [μ.ADCC] [μ.IsConvex] [μ.Admissible],\n ∃ s,\n s.head = ⊥ ∧\n s.last = ⊤ ∧\n ∀ (i : ℕ) (hi : i + 1 < s.length),\n ¬μ.A { left := s.toFun ↑i, right := s.toFun ↑(i + 1), lt := ⋯ } ≤\n μ.A { left := s.toFun ↑(i + 1), right := s.toFun ↑(i + 2), lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.rec", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {S : Type u_2} →\n {motive : HarderNarasimhan.PayoffFunction ℒ S → Sort u} →\n ((toFun : HarderNarasimhan.StrictIntvl ℒ → S) → motive { toFun := toFun }) →\n (t : HarderNarasimhan.PayoffFunction ℒ S) → motive t", "constCategory": "Other"}, {"references": ["Inhabited"], "name": "Inhabited.mk", "constType": "{α : Sort u} → α → Inhabited α", "constCategory": "Other"}, {"references": ["Set.Nonempty", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.breakpoints", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.breakpoints_nonempty", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} [hwf : WellFoundedGT ℒ]\n [hμDCC : μ.ADCC], μ.IsConvexOn I → (μ.breakpoints I).Nonempty", "constCategory": "Theorem"}, {"references": ["SMul", "SMulZeroClass", "Zero"], "name": "SMulZeroClass.toSMul", "constType": "{M : Type u_12} → {A : Type u_13} → {inst : Zero A} → [self : SMulZeroClass M A] → SMul M A", "constCategory": "Definition"}, {"references": ["HEq", "Eq"], "name": "heq_of_eq", "constType": "∀ {α : Sort u_1} {a a' : α}, a = a' → a ≍ a'", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.StrongDCC", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_top_eq_max_top_iff_hasNashEquilibrium", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike] [h₁ : μ.WeakACC]\n [h₂ : μ.StrongDCC], μ.min ⊤ = μ.max ⊤ ↔ μ.HasNashEquilibrium", "constCategory": "Theorem"}, {"references": [], "name": "Equiv", "constType": "Sort u_1 → Sort u_2 → Sort (max (max 1 u_1) u_2)", "constCategory": "Other"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.rec", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.EventuallyTopDCC → Sort u} →\n (t : μ.EventuallyTopDCC) →\n ((exists_eq_top :\n ∀ (x : ℕ → ℒ) (hx : StrictAnti x), ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "OrderTop.toTop", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instIsConvexOfIsConvexOnTopStrictIntvl", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_2 : Nontrivial ℒ] [inst_3 : BoundedOrder ℒ] [μ.IsConvexOn ⊤],\n μ.IsConvex", "constCategory": "Theorem"}, {"references": ["SupSet", "ConditionallyCompletePartialOrderSup"], "name": "ConditionallyCompletePartialOrderSup.toSupSet", "constType": "{α : Type u_3} → [self : ConditionallyCompletePartialOrderSup α] → SupSet α", "constCategory": "Definition"}, {"references": ["Nat", "RelSeries", "SetRel"], "name": "RelSeries.length", "constType": "{α : Type u_1} → {r : SetRel α α} → RelSeries r → ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {x : ℒ} →\n {motive : μ.IsBreakpoint I x → Sort u} →\n ((mem : x ∈ I) →\n (ne_left : I.left ≠ x) →\n (not_lt :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } <\n μ.A { left := I.left, right := y, lt := ⋯ }) →\n (le_of_eq :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } =\n μ.A { left := I.left, right := x, lt := ⋯ } →\n y ≤ x) →\n motive ⋯) →\n (t : μ.IsBreakpoint I x) → motive t", "constCategory": "Other"}, {"references": ["IsNoetherian", "AddCommMonoid", "Module", "Module.Finite", "Semiring"], "name": "Module.IsNoetherian.finite", "constType": "∀ (R : Type u_1) (M : Type u_3) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : _root_.Module R M]\n [IsNoetherian R M], Module.Finite R M", "constCategory": "Theorem"}, {"references": ["DFunLike"], "name": "FunLike", "constType": "Sort u_1 → Sort u_2 → Sort u_3 → Sort (max (max (max 1 u_1) u_2) u_3)", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "Module", "Submodule.instTop", "CommSemiring.toSemiring", "HarderNarasimhan.CoprimaryFiltration.length", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "LE.le", "Top.top", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "instLENat", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.eq_top_of_length_le", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n {F : HarderNarasimhan.CoprimaryFiltration R M} {m : ℕ}, F.length ≤ m → F m = ⊤", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Subtype", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "PartialOrder", "LE.le", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE", "Subtype.val"], "name": "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_2", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} (a : { x // x ∈ I }), ↑a ≤ I.right", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Preorder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.apply_le_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, μ I ≤ μ.max I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "OrderBot.toBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "SemilatticeInf.toPartialOrder", "bot_lt_iff_ne_bot", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Iff.mpr", "Lattice", "Nontrivial", "Ne", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.isSemistable_of_hasNashEquilibrium", "constType": "∀ {ℒ : Type u_3} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] {S : Type u_4}\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℒ) (hx : x ≠ ⊥), (μ.restrict { left := ⊥, right := x, lt := ⋯ }).WeakACC) →\n (∀ (x : ℒ) (hx : x ≠ ⊥), (μ.restrict { left := ⊥, right := x, lt := ⋯ }).WeakSlopeLikeAtTop) →\n μ.HasNashEquilibrium → μ.IsSemistable", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Iff.mpr", "Iff", "Nontrivial", "LE.le", "HarderNarasimhan.PayoffFunction.max", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "lt_top_iff_ne_top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.hasNashEquilibrium_iff_le_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.StrongDCC] [μ.WeakSlopeLikeAtBot],\n μ.HasNashEquilibrium ↔ ∀ (y : ℒ) (hy : y ≠ ⊤), μ.max ⊤ ≤ μ.max { left := y, right := ⊤, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "associatedPrimes", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "CommRing", "Set.instMembership", "HarderNarasimhan.IsCoprimary", "Ideal", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.existsUnique_associatedPrime", "constType": "∀ {R : Type u_1} {inst : CommRing R} {M : Type u_2} {inst_1 : AddCommGroup M} {inst_2 : _root_.Module R M}\n [self : HarderNarasimhan.IsCoprimary R M], ∃! p, p ∈ associatedPrimes R M", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.preorder", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction", "Eq", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, (μ.restrict I).B = μ.B.restrict I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.left", "True", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.StrictIntvl.left_mem._simp_1", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), (I.left ∈ I) = True", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.PayoffFunction.StrongDCC.rec", "HarderNarasimhan.PayoffFunction.StrongDCC.mk", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.StrongDCC → Sort u} →\n (t : μ.StrongDCC) →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["le_refl", "OrderDual", "Preorder", "lt_iff_le_not_ge", "OrderDual.instLT", "LE.le", "Preorder.toLT", "OrderDual.instLE", "Preorder.mk", "Preorder.toLE", "LE.le.trans"], "name": "OrderDual.instPreorder", "constType": "(α : Type u_2) → [Preorder α] → Preorder αᵒᵈ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_nonempty", "Finset.min'", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.instMembership", "HarderNarasimhan.StrictIntvl", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "associatedPrimes", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.module", "OrderHom", "DFunLike.coe", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "Set", "HarderNarasimhan.StrictIntvl.right", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "toLinearExtension", "Set.toFinset", "AddCommGroup", "CommRing", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "ExistsUnique", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.toLinearExtension_eq_min'", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (∃! p, p ∈ associatedPrimes R (↥I.right ⧸ I.left.submoduleOf I.right)) →\n ∀ {p : PrimeSpectrum R},\n p.asIdeal ∈ associatedPrimes R (↥I.right ⧸ I.left.submoduleOf I.right) →\n toLinearExtension p = (HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I).toFinset.min' ⋯", "constCategory": "Theorem"}, {"references": ["Nat", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.ctorIdx", "constType": "{ℒ : Type u_1} → {inst : LT ℒ} → {S : Type u_2} → HarderNarasimhan.PayoffFunction ℒ S → ℕ", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsStable", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "CompleteLinearOrder", "Nat.instIsOrderedAddMonoid", "SemilatticeInf.toPartialOrder", "instLTNat", "Nat.instPartialOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl", "Nat", "Iff", "Lattice", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "WellFoundedGT", "Preorder.toLE", "Nat.instAddMonoid", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "CompletelyDistribLattice.toCompleteLattice", "HAdd.hAdd", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "lt_add_one", "LE.le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "instLENat"], "name": "HarderNarasimhan.PayoffFunction.piecewise_isStable_iff", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] [WellFoundedGT ℒ] {S : Type u_2} [inst_2 : CompleteLinearOrder S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) [μ.IsSlopeLike] [μ.EventuallyTopDCC] (f : ℕ → ℒ) {n : ℕ}\n (hsa : ∀ (i j : ℕ), i < j → j ≤ n → f j < f i),\n (∀ (i : ℕ) (hi : i < n), (μ.restrict { left := f (i + 1), right := f i, lt := ⋯ }).IsStable) ↔\n ∀ (i : ℕ) (hi : i < n) (z : ℒ) (h' : f (i + 1) < z),\n z < f i → μ { left := f (i + 1), right := z, lt := h' } < μ { left := f (i + 1), right := f i, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Bot.mk", "Module", "Singleton.singleton", "Submodule.instBot._proof_1", "AddCommMonoid.toAddMonoid", "Submodule", "AddSubsemigroup.mk", "AddCommMonoid", "Bot", "Zero.toOfNat0", "Set", "AddSubmonoid.mk", "AddZero.toAdd", "AddZeroClass.toAddZero", "Set.instSingletonSet", "Bot.bot", "OfNat.ofNat", "Submodule.instBot._proof_3", "Submodule.mk", "AddSubmonoid", "AddSubmonoid.instBot", "Submodule.instBot._proof_2", "AddZero.toZero", "AddMonoid.toAddZeroClass", "Semiring"], "name": "Submodule.instBot", "constType": "{R : Type u_1} →\n {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → Bot (Submodule R M)", "constCategory": "Definition"}, {"references": ["LT.lt", "PartialOrder.toPreorder", "Iff", "PartialOrder", "Top.top", "Preorder.toLT", "OrderTop", "Ne", "Preorder.toLE", "OrderTop.toTop"], "name": "lt_top_iff_ne_top", "constType": "∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderTop α] {a : α}, a < ⊤ ↔ a ≠ ⊤", "constCategory": "Theorem"}, {"references": ["Unique", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Unique.mk", "Preorder.toLT", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "_private.HarderNarasimhan.Filtration.Unique.0.HarderNarasimhan.PayoffFunction.instUniqueHarderNarasimhanFiltration._proof_1", "HarderNarasimhan.PayoffFunction.instInhabitedHarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.IsConvex", "CompletelyDistribLattice.toCompleteLattice", "_private.HarderNarasimhan.Filtration.Unique.0.HarderNarasimhan.PayoffFunction.eq_hnFiltration", "Nontrivial", "Lattice", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "CompleteLinearOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instUniqueHarderNarasimhanFiltration", "constType": "{ℒ : Type u_1} →\n [Nontrivial ℒ] →\n [inst : Lattice ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [WellFoundedGT ℒ] →\n {S : Type u_2} →\n [inst_3 : CompleteLinearOrder S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} → [μ.ADCC] → [μ.IsConvex] → Unique μ.HarderNarasimhanFiltration", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.bot_lt_of_lt", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "Eq", "instLTNat", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.payoff_bot_eq_top_payoff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hsl : μ.IsSlopeLike]\n (F : μ.JordanHolderFiltration) (i : ℕ) (hi : i < F.length), μ { left := ⊥, right := F i, lt := ⋯ } = μ ⊤", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })) →\n μ.IsSlopeLike", "constCategory": "Definition"}, {"references": ["Prod.casesOn", "Prod", "Prod.mk"], "name": "HarderNarasimhan.PayoffFunction.semistableRel.match_1", "constType": "{ℒ : Type u_1} → (motive : ℒ × ℒ → Sort u_2) → (x : ℒ × ℒ) → ((x y : ℒ) → motive (x, y)) → motive x", "constCategory": "Definition"}, {"references": ["instLTNat", "CommRing.toCommSemiring", "Module", "Submodule.instTop", "CommSemiring.toSemiring", "HarderNarasimhan.CoprimaryFiltration.length", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "LT.lt", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "Top.top", "AddCommGroup.toAddCommMonoid", "Ne", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.ne_top_of_lt", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n {F : HarderNarasimhan.CoprimaryFiltration R M} {m : ℕ}, m < F.length → F m ≠ ⊤", "constCategory": "Theorem"}, {"references": ["Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ConditionallyCompleteLattice.toLattice", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n (HarderNarasimhan.Coprimary.payoff R M).IsConvex", "constCategory": "Theorem"}, {"references": ["instAddNat", "HAdd.hAdd", "NeZero", "Nat", "Zero.ofOfNat0", "instHAdd", "instOfNatNat", "OfNat.ofNat"], "name": "Nat.instNeZeroSucc", "constType": "∀ {n : ℕ}, NeZero (n + 1)", "constCategory": "Theorem"}, {"references": ["Set", "Add", "AddSubsemigroup"], "name": "AddSubsemigroup.carrier", "constType": "{M : Type u_3} → [inst : Add M] → AddSubsemigroup M → Set M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "And", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk.injEq", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] (left right : ℒ) (lt : left < right) (left_1 right_1 : ℒ) (lt_1 : left_1 < right_1),\n ({ left := left, right := right, lt := lt } = { left := left_1, right := right_1, lt := lt_1 }) =\n (left = left_1 ∧ right = right_1)", "constCategory": "Theorem"}, {"references": ["FunLike", "DFunLike.mk", "EquivLike.toFunLike._proof_1", "EquivLike.coe", "EquivLike"], "name": "EquivLike.toFunLike", "constType": "{E : Sort u_1} → {α : Sort u_3} → {β : Sort u_4} → [EquivLike E α β] → FunLike E α β", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn.rec", "HarderNarasimhan.PayoffFunction.IsConvexOn.mk", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "Lattice", "LE.le", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {motive : μ.IsConvexOn I → Sort u} →\n (t : μ.IsConvexOn I) →\n ((le :\n ∀ (x y : ℒ),\n x ∈ I →\n y ∈ I →\n ∀ (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Unique", "CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "Unique.mk", "AddCommGroup", "CommRing", "_private.HarderNarasimhan.Coprimary.Filtration.0.HarderNarasimhan.CoprimaryFiltration.instUnique._proof_1", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.Coprimary.instInhabitedCoprimaryFiltration", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.instUnique", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] → Unique (HarderNarasimhan.CoprimaryFiltration R M)", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsStable", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Ne", "OrderBot.toBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.ne", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsStable] (x : ℒ) (hx : ⊥ < x),\n x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration._sizeOf_inst", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "SizeOf", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instSizeOfNat", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "SizeOf.sizeOf", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "instSizeOfDefault", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk.sizeOf_spec", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_4 : SizeOf ℒ] [inst_5 : SizeOf S]\n (toFun : ℕ → ℒ) (length : ℕ) (antitone : Antitone toFun) (head_eq_top : toFun 0 = ⊤)\n (length_eq_bot : toFun length = ⊥) (strictAntiOn : StrictAntiOn toFun (Set.Iic length))\n (step_payoff_eq : ∀ (i : ℕ) (hi : i < length), μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤)\n (payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } < μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }),\n sizeOf\n { toFun := toFun, length := length, antitone := antitone, head_eq_top := head_eq_top,\n length_eq_bot := length_eq_bot, strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq,\n payoff_lt_of_between := payoff_lt_of_between } =\n 1 + sizeOf length + sizeOf head_eq_top + sizeOf length_eq_bot", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "And", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk.inj", "constType": "∀ {ℒ : Type u_1} {inst : LT ℒ} {left right : ℒ} {lt : left < right} {left_1 right_1 : ℒ} {lt_1 : left_1 < right_1},\n { left := left, right := right, lt := lt } = { left := left_1, right := right_1, lt := lt_1 } →\n left = left_1 ∧ right = right_1", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "lt_of_le_of_lt'", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, b ≤ a → c < b → c < a", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsStable", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "CompleteLinearOrder", "Nat.instIsOrderedAddMonoid", "SemilatticeInf.toPartialOrder", "instLTNat", "Nat.instPartialOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl", "Nat", "Lattice", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "WellFoundedGT", "Preorder.toLE", "Nat.instAddMonoid", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "lt_add_one", "LE.le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "instLENat"], "name": "HarderNarasimhan.PayoffFunction.payoff_lt_of_piecewise_isStable", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] [WellFoundedGT ℒ] {S : Type u_2} [inst_2 : CompleteLinearOrder S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) [μ.IsSlopeLike] [μ.EventuallyTopDCC] (f : ℕ → ℒ) {n : ℕ}\n (hsa : ∀ (i j : ℕ), i < j → j ≤ n → f j < f i),\n (∀ (i : ℕ) (hi : i < n), (μ.restrict { left := f (i + 1), right := f i, lt := ⋯ }).IsStable) →\n ∀ (i : ℕ) (hi : i < n) (z : ℒ) (h' : f (i + 1) < z),\n z < f i → μ { left := f (i + 1), right := z, lt := h' } < μ { left := f (i + 1), right := f i, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Subtype", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Nontrivial", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.PayoffFunction.semistableRel._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] (x y : ℒ) (h : x < y),\n Nontrivial { x_1 // x_1 ∈ { left := x, right := y, lt := h } }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.rec", "LT"], "name": "HarderNarasimhan.StrictIntvl.recOn", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {motive : HarderNarasimhan.StrictIntvl ℒ → Sort u} →\n (t : HarderNarasimhan.StrictIntvl ℒ) →\n ((left right : ℒ) → (lt : left < right) → motive { left := left, right := right, lt := lt }) → motive t", "constCategory": "Definition"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.left_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ},\n μ.IsBreakpoint I x → I.left < x", "constCategory": "Theorem"}, {"references": ["Finset.univ", "Finset.map", "Finset", "Subtype", "Set", "Membership.mem", "Function.Embedding.subtype", "Fintype", "Set.Elem", "Set.instMembership"], "name": "Set.toFinset", "constType": "{α : Type u_1} → (s : Set α) → [Fintype ↑s] → Finset α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.semistableRel", "instAddNat", "instLTNat", "Nat.cast", "RelSeries.toFun", "PartialOrder.toPreorder", "instHAdd", "Preorder.toLT", "Fin", "RelSeries", "instNeZeroNatHAdd_1", "OfNat.ofNat", "Nat.instNeZeroSucc", "LT.lt", "HAdd.hAdd", "Nat", "Fin.NatCast.instNatCast", "instOfNatNat", "RelSeries.length", "PartialOrder", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.relSeries_succ_step_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (s : RelSeries μ.semistableRel) {i : ℕ},\n i + 1 < s.length → s.toFun ↑(i + 1) < s.toFun ↑(i + 2)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSemistable → Sort u} →\n ((not_lt : ∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) → motive ⋯) →\n (t : μ.IsSemistable) → motive t", "constCategory": "Other"}, {"references": ["CompletelyDistribLattice", "CompleteLattice"], "name": "CompletelyDistribLattice.toCompleteLattice", "constType": "{α : Type u} → [self : CompletelyDistribLattice α] → CompleteLattice α", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }) →\n μ.StrongDCC", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "HarderNarasimhan.CoprimaryFiltration.rec", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Bot.bot", "SizeOf", "Set.Iic", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "associatedPrimes", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instSizeOfNat", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "SizeOf.sizeOf", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "instSizeOfDefault", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration._sizeOf_1", "constType": "{R : Type u_1} →\n {inst : CommRing R} →\n {inst_1 : IsNoetherianRing R} →\n {M : Type u_2} →\n {inst_2 : Nontrivial M} →\n {inst_3 : AddCommGroup M} →\n {inst_4 : _root_.Module R M} →\n {inst_5 : Module.Finite R M} → [SizeOf R] → [SizeOf M] → HarderNarasimhan.CoprimaryFiltration R M → ℕ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "HarderNarasimhan.PayoffFunction", "DFunLike.coe", "LT"], "name": "HarderNarasimhan.PayoffFunction.ext_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : LT ℒ] {μ ν : HarderNarasimhan.PayoffFunction ℒ S},\n μ = ν ↔ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ I = ν I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.mk", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] → {S : Type u_2} → (HarderNarasimhan.StrictIntvl ℒ → S) → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.rec", "LT"], "name": "HarderNarasimhan.StrictIntvl.casesOn", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {motive : HarderNarasimhan.StrictIntvl ℒ → Sort u} →\n (t : HarderNarasimhan.StrictIntvl ℒ) →\n ((left right : ℒ) → (lt : left < right) → motive { left := left, right := right, lt := lt }) → motive t", "constCategory": "Definition"}, {"references": ["Preorder", "LT"], "name": "Preorder.toLT", "constType": "{α : Type u_2} → [self : Preorder α] → LT α", "constCategory": "Definition"}, {"references": ["LT.lt", "LT"], "name": "GT.gt", "constType": "{α : Type u} → [LT α] → α → α → Prop", "constCategory": "Definition"}, {"references": ["associatedPrimes", "Submodule.instNontrivial", "Finset", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Membership.mem", "Submodule", "instDistribLatticeOfLinearOrder", "Ideal", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "Set", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "Set.instMembership", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Iff", "LE.le", "Nontrivial", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ExistsUnique", "CompleteLattice.toBoundedOrder", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.isSemistable_iff_existsUnique_associatedPrime", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] [inst_5 : Nontrivial M],\n (HarderNarasimhan.Coprimary.payoff R M).IsSemistable ↔ ∃! p, p ∈ associatedPrimes R M", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "StrictMonoOn", "Preorder.toLT", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "Set.Iic", "HarderNarasimhan.PayoffFunction.IsConvex", "Nat.instPreorder", "Nat", "Nontrivial", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_strictMonoOn", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible],\n StrictMonoOn (HarderNarasimhan.PayoffFunction.HNFil✝ μ) (Set.Iic (HarderNarasimhan.PayoffFunction.HNlen✝ μ))", "constCategory": "Theorem"}, {"references": ["CommSemiring", "CommSemiring.toSemiring", "Distrib.toMul", "instDistribOfSemiring", "instSMulOfMul", "Algebra.algebraMap", "Algebra", "SMul", "Semiring.toNonAssocSemiring", "Algebra.id._proof_1", "Algebra.id._proof_2", "RingHom.id", "Algebra.mk", "RingHom.toAlgebra"], "name": "Algebra.id", "constType": "(R : Type u) → [inst : CommSemiring R] → Algebra R R", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "Set.Ioc", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.max_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {s : S},\n (∀ (u : ℒ) (hu : u ∈ Set.Ioc I.left I.right), μ { left := I.left, right := u, lt := ⋯ } ≤ s) → μ.max I ≤ s", "constCategory": "Theorem"}, {"references": ["IsNoetherian", "AddCommMonoid", "PartialOrder.toPreorder", "Module", "Submodule.instPartialOrder", "Preorder.toLT", "WellFoundedGT", "Submodule", "Semiring"], "name": "wellFoundedGT", "constType": "∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : _root_.Module R M]\n [h : IsNoetherian R M], WellFoundedGT (Submodule R M)", "constCategory": "Theorem"}, {"references": ["Preorder", "AddCommMonoid"], "name": "IsOrderedAddMonoid", "constType": "(α : Type u_2) → [AddCommMonoid α] → [Preorder α] → Prop", "constCategory": "Other"}, {"references": ["Semiring.toAddCommMonoid", "PrimeSpectrum", "Ideal", "PartialOrder", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "CommSemiring", "PrimeSpectrum.ext", "PrimeSpectrum.asIdeal", "Semiring.toModule", "PartialOrder.lift"], "name": "PrimeSpectrum.instPartialOrder", "constType": "{R : Type u_1} → [inst : CommSemiring R] → PartialOrder (PrimeSpectrum R)", "constCategory": "Definition"}, {"references": ["Semiring.toAddCommMonoid", "Module.mk", "Semiring.toModule._proof_2", "Semiring.zero_mul", "Semiring.toMonoid", "Module", "DistribMulAction.mk", "Semiring.toModule._proof_1", "AddCommMonoid.toAddMonoid", "Monoid.toMulAction", "Semiring.mul_zero", "Semiring"], "name": "Semiring.toModule", "constType": "{R : Type u_1} → [inst : Semiring R] → _root_.Module R R", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "PartialOrder.toPreorder", "Module", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Nat.instPreorder", "Submodule", "HarderNarasimhan.CoprimaryFiltration.toFun", "IsNoetherianRing", "Nat", "Monotone", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.monotone", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M), Monotone self.toFun", "constCategory": "Theorem"}, {"references": ["Equiv.refl", "Colex", "Equiv"], "name": "toColex", "constType": "{α : Type u_1} → α ≃ Colex α", "constCategory": "Definition"}, {"references": [], "name": "Real", "constType": "Type", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.PayoffFunction.min._proof_1", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "iInf", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf", "ConditionallyCompletePartialOrderInf.toInfSet", "Set.Ico", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Definition"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "Preorder.toLT", "DFunLike.coe", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_total_eq_right_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := z, lt := ⋯ } = μ { left := y, right := z, lt := h₂ } ↔\n μ { left := x, right := y, lt := h₁ } = μ { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": [], "name": "Nonempty", "constType": "Sort u → Prop", "constCategory": "Other"}, {"references": ["LT.lt", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "Top.top", "OrderTop", "Ne", "Preorder.toLE", "OrderTop.toTop"], "name": "Ne.lt_top", "constType": "∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderTop α] {a : α}, a ≠ ⊤ → a < ⊤", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "HarderNarasimhan.CoprimaryFiltration._sizeOf_inst", "Set.Iic", "Bot.bot", "SizeOf", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instSizeOfNat", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "SizeOf.sizeOf", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "instSizeOfDefault", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk.sizeOf_spec", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M] [inst_6 : SizeOf R]\n [inst_7 : SizeOf M] (toFun : ℕ → Submodule R M) (length : ℕ) (monotone : Monotone toFun) (head_eq_bot : toFun 0 = ⊥)\n (length_eq_top : toFun length = ⊤) (strictMonoOn : StrictMonoOn toFun (Set.Iic length))\n (piecewise_isCoprimary :\n ∀ i < length, HarderNarasimhan.IsCoprimary R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))))\n (associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q),\n sizeOf\n { toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot,\n length_eq_top := length_eq_top, strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt } =\n 1 + sizeOf length + sizeOf head_eq_bot + sizeOf length_eq_top", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Submodule", "Nat.instPreorder", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n (toFun : ℕ → Submodule R M) →\n (length : ℕ) →\n Monotone toFun →\n toFun 0 = ⊥ →\n toFun length = ⊤ →\n StrictMonoOn toFun (Set.Iic length) →\n (∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))) →\n (∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q) →\n HarderNarasimhan.CoprimaryFiltration R M", "constCategory": "Other"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.StrictIntvl.ofSub._proof_1", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder", "HarderNarasimhan.StrictIntvl.ofSub", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.StrictIntvl.ofSub.eq_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}\n (J : HarderNarasimhan.StrictIntvl { x // x ∈ I }),\n HarderNarasimhan.StrictIntvl.ofSub J = { left := ↑J.left, right := ↑J.right, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.PayoffFunction.min._proof_1", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "iInf", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf", "ConditionallyCompletePartialOrderInf.toInfSet", "Set.Ico", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.PayoffFunction.StrongDCC.mk", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }) →\n μ.StrongDCC", "constCategory": "Definition"}, {"references": ["ConditionallyCompleteLinearOrderBot", "ConditionallyCompleteLinearOrder"], "name": "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "constType": "{α : Type u_5} → [self : ConditionallyCompleteLinearOrderBot α] → ConditionallyCompleteLinearOrder α", "constCategory": "Definition"}, {"references": ["ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "iSup", "HarderNarasimhan.StrictIntvl.left", "ConditionallyCompletePartialOrderSup.toSupSet", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "HarderNarasimhan.PayoffFunction.B", "Set", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.PayoffFunction.min", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "Set.Ioc", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) (I : HarderNarasimhan.StrictIntvl ℒ),\n μ.B I = ⨆ b, ⨆ (hb : b ∈ Set.Ioc I.left I.right), μ.min { left := I.left, right := b, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Preorder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible", "constType": "{ℒ : Type u_1} → {S : Type u_2} → [inst : Preorder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["SemilatticeInf", "SemilatticeInf.inf", "Min", "Min.mk"], "name": "SemilatticeInf.toMin", "constType": "{α : Type u} → [SemilatticeInf α] → Min α", "constCategory": "Definition"}, {"references": ["NNReal.instPartialOrder._aux_1", "NNReal.instPartialOrder._aux_3", "PartialOrder.mk", "NNReal.instPartialOrder._proof_6", "PartialOrder", "NNReal.instPartialOrder._proof_7", "LE.mk", "NNReal", "NNReal.instPartialOrder._proof_8", "NNReal.instPartialOrder._proof_5", "LT.mk", "Preorder.mk"], "name": "NNReal.instPartialOrder", "constType": "PartialOrder NNReal", "constCategory": "Definition"}, {"references": ["Exists", "Set.ofPred", "Set", "Eq"], "name": "Set.range", "constType": "{α : Type u} → {ι : Sort u_1} → (ι → α) → Set α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "Nontrivial", "LE.le", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "instLENat", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.eq_bot_of_length_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n F.length ≤ m → F m = ⊥", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "List.TFAE", "Preorder.toLE", "Eq", "List.cons", "List.nil", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.min", "BoundedOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "HarderNarasimhan.PayoffFunction.max", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.nashEquilibrium_tfae", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike] [μ.WeakACC] [μ.StrongDCC],\n [μ.max ⊤ = μ ⊤, μ.min ⊤ = μ ⊤, μ.min ⊤ = μ.max ⊤, μ.HasNashEquilibrium].TFAE", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.rec", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HarderNarasimhanFiltration → Sort u} →\n (t : μ.HarderNarasimhanFiltration) →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable) →\n (not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ }) →\n motive t", "constCategory": "Definition"}, {"references": ["OrderDual", "Nontrivial"], "name": "OrderDual.instNontrivial", "constType": "∀ {α : Type u_1} [h : Nontrivial α], Nontrivial αᵒᵈ", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "Module", "Submodule.instTop", "CommSemiring.toSemiring", "HarderNarasimhan.CoprimaryFiltration.length", "AddCommGroup", "CommRing", "Submodule", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration.toFun", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "Top.top", "AddCommGroup.toAddCommMonoid", "Eq", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.length_eq_top", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M), self.toFun self.length = ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.hnFiltration", "Preorder.toLT", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.PayoffFunction.instAdmissible", "CompletelyDistribLattice.toCompleteLattice", "Nontrivial", "Lattice", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "CompleteLinearOrder", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Unique.0.HarderNarasimhan.PayoffFunction.eq_hnFiltration", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [inst_3 : WellFoundedGT ℒ]\n {S : Type u_2} [inst_4 : CompleteLinearOrder S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_5 : μ.ADCC]\n [inst_6 : μ.IsConvex] (F : μ.HarderNarasimhanFiltration), F = μ.hnFiltration", "constCategory": "Theorem"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.exists_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.EventuallyTopDCC] (x : ℕ → ℒ) (hx : StrictAnti x),\n ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤", "constCategory": "Theorem"}, {"references": ["AddCommMagma", "AddCommSemigroup.toAddSemigroup", "AddCommSemigroup.add_comm", "AddCommSemigroup", "AddCommMagma.mk", "AddSemigroup.toAdd"], "name": "AddCommSemigroup.toAddCommMagma", "constType": "{G : Type u} → [self : AddCommSemigroup G] → AddCommMagma G", "constCategory": "Definition"}, {"references": ["Nat", "SizeOf"], "name": "SizeOf.mk", "constType": "{α : Sort u} → (α → ℕ) → SizeOf α", "constCategory": "Other"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsAffine.mk", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsAffine → Sort u} →\n ((eq :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.IsAffine) → motive t", "constCategory": "Other"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "Nat.instIsOrderedAddMonoid", "SemilatticeInf.toPartialOrder", "Exists", "Nat.instPartialOrder", "CompleteLattice.toLattice", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Nat", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "CompleteLattice.toBoundedOrder", "AddMonoid.toAddZeroClass", "SemilatticeSup.toPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Eq", "Preorder.toLE", "Nat.instAddMonoid", "Lattice.toSemilatticeInf", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "One.toOfNat1", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.adcc_of_exists_A_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_2 : Nontrivial ℒ] [inst_3 : BoundedOrder ℒ],\n μ.IsConvexOn ⊤ →\n (∀ (f : ℕ → ℒ) (h : StrictAnti f), ∃ N, μ.A { left := f (N + 1), right := f N, lt := ⋯ } = ⊤) → μ.ADCC", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "AddZeroClass", "instHAdd", "AddLeftStrictMono", "Preorder.toLT", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "ZeroLEOneClass", "HAdd.hAdd", "LT.lt", "NeZero", "One.toOfNat1", "PartialOrder", "One", "Preorder.toLE", "AddZero.toZero"], "name": "lt_add_one", "constType": "∀ {α : Type u_1} [inst : One α] [inst_1 : AddZeroClass α] [inst_2 : PartialOrder α] [ZeroLEOneClass α] [NeZero 1]\n [AddLeftStrictMono α] (a : α), a < a + 1", "constCategory": "Theorem"}, {"references": ["OfNat", "OfNat.mk", "One.one", "One"], "name": "One.toOfNat1", "constType": "{α : Type u_1} → [One α] → OfNat α 1", "constCategory": "Definition"}, {"references": ["SemilatticeSup.sup", "SemilatticeSup", "Max", "Max.mk"], "name": "SemilatticeSup.toMax", "constType": "{α : Type u} → [SemilatticeSup α] → Max α", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "HarderNarasimhan.PayoffFunction.IsStable", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "CompleteLinearOrder", "Nat.instIsOrderedAddMonoid", "SemilatticeInf.toPartialOrder", "instLTNat", "Nat.instPartialOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "Lattice", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "WellFoundedGT", "Preorder.toLE", "Nat.instAddMonoid", "Lattice.toSemilatticeInf", "instHAdd", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "HAdd.hAdd", "LT.lt", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "lt_add_one", "LE.le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "instLENat"], "name": "HarderNarasimhan.PayoffFunction.piecewise_isStable_of_payoff_lt", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] [WellFoundedGT ℒ] {S : Type u_2} [inst_2 : CompleteLinearOrder S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) [μ.IsSlopeLike] [μ.EventuallyTopDCC] (f : ℕ → ℒ) {n : ℕ}\n (hsa : ∀ (i j : ℕ), i < j → j ≤ n → f j < f i),\n (∀ (i : ℕ) (hi : i < n) (z : ℒ) (h' : f (i + 1) < z),\n z < f i → μ { left := f (i + 1), right := z, lt := h' } < μ { left := f (i + 1), right := f i, lt := ⋯ }) →\n ∀ (i : ℕ) (hi : i < n), (μ.restrict { left := f (i + 1), right := f i, lt := ⋯ }).IsStable", "constCategory": "Theorem"}, {"references": ["Subtype"], "name": "Subtype.mk", "constType": "{α : Sort u} → {p : α → Prop} → (val : α) → p val → Subtype p", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.IsStable", "PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "Eq", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.congr_simp", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] (μ μ_1 : HarderNarasimhan.PayoffFunction ℒ S), μ = μ_1 → μ.IsStable = μ_1.IsStable", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.module", "Submodule.Quotient.module", "Submodule.map", "Membership.mem", "Submodule.subtype", "Submodule", "Semiring.toNonAssocSemiring", "RingHom.id", "Preorder.toLE", "Submodule.mkQ", "CommRing.toCommSemiring", "SetLike.instMembership", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "CommRing", "Bot.bot", "RingHomSurjective.ids", "Ring.toSemiring", "CommRing.toRing", "Submodule.comap", "Submodule.setLike", "Submodule.instBot", "LE.le", "AddCommGroup.toAddCommMonoid", "Ne", "Submodule.submoduleOf"], "name": "HarderNarasimhan.Coprimary.map_comap_ne_bot", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n {N₁ N₂ W : Submodule R M},\n N₁ ≤ W → W ≤ N₂ → W ≠ N₁ → Submodule.map (N₁.submoduleOf N₂).mkQ (Submodule.comap N₂.subtype W) ≠ ⊥", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "eq_of_heq", "PrimeSpectrum.instPartialOrder", "Eq.ndrec", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Bot.bot", "Set.Iic", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Eq.refl", "Submodule.instBot", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HEq", "Top.top", "HarderNarasimhan.CoprimaryFiltration.casesOn", "Submodule.submoduleOf", "OrderHom.instFunLike", "associatedPrimes", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "HEq.refl", "LinearExtension", "Set", "instHAdd", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.instTop", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "HarderNarasimhan.CoprimaryFiltration.noConfusionType", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.noConfusion", "constType": "{P : Sort u} →\n {R : Type u_1} →\n {inst : CommRing R} →\n {inst_1 : IsNoetherianRing R} →\n {M : Type u_2} →\n {inst_2 : Nontrivial M} →\n {inst_3 : AddCommGroup M} →\n {inst_4 : _root_.Module R M} →\n {inst_5 : Module.Finite R M} →\n {t : HarderNarasimhan.CoprimaryFiltration R M} →\n {R' : Type u_1} →\n {inst' : CommRing R'} →\n {inst'_1 : IsNoetherianRing R'} →\n {M' : Type u_2} →\n {inst'_2 : Nontrivial M'} →\n {inst'_3 : AddCommGroup M'} →\n {inst'_4 : _root_.Module R' M'} →\n {inst'_5 : Module.Finite R' M'} →\n {t' : HarderNarasimhan.CoprimaryFiltration R' M'} →\n R = R' →\n inst ≍ inst' →\n inst_1 ≍ inst'_1 →\n M = M' →\n inst_2 ≍ inst'_2 →\n inst_3 ≍ inst'_3 →\n inst_4 ≍ inst'_4 →\n inst_5 ≍ inst'_5 →\n t ≍ t' →\n HarderNarasimhan.CoprimaryFiltration.noConfusionType P t t'", "constCategory": "Definition"}, {"references": ["Preorder", "PartialOrder.toPreorder", "Real", "inferInstance", "Real.partialOrder"], "name": "Real.instPreorder", "constType": "Preorder ℝ", "constCategory": "Definition"}, {"references": ["Bot"], "name": "Bot.mk", "constType": "{α : Type u_1} → α → Bot α", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "And", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Submodule", "Nat.instPreorder", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk.inj", "constType": "∀ {R : Type u_1} {inst : CommRing R} {inst_1 : IsNoetherianRing R} {M : Type u_2} {inst_2 : Nontrivial M}\n {inst_3 : AddCommGroup M} {inst_4 : _root_.Module R M} {inst_5 : Module.Finite R M} {toFun : ℕ → Submodule R M}\n {length : ℕ} {monotone : Monotone toFun} {head_eq_bot : toFun 0 = ⊥} {length_eq_top : toFun length = ⊤}\n {strictMonoOn : StrictMonoOn toFun (Set.Iic length)}\n {piecewise_isCoprimary :\n ∀ i < length, HarderNarasimhan.IsCoprimary R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))}\n {associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q}\n {toFun_1 : ℕ → Submodule R M} {length_1 : ℕ} {monotone_1 : Monotone toFun_1} {head_eq_bot_1 : toFun_1 0 = ⊥}\n {length_eq_top_1 : toFun_1 length_1 = ⊤} {strictMonoOn_1 : StrictMonoOn toFun_1 (Set.Iic length_1)}\n {piecewise_isCoprimary_1 :\n ∀ i < length_1, HarderNarasimhan.IsCoprimary R (↥(toFun_1 (i + 1)) ⧸ (toFun_1 i).submoduleOf (toFun_1 (i + 1)))}\n {associatedPrime_succ_lt_1 :\n ∀ (i : ℕ),\n i + 1 < length_1 →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(toFun_1 (i + 2)) ⧸ (toFun_1 (i + 1)).submoduleOf (toFun_1 (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(toFun_1 (i + 1)) ⧸ (toFun_1 i).submoduleOf (toFun_1 (i + 1))) →\n toLinearExtension p < toLinearExtension q},\n { toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt } =\n { toFun := toFun_1, length := length_1, monotone := monotone_1, head_eq_bot := head_eq_bot_1,\n length_eq_top := length_eq_top_1, strictMonoOn := strictMonoOn_1,\n piecewise_isCoprimary := piecewise_isCoprimary_1, associatedPrime_succ_lt := associatedPrime_succ_lt_1 } →\n toFun = toFun_1 ∧ length = length_1", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.PayoffFunction.IsSemistable.rec", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSemistable → Sort u} →\n (t : μ.IsSemistable) →\n ((not_lt : ∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) → motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration._sizeOf_1", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "SizeOf", "SizeOf.mk", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration._sizeOf_inst", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : Nontrivial ℒ} →\n {inst_1 : PartialOrder ℒ} →\n {inst_2 : BoundedOrder ℒ} →\n {inst_3 : CompleteLattice S} →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) → [SizeOf ℒ] → [SizeOf S] → SizeOf μ.JordanHolderFiltration", "constCategory": "Definition"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.le_or_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : BoundedOrder ℒ} {inst_2 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.WeakSlopeLikeAtBot] (z : HarderNarasimhan.StrictIntvl ℒ)\n (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.casesOn", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "HEq", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.noConfusionType", "constType": "Sort u →\n {ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n μ.JordanHolderFiltration →\n {ℒ' : Type u_1} →\n {S' : Type u_2} →\n [inst' : Nontrivial ℒ'] →\n [inst'_1 : PartialOrder ℒ'] →\n [inst'_2 : BoundedOrder ℒ'] →\n [inst'_3 : CompleteLattice S'] →\n {μ' : HarderNarasimhan.PayoffFunction ℒ' S'} → μ'.JordanHolderFiltration → Sort u", "constCategory": "Definition"}, {"references": ["Submodule.toAddSubmonoid", "Submodule.Quotient.module", "Membership.mem", "Inter.inter", "Algebra.id", "Semiring.toNonAssocSemiring", "RingHom.id", "DistribMulAction.toMulAction", "IsScalarTower.right", "Semiring.toModule", "Set.instInter", "LinearMap.ker", "LocalizedModule", "Submonoid.instSetLike", "Submodule.Quotient.addCommMonoid", "And", "AddZeroClass.toAddZero", "CommSemiring.toCommMonoid", "Set.instMembership", "Submonoid", "SetLike.coe", "AddCommGroup.toAddCommMonoid", "AddSubmonoid.toAddSubsemigroup", "AddMonoid.toAddZeroClass", "Submodule.hasQuotient", "associatedPrimes", "HasQuotient.Quotient", "Set.ofPred", "OreLocalization.oreSetComm", "Module", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "Set.instEmptyCollection", "EmptyCollection.emptyCollection", "Submodule", "LocalizedModule.mkLinearMap", "AddSubsemigroup.carrier", "Ideal", "instMulZeroOneClassOfSemiring", "Eq", "CommRing.toCommSemiring", "Semiring.toMonoid", "OreLocalization.instAddCommMonoidOreLocalization", "IsScalarTower.left", "Set", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "AddCommGroup", "CommRing", "AddZero.toAdd", "Semiring.toAddCommMonoid", "CommRing.toRing", "Module.toDistribMulAction", "IsNoetherianRing"], "name": "HarderNarasimhan.associatedPrimes_quot_ker_mkLinearMap", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (S : Submonoid R) [IsNoetherianRing R],\n associatedPrimes R (M ⧸ (LocalizedModule.mkLinearMap S M).ker) = {p | p ∈ associatedPrimes R M ∧ p.carrier ∩ ↑S = ∅}", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "associatedPrimes", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Preorder.toLT", "Submodule", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "SetLike.instMembership", "LinearExtension", "Set", "HarderNarasimhan.StrictIntvl.right", "Submodule.addCommGroup", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.Quotient.addCommMonoid", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "AddCommGroup", "CommRing", "Set.instMembership", "PrimeSpectrum", "CommRing.toRing", "HarderNarasimhan.StrictIntvl", "Submodule.setLike", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf"], "name": "HarderNarasimhan.Coprimary.mem_subquotientAssociatedPrimes._simp_1", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n {I : HarderNarasimhan.StrictIntvl (Submodule R M)} {q : LinearExtension (PrimeSpectrum R)},\n (q ∈ HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I) =\n (q.asIdeal ∈ associatedPrimes R (↥I.right ⧸ I.left.submoduleOf I.right))", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Lattice ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsConvex] (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n (toFun : ℕ → ℒ) →\n (length : ℕ) →\n Monotone toFun →\n toFun 0 = ⊥ →\n toFun length = ⊤ →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable) →\n (∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }) →\n μ.HarderNarasimhanFiltration", "constCategory": "Other"}, {"references": ["Not", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "Iff", "PartialOrder", "LE.le", "And", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] (a b : HarderNarasimhan.StrictIntvl ℒ),\n (fun I J => J.left ≤ I.left ∧ I.right ≤ J.right) a b ∧ ¬(fun I J => J.left ≤ I.left ∧ I.right ≤ J.right) b a ↔\n (b.left ≤ a.left ∧ a.right ≤ b.right) ∧ ¬(a.left ≤ b.left ∧ b.right ≤ a.right)", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "Eq", "DFunLike.coe", "LT"], "name": "HarderNarasimhan.PayoffFunction.coe_mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : LT ℒ] (f : HarderNarasimhan.StrictIntvl ℒ → S), ⇑{ toFun := f } = f", "constCategory": "Theorem"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Subtype.preorder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_restrict_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}\n {J : HarderNarasimhan.StrictIntvl { x // x ∈ I }}, (μ.restrict I).A J = μ.A (HarderNarasimhan.StrictIntvl.ofSub J)", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "associatedPrimes", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Preorder.toLT", "Submodule", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "CommRing.toCommSemiring", "SetLike.instMembership", "LinearExtension", "Set", "HarderNarasimhan.StrictIntvl.right", "Submodule.addCommGroup", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.Quotient.addCommMonoid", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "AddCommGroup", "CommRing", "Set.instMembership", "PrimeSpectrum", "CommRing.toRing", "HarderNarasimhan.StrictIntvl", "Submodule.setLike", "Iff", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf"], "name": "HarderNarasimhan.Coprimary.mem_subquotientAssociatedPrimes", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n {I : HarderNarasimhan.StrictIntvl (Submodule R M)} {q : LinearExtension (PrimeSpectrum R)},\n q ∈ HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I ↔\n q.asIdeal ∈ associatedPrimes R (↥I.right ⧸ I.left.submoduleOf I.right)", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HEq", "HarderNarasimhan.PayoffFunction.casesOn", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.noConfusionType", "constType": "Sort u →\n {ℒ : Type u_1} →\n [inst : LT ℒ] →\n {S : Type u_2} →\n HarderNarasimhan.PayoffFunction ℒ S →\n {ℒ' : Type u_1} → [inst' : LT ℒ'] → {S' : Type u_2} → HarderNarasimhan.PayoffFunction ℒ' S' → Sort u", "constCategory": "Definition"}, {"references": ["Subtype", "Real.instZero", "Real", "LE.le", "Zero.toOfNat0", "OfNat.ofNat", "Real.instLE"], "name": "NNReal", "constType": "Type", "constCategory": "Definition"}, {"references": [], "name": "SizeOf", "constType": "Sort u → Sort (max 1 u)", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "Bot.bot", "BoundedOrder.toOrderBot", "PartialOrder", "Nontrivial", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length_eq_bot", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration),\n self.toFun self.length = ⊥", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn.rec", "HarderNarasimhan.PayoffFunction.IsConvexOn.mk", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "Lattice", "LE.le", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {motive : μ.IsConvexOn I → Sort u} →\n (t : μ.IsConvexOn I) →\n ((le :\n ∀ (x y : ℒ),\n x ∈ I →\n y ∈ I →\n ∀ (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.preorder", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction", "Eq", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.max_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, (μ.restrict I).max = μ.max.restrict I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Subtype", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Nontrivial", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, Nontrivial { x // x ∈ I }", "constCategory": "Theorem"}, {"references": ["LE", "BoundedOrder", "OrderBot"], "name": "BoundedOrder.toOrderBot", "constType": "{α : Type u} → {inst : LE α} → [self : BoundedOrder α] → OrderBot α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.not_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsSemistable] (x : ℒ) (hx : ⊥ < x),\n ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }", "constCategory": "Theorem"}, {"references": [], "name": "Lattice", "constType": "Type u → Type u", "constCategory": "Other"}, {"references": ["AddZeroClass", "AddSubmonoid", "AddSubsemigroup", "AddZeroClass.toAddZero", "AddZero.toAdd"], "name": "AddSubmonoid.toAddSubsemigroup", "constType": "{M : Type u_3} → [inst : AddZeroClass M] → AddSubmonoid M → AddSubsemigroup M", "constCategory": "Definition"}, {"references": ["Top"], "name": "Top.mk", "constType": "{α : Type u_1} → α → Top α", "constCategory": "Other"}, {"references": ["Not", "LT.lt", "Preorder", "Preorder.lt_iff_le_not_ge._autoParam", "Iff", "LE.le", "And", "LE", "autoParam", "LT"], "name": "Preorder.mk", "constType": "{α : Type u_2} →\n [toLE : LE α] →\n [toLT : LT α] →\n (∀ (a : α), a ≤ a) →\n (∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c) →\n autoParam (∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a) Preorder.lt_iff_le_not_ge._autoParam → Preorder α", "constCategory": "Other"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.rec", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtTop → Sort u} →\n (t : μ.WeakSlopeLikeAtTop) →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["AddMonoid.zero_add", "AddMonoid.toZero", "AddZero.mk", "AddMonoid.add_zero", "AddZeroClass", "AddMonoid.toAddSemigroup", "AddMonoid", "AddZeroClass.mk", "AddSemigroup.toAdd"], "name": "AddMonoid.toAddZeroClass", "constType": "{M : Type u} → [self : AddMonoid M] → AddZeroClass M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.semistableRel.match_1", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Exists", "Subtype", "HarderNarasimhan.PayoffFunction.semistableRel._proof_1", "Set.ofPred", "HarderNarasimhan.PayoffFunction.IsSemistable", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "SetRel", "HarderNarasimhan.StrictIntvl.mk", "Prod", "LT.lt", "HarderNarasimhan.StrictIntvl", "PartialOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.semistableRel", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} → [inst : PartialOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → SetRel ℒ ℒ", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn.mk", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "Lattice", "LE.le", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {motive : μ.IsConvexOn I → Sort u} →\n ((le :\n ∀ (x y : ℒ),\n x ∈ I →\n y ∈ I →\n ∀ (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.IsConvexOn I) → motive t", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "Eq", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.congr_simp", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] (μ μ_1 : HarderNarasimhan.PayoffFunction ℒ S),\n μ = μ_1 → μ.FiniteTotalPayoff = μ_1.FiniteTotalPayoff", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.right", "Iff", "And", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.ext_iff", "constType": "∀ {ℒ : Type u_1} {inst : LT ℒ} {x y : HarderNarasimhan.StrictIntvl ℒ}, x = y ↔ x.left = y.left ∧ x.right = y.right", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.ADCC.mk", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC.rec", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.ADCC → Sort u} →\n (t : μ.ADCC) →\n ((dcc :\n ∀ (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f →\n ∃ N,\n ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["CompleteLinearOrder.toCompletelyDistribLattice._proof_7", "CompletelyDistribLattice.mk", "CompleteLinearOrder.himp_bot", "CompleteLinearOrder.top_sdiff", "CompleteLinearOrder.toHNot", "CompleteLinearOrder.toHImp", "CompleteLinearOrder.toCompl", "CompletelyDistribLattice", "CompleteLinearOrder.sdiff_le_iff", "CompleteLinearOrder.toCompleteLattice", "CompleteLinearOrder", "CompleteLinearOrder.le_himp_iff", "CompleteLinearOrder.toSDiff"], "name": "CompleteLinearOrder.toCompletelyDistribLattice", "constType": "{α : Type u} → [CompleteLinearOrder α] → CompletelyDistribLattice α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "Set.Ioc", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.le_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {u : ℒ}\n (hu : u ∈ Set.Ioc I.left I.right), μ { left := I.left, right := u, lt := ⋯ } ≤ μ.max I", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "Real.instPreorder", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "Membership.mem", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "Set.range", "Set.Elem", "Monotone", "HarderNarasimhan.StrictIntvl.left", "BoundedOrder.toOrderTop", "PartialOrder", "Real.instLT", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Eq", "Subtype.instLT", "Real", "Set", "CompleteLattice.toLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Set.instMembership", "LT.lt", "IsWellOrder", "HarderNarasimhan.StrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.strongDCC_of_wellOrderedRank", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) (r : ℒ → ℝ),\n Monotone r →\n (IsWellOrder ↑(Set.range r) fun x1 x2 => x1 < x2) →\n (∀ (z : HarderNarasimhan.StrictIntvl ℒ), r z.left = r z.right → μ z = ⊤) → μ.StrongDCC", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn", "HarderNarasimhan.PayoffFunction.max", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, μ.IsConvexOn I → μ.max.IsConvexOn I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "DedekindCut.instCompleteLinearOrder", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Colex", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "CompletelyDistribLattice.toCompleteLattice", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ConditionallyCompleteLattice.toLattice", "Finset.Colex.instLinearOrder", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M], (HarderNarasimhan.Coprimary.payoff R M).Admissible", "constCategory": "Theorem"}, {"references": ["Prod", "Prod.mk", "Prod.rec"], "name": "Prod.casesOn", "constType": "{α : Type u} →\n {β : Type v} → {motive : α × β → Sort u_1} → (t : α × β) → ((fst : α) → (snd : β) → motive (fst, snd)) → motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Singleton.singleton", "Preorder.toLT", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_nonempty", "Equiv", "Finset.min'", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "toColex", "Bot.bot", "HarderNarasimhan.StrictIntvl", "Finset.instSingleton", "Iff", "Nontrivial", "Submodule.instBot", "Top.top", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.A", "CompleteLattice.toBoundedOrder", "Submodule.instNontrivial", "Equiv.instEquivLike", "Module", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "EquivLike.toFunLike", "HarderNarasimhan.Coprimary.payoff", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Set.toFinset", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "IsNoetherianRing", "HarderNarasimhan.StrictIntvl.instOrderTop", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "DedekindCut.principal", "LE.le", "Submodule.completeLattice", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.isSemistable_iff_A_const", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] [inst_5 : Nontrivial M],\n (HarderNarasimhan.Coprimary.payoff R M).IsSemistable ↔\n ∀ (N : Submodule R M) (hN : ⊥ < N),\n (HarderNarasimhan.Coprimary.payoff R M).A { left := ⊥, right := N, lt := hN } =\n DedekindCut.principal (toColex {(HarderNarasimhan.Coprimary.subquotientAssociatedPrimes ⊤).toFinset.min' ⋯})", "constCategory": "Theorem"}, {"references": ["SemilatticeInf.toMin", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "And", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "Min.min", "Lattice", "LE.le", "Preorder.toLE", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem._proof_2", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} (x x_1 : ℒ),\n x ∈ I → x_1 ∈ I → I.left ≤ x ⊓ x_1 ∧ x ⊓ x_1 ≤ I.right", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsBreakpoint.rec", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {x : ℒ} →\n {motive : μ.IsBreakpoint I x → Sort u} →\n (t : μ.IsBreakpoint I x) →\n ((mem : x ∈ I) →\n (ne_left : I.left ≠ x) →\n (not_lt :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } <\n μ.A { left := I.left, right := y, lt := ⋯ }) →\n (le_of_eq :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } =\n μ.A { left := I.left, right := x, lt := ⋯ } →\n y ≤ x) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "PartialOrder", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration), self.toFun self.length = ⊤", "constCategory": "Theorem"}, {"references": ["AddCommMagma", "Add"], "name": "AddCommMagma.toAdd", "constType": "{G : Type u} → [self : AddCommMagma G] → Add G", "constCategory": "Definition"}, {"references": ["Concept.instCompleteLattice._proof_2", "Concept", "Concept.instSupSet", "CompleteLattice.mk", "Set", "Concept.instBoundedOrderConcept", "Concept.instCompleteLattice._proof_1", "Concept.instInfSet", "CompleteLattice", "Concept.instLattice"], "name": "Concept.instCompleteLattice", "constType": "{α : Type u_2} → {β : Type u_3} → {r : α → β → Prop} → CompleteLattice (Concept α β r)", "constCategory": "Definition"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "OfNat.ofNat", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "LT.lt", "Nat", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "LE.le", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "instLENat", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.apply_lt_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n 0 < m → m ≤ F.length → F m < ⊤", "constCategory": "Theorem"}, {"references": ["Preorder", "LE"], "name": "Preorder.toLE", "constType": "{α : Type u_2} → [self : Preorder α] → LE α", "constCategory": "Definition"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.left", "Set.Ioc", "Set", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "Preorder.toLT", "Set.instMembership"], "name": "HarderNarasimhan.PayoffFunction.max._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), ∀ u ∈ Set.Ioc I.left I.right, I.left < u", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "Nat", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "Function.Injective"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat._proof_1", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n Function.Injective HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "constCategory": "Theorem"}, {"references": ["LE.le", "LE", "Bot", "Bot.bot", "OrderBot"], "name": "OrderBot.mk", "constType": "{α : Type u} → [inst : LE α] → [toBot : Bot α] → (∀ (a : α), ⊥ ≤ a) → OrderBot α", "constCategory": "Other"}, {"references": ["Not", "LT.lt", "SemilatticeInf", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Min.min", "Iff", "LE.le", "Preorder.toLT", "Preorder.toLE", "SemilatticeInf.toPartialOrder"], "name": "inf_lt_left", "constType": "∀ {α : Type u} [inst : SemilatticeInf α] {a b : α}, a ⊓ b < a ↔ ¬a ≤ b", "constCategory": "Theorem"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtTop → Sort u} →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.WeakSlopeLikeAtTop) → motive t", "constCategory": "Other"}, {"references": ["Nat", "inferInstance", "AddMonoid", "AddCommMonoid.toAddMonoid", "Nat.instAddCommMonoid"], "name": "Nat.instAddMonoid", "constType": "AddMonoid ℕ", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.strictAntiOn", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Nat.instIsOrderedAddMonoid", "Preorder.toLE", "Eq", "LT.lt.le", "instLTNat", "Nat.instAddMonoid", "Nat.instPartialOrder", "instHAdd", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "HAdd.hAdd", "lt_add_one", "Nat", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.step_payoff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration) {i : ℕ}\n (hi : i < F.length), μ { left := F (i + 1), right := F i, lt := ⋯ } = μ ⊤", "constCategory": "Theorem"}, {"references": ["Monoid", "Semiring"], "name": "Semiring.toMonoid", "constType": "{α : Type u} → [self : Semiring α] → Monoid α", "constCategory": "Definition"}, {"references": ["MulOneClass.toMulOne", "MulOne.toMul", "MulAction", "IsScalarTower", "Monoid.toMulOneClass", "Monoid", "SemigroupAction.toSMul", "instSMulOfMul", "Monoid.toSemigroup", "MulAction.toSemigroupAction"], "name": "IsScalarTower.left", "constType": "∀ (M : Type u_1) {α : Type u_5} [inst : Monoid M] [inst_1 : MulAction M α], IsScalarTower M M α", "constCategory": "Theorem"}, {"references": ["Submodule.instTop._proof_1", "AddSubmonoid.instTop", "Set", "Module", "Top", "Membership.mem", "AddSubmonoid.mk", "AddCommMonoid.toAddMonoid", "AddZeroClass.toAddZero", "AddZero.toAdd", "Set.instMembership", "Submodule", "Submodule.instTop._proof_2", "Set.univ", "AddSubsemigroup.mk", "AddCommMonoid", "AddSubmonoid", "Submodule.mk", "Top.top", "Top.mk", "trivial", "Semiring", "AddMonoid.toAddZeroClass"], "name": "Submodule.instTop", "constType": "{R : Type u_1} →\n {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → Top (Submodule R M)", "constCategory": "Definition"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.JordanHolderFiltration → ℕ → ℒ", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "SemilatticeSup.toMax", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "ConditionallyCompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction", "lt_sup_of_lt_left", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.inf_A_le_A_sup", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y u : ℒ},\n x ∈ I →\n y ∈ I →\n u ∈ I →\n ∀ (h₁ : u < x) (h₂ : u < y),\n μ.A { left := u, right := x, lt := h₁ } ⊓ μ.A { left := u, right := y, lt := h₂ } ≤\n μ.A { left := u, right := x ⊔ y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := y, lt := h₁ } < μ { left := x, right := z, lt := ⋯ } ∧\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := h₂ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := h₁ } ∧\n μ { left := y, right := z, lt := h₂ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := x, right := y, lt := h₁ } = μ { left := x, right := z, lt := ⋯ } ∧\n μ { left := x, right := z, lt := ⋯ } = μ { left := y, right := z, lt := h₂ }", "constCategory": "Theorem"}, {"references": ["DFunLike", "outParam", "Function.Injective"], "name": "DFunLike.mk", "constType": "{F : Sort u_1} →\n {α : outParam (Sort u_2)} →\n {β : outParam (α → Sort u_3)} → (coe : F → (a : α) → β a) → Function.Injective coe → DFunLike F α β", "constCategory": "Other"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl.noConfusionType", "HarderNarasimhan.StrictIntvl", "HEq.refl", "eq_of_heq", "HarderNarasimhan.StrictIntvl.casesOn", "HEq", "Eq.ndrec", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.noConfusion", "constType": "{P : Sort u} →\n {ℒ : Type u_1} →\n {inst : LT ℒ} →\n {t : HarderNarasimhan.StrictIntvl ℒ} →\n {ℒ' : Type u_1} →\n {inst' : LT ℒ'} →\n {t' : HarderNarasimhan.StrictIntvl ℒ'} →\n ℒ = ℒ' → inst ≍ inst' → t ≍ t' → HarderNarasimhan.StrictIntvl.noConfusionType P t t'", "constCategory": "Definition"}, {"references": ["SubtractionMonoid", "SubtractionMonoid.mk", "AddCommGroup.add_comm", "SubtractionMonoid.neg_eq_of_add", "AddCommGroup.toAddGroup", "SubtractionCommMonoid", "SubtractionCommMonoid.mk", "AddCommGroup", "SubtractionMonoid.neg_neg", "AddGroup.toSubNegMonoid", "SubtractionMonoid.neg_add_rev", "AddGroup.toSubtractionMonoid"], "name": "AddCommGroup.toDivisionAddCommMonoid", "constType": "{G : Type u_1} → [AddCommGroup G] → SubtractionCommMonoid G", "constCategory": "Definition"}, {"references": ["Not", "Finset.instSetLike", "Equiv.instEquivLike", "Finset", "SetLike.instMembership", "ofColex", "Membership.mem", "And", "Finset.Colex.instPartialOrder._proof_5", "Finset.Colex.instPartialOrder._proof_1", "DFunLike.coe", "Equiv", "Finset.Colex.instPartialOrder._proof_2", "PartialOrder.mk", "PartialOrder", "EquivLike.toFunLike", "Finset.Colex.instLE", "LE.le", "Finset.Colex.instPartialOrder._proof_6", "Colex", "Preorder.mk", "LT.mk"], "name": "Finset.Colex.instPartialOrder", "constType": "{α : Type u_1} → [PartialOrder α] → PartialOrder (Colex (Finset α))", "constCategory": "Definition"}, {"references": ["Submodule.completeLattice._proof_6", "PartialOrder.toPreorder", "Set.ofPred", "Submodule.instInfSet", "Module", "Submodule.isGLB_sInf", "Membership.mem", "Lattice.mk", "SupSet.mk", "Submodule.completeLattice._proof_1", "BoundedOrder.mk", "Submodule", "Submodule.completeLattice._proof_5", "AddCommMonoid", "InfSet.sInf", "Submodule.instOrderBot", "Preorder.toLE", "Submodule.instOrderTop", "Submodule.completeLattice._proof_3", "CompleteLattice.mk", "Set", "Submodule.completeLattice._proof_7", "Submodule.instPartialOrder", "Submodule.completeLattice._proof_2", "And", "Set.instMembership", "Submodule.completeLattice._proof_4", "Min.min", "LE.le", "SemilatticeSup.mk", "Submodule.instMin", "CompleteLattice", "Semiring"], "name": "Submodule.completeLattice", "constType": "{R : Type u_1} →\n {M : Type u_3} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → CompleteLattice (Submodule R M)", "constCategory": "Definition"}, {"references": ["Not", "Decidable", "Decidable.casesOn"], "name": "dite", "constType": "{α : Sort u} → (c : Prop) → [h : Decidable c] → (c → α) → (¬c → α) → α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Subtype", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Lattice", "Preorder.toLT", "IsModularLattice", "Preorder.toLE", "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.StrictIntvl.instIsModularLatticeSubtypeMem", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} [iml : IsModularLattice ℒ],\n IsModularLattice { x // x ∈ I }", "constCategory": "Theorem"}, {"references": ["CompletelyDistribLattice.toCompleteLattice", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "PartialOrder", "Preorder.toLT", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "CompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE"], "name": "HarderNarasimhan.PayoffFunction.instWeakSlopeLikeAtBotOfIsSlopeLike", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] {S : Type u_3} [inst_2 : CompleteLinearOrder S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike], μ.WeakSlopeLikeAtBot", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "Preorder.toLT"], "name": "lt_trans", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a < b → b < c → a < c", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f → ∃ N, ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }) →\n μ.ADCC", "constCategory": "Other"}, {"references": ["instAddNat", "HAdd.hAdd", "Nat", "instOfNatNat", "instHAdd", "RelSeries.length", "Fin", "RelSeries", "SetRel", "OfNat.ofNat"], "name": "RelSeries.toFun", "constType": "{α : Type u_1} → {r : SetRel α α} → (self : RelSeries r) → Fin (self.length + 1) → α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSemistable", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsAffine", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "Nat", "Nontrivial", "Lattice", "WellFoundedGT", "IsModularLattice", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "CompleteLinearOrder", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length_eq", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [WellFoundedGT ℒ]\n [IsModularLattice ℒ] {S : Type u_2} [inst_5 : CompleteLinearOrder S] {μ : HarderNarasimhan.PayoffFunction ℒ S}\n [μ.FiniteTotalPayoff] [μ.IsSlopeLike] [μ.IsSemistable] [μ.EventuallyTopDCC] [μ.IsAffine]\n (F G : μ.JordanHolderFiltration), F.length = G.length", "constCategory": "Theorem"}, {"references": ["instAddNat", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Set.iUnion", "HarderNarasimhan.CoprimaryFiltration.length", "DFunLike.coe", "Submodule", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "instLTNat", "CommRing.toCommSemiring", "SetLike.instMembership", "instHAdd", "Set", "Submodule.addCommGroup", "CommSemiring.toSemiring", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "CommRing", "OfNat.ofNat", "HAdd.hAdd", "LT.lt", "CommRing.toRing", "IsNoetherianRing", "Nat", "Submodule.setLike", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.associatedPrimes_eq_iUnion", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (F : HarderNarasimhan.CoprimaryFiltration R M),\n associatedPrimes R M = ⋃ i, ⋃ (_ : i < F.length), associatedPrimes R (↥(F (i + 1)) ⧸ (F i).submoduleOf (F (i + 1)))", "constCategory": "Theorem"}, {"references": ["LE.le", "LE", "OrderBot.toBot", "Bot.bot", "OrderBot"], "name": "bot_le", "constType": "∀ {α : Type u} [inst : LE α] [inst_1 : OrderBot α] {a : α}, ⊥ ≤ a", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })) →\n μ.IsSlopeLike", "constCategory": "Other"}, {"references": ["Set.instSupSet", "iSup", "Set"], "name": "Set.iUnion", "constType": "{α : Type u} → {ι : Sort v} → (ι → Set α) → Set α", "constCategory": "Definition"}, {"references": ["Nat", "CommRing.toCommSemiring", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Module", "Nontrivial", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.ctorIdx", "constType": "{R : Type u_1} →\n {inst : CommRing R} →\n {inst_1 : IsNoetherianRing R} →\n {M : Type u_2} →\n {inst_2 : Nontrivial M} →\n {inst_3 : AddCommGroup M} →\n {inst_4 : _root_.Module R M} → {inst_5 : Module.Finite R M} → HarderNarasimhan.CoprimaryFiltration R M → ℕ", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsAttained", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "Relation.SymmGen", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "SemilatticeSup.toMax", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "lt_sup_of_lt_left", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_le_A_sup_or", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y u : ℒ},\n x ∈ I →\n y ∈ I →\n u ∈ I →\n ∀ (h₁ : u < x) (h₂ : u < y),\n Relation.SymmGen (fun x1 x2 => x1 ≤ x2) (μ.A { left := u, right := x, lt := h₁ })\n (μ.A { left := u, right := y, lt := h₂ }) ∨\n μ.IsAttained { left := u, right := x ⊔ y, lt := ⋯ } →\n μ.A { left := u, right := x, lt := h₁ } ≤ μ.A { left := u, right := x ⊔ y, lt := ⋯ } ∨\n μ.A { left := u, right := y, lt := h₂ } ≤ μ.A { left := u, right := x ⊔ y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.PayoffFunction.StrongDCC", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_top_le_B_top_of_strongDCC", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.StrongDCC] [μ.WeakSlopeLikeAtBot],\n μ.A ⊤ ≤ μ.B ⊤", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.lt", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }) →\n μ.WeakSlopeLikeAtTop", "constCategory": "Definition"}, {"references": ["semiOutParam", "CoeOut"], "name": "CoeOut.mk", "constType": "{α : Sort u} → {β : semiOutParam (Sort v)} → (α → β) → CoeOut α β", "constCategory": "Other"}, {"references": ["OrderDual", "PartialOrder.toPreorder", "Equiv.instEquivLike", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Equiv", "OrderDual.ofDual", "OrderDual.instNontrivial", "PartialOrder", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderDual.instCompleteLattice", "OrderDual.instPartialOrder", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.dual", "BoundedOrder", "OrderDual.instPreorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "OrderDual.instBoundedOrder", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_top_dual", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, OrderDual.ofDual (μ.dual.B ⊤) = μ.A ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "PartialOrder", "Iff", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.ext_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F G : μ.JordanHolderFiltration},\n F = G ↔ ∀ (n : ℕ), F n = G n", "constCategory": "Theorem"}, {"references": [], "name": "And", "constType": "Prop → Prop → Prop", "constCategory": "Other"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "SizeOf", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration._sizeOf_1", "SizeOf.mk", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration._sizeOf_inst", "constType": "(R : Type u_1) →\n {inst : CommRing R} →\n {inst_1 : IsNoetherianRing R} →\n (M : Type u_2) →\n {inst_2 : Nontrivial M} →\n {inst_3 : AddCommGroup M} →\n {inst_4 : _root_.Module R M} →\n {inst_5 : Module.Finite R M} → [SizeOf R] → [SizeOf M] → SizeOf (HarderNarasimhan.CoprimaryFiltration R M)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "Inhabited.mk", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.hnFiltration", "Preorder.toLT", "BoundedOrder", "Inhabited", "HarderNarasimhan.PayoffFunction.IsConvex", "Nontrivial", "Lattice", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instInhabitedHarderNarasimhanFiltration", "constType": "{ℒ : Type u_1} →\n [Nontrivial ℒ] →\n [inst : Lattice ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [hwf : WellFoundedGT ℒ] →\n {S : Type u_2} →\n [inst_2 : CompleteLattice S] →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) →\n [μ.ADCC] → [μ.IsConvex] → [hadm : μ.Admissible] → Inhabited μ.HarderNarasimhanFiltration", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.JordanHolderFiltration → Sort u} →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (antitone : Antitone toFun) →\n (head_eq_top : toFun 0 = ⊤) →\n (length_eq_bot : toFun length = ⊥) →\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length)) →\n (step_payoff_eq :\n ∀ (i : ℕ) (hi : i < length),\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤) →\n (payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, antitone := antitone,\n head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq,\n payoff_lt_of_between := payoff_lt_of_between }) →\n (t : μ.JordanHolderFiltration) → motive t", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "IsGreatest", "instOfNatNat", "BoundedOrder.toOrderTop", "WellFoundedGT", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.breakpoints", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "instHAdd", "HarderNarasimhan.PayoffFunction.hnFiltration", "BoundedOrder", "Ne.lt_top", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.ADCC", "Ne", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.hnFiltration_succ_isGreatest_breakpoints", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible] {n : ℕ} (h : μ.hnFiltration n ≠ ⊤),\n IsGreatest (μ.breakpoints { left := μ.hnFiltration n, right := ⊤, lt := ⋯ }) (μ.hnFiltration (n + 1))", "constCategory": "Theorem"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_total_lt_right_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := h₂ } ↔\n μ { left := x, right := y, lt := h₁ } < μ { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "Preorder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "iSup", "Set.Ioc", "ConditionallyCompletePartialOrderSup.toSupSet", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.max._proof_1", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.max", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.eq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Lattice ℒ} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsAffine]\n (x y : ℒ) (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsStable", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [toIsSemistable : μ.IsSemistable],\n (∀ (x : ℒ) (hx : ⊥ < x), x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤) → μ.IsStable", "constCategory": "Other"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.left", "Set", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "Preorder.toLT", "Set.Ico", "Set.instMembership"], "name": "HarderNarasimhan.PayoffFunction.min._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), ∀ u ∈ Set.Ico I.left I.right, u < I.right", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["AddMonoid.toZero", "SubNegMonoid.toAddMonoid", "SubNegMonoid.toNeg", "SubNegZeroMonoid.toSubNegMonoid", "NegZeroClass", "NegZeroClass.mk", "SubNegZeroMonoid", "SubNegZeroMonoid.neg_zero"], "name": "SubNegZeroMonoid.toNegZeroClass", "constType": "{G : Type u_2} → [self : SubNegZeroMonoid G] → NegZeroClass G", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Membership.mem", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.IsCoprimary", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "CompleteLattice.toBoundedOrder", "Submodule.submoduleOf", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.module", "DFunLike.coe", "Submodule", "Submodule.Quotient.addCommGroup", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "instOfNatNat", "HarderNarasimhan.Coprimary.payoff", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "AddCommGroup", "DedekindCut", "CommRing", "OfNat.ofNat", "PrimeSpectrum", "LT.lt", "HAdd.hAdd", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Submodule.setLike", "LE.le", "Submodule.completeLattice", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.piecewise_isCoprimary", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M]\n (F : (HarderNarasimhan.Coprimary.payoff R M).HarderNarasimhanFiltration),\n ∀ i < F.length, HarderNarasimhan.IsCoprimary R (↥(F (i + 1)) ⧸ (F i).submoduleOf (F (i + 1)))", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.left", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → HarderNarasimhan.StrictIntvl ℒ → ℒ", "constCategory": "Definition"}, {"references": ["Order.Frame.mk", "CompletelyDistribLattice.himp_bot", "CompletelyDistribLattice.toCompl", "CompletelyDistribLattice.toHNot", "CompletelyDistribLattice.sdiff_le_iff", "CompleteDistribLattice", "CompletelyDistribLattice.toCompleteLattice", "CompletelyDistribLattice.top_sdiff", "CompletelyDistribLattice.toHImp", "CompletelyDistribLattice.toSDiff", "CompletelyDistribLattice", "CompleteDistribLattice.mk", "CompletelyDistribLattice.le_himp_iff"], "name": "CompletelyDistribLattice.toCompleteDistribLattice", "constType": "{α : Type u} → [CompletelyDistribLattice α] → CompleteDistribLattice α", "constCategory": "Definition"}, {"references": ["MulOneClass.toMulOne", "Nat", "MulOne.toOne", "Nat.instMulOneClass", "One", "inferInstance"], "name": "Nat.instOne", "constType": "One ℕ", "constCategory": "Definition"}, {"references": ["Real", "_private.Mathlib.Data.Real.Basic.0.Real.lt", "LT.mk", "LT"], "name": "Real.instLT", "constType": "LT ℝ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "BoundedOrder.toOrderTop", "PartialOrder", "Iff", "Nontrivial", "Top.top", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.isSemistable_iff_isBreakpoint_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.IsSemistable ↔ μ.IsBreakpoint ⊤ ⊤", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "PartialOrder.toPreorder", "Module", "Submodule.instBot", "Submodule.instPartialOrder", "Submodule.instOrderBot._proof_2", "Preorder.toLE", "OrderBot.mk", "OrderBot", "Submodule", "Semiring"], "name": "Submodule.instOrderBot", "constType": "{R : Type u_1} →\n {M : Type u_3} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → OrderBot (Submodule R M)", "constCategory": "Definition"}, {"references": ["Set.Subset", "Set", "LE.mk", "LE"], "name": "Set.instLE", "constType": "{α : Type u} → LE (Set α)", "constCategory": "Definition"}, {"references": ["Preorder", "SMul", "Zero"], "name": "PosSMulStrictMono", "constType": "(α : Type u_1) → (β : Type u_2) → [SMul α β] → [Preorder α] → [Preorder β] → [Zero α] → Prop", "constCategory": "Other"}, {"references": ["SemilatticeInf", "Lattice.toSemilatticeSup", "Lattice.inf_le_right", "Lattice.inf_le_left", "Lattice", "SemilatticeSup.toPartialOrder", "Lattice.inf", "SemilatticeInf.mk", "Lattice.le_inf"], "name": "Lattice.toSemilatticeInf", "constType": "{α : Type u} → [self : Lattice α] → SemilatticeInf α", "constCategory": "Definition"}, {"references": ["HEq"], "name": "HEq.refl", "constType": "∀ {α : Sort u} (a : α), a ≍ a", "constCategory": "Other"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.le_or_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : BoundedOrder ℒ} {inst_2 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.WeakSlopeLikeAtTop] (z : HarderNarasimhan.StrictIntvl ℒ)\n (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }", "constCategory": "Theorem"}, {"references": [], "name": "CommRing", "constType": "Type u → Type u", "constCategory": "Other"}, {"references": ["Min"], "name": "Min.min", "constType": "{α : Type u} → [self : Min α] → α → α → α", "constCategory": "Definition"}, {"references": ["Submodule.toAddSubmonoid", "Set", "Module", "AddCommMonoid.toAddMonoid", "AddZero.toAdd", "AddZeroClass.toAddZero", "Submodule.setLike._proof_1", "Submodule", "AddSubsemigroup.carrier", "SetLike.mk", "AddCommMonoid", "SetLike", "AddSubmonoid.toAddSubsemigroup", "Eq", "Semiring", "AddMonoid.toAddZeroClass"], "name": "Submodule.setLike", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → SetLike (Submodule R M) M", "constCategory": "Definition"}, {"references": ["Submodule.hasQuotient", "associatedPrimes", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Preorder.toLT", "Submodule", "Submodule.Quotient.addCommGroup", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "CommRing.toCommSemiring", "Set.preimage", "SetLike.instMembership", "HarderNarasimhan.StrictIntvl.right", "Submodule.addCommGroup", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "PrimeSpectrum", "CommRing.toRing", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "Submodule.setLike", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Set.Finite", "Submodule.submoduleOf", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite._proof_1", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n [IsNoetherianRing R] [Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (PrimeSpectrum.asIdeal ⁻¹' associatedPrimes R (↥I.right ⧸ I.left.submoduleOf I.right)).Finite", "constCategory": "Theorem"}, {"references": ["associatedPrimes", "Submodule.toAddSubmonoid", "Module", "OreLocalization.oreSetComm", "Inter.inter", "Membership.mem", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "Set.instEmptyCollection", "EmptyCollection.emptyCollection", "Algebra.id", "AddSubsemigroup.carrier", "Ideal", "instMulZeroOneClassOfSemiring", "DistribMulAction.toMulAction", "IsScalarTower.right", "Eq", "Semiring.toModule", "Set.instInter", "CommRing.toCommSemiring", "LocalizedModule", "OreLocalization.instAddCommMonoidOreLocalization", "Semiring.toMonoid", "IsScalarTower.left", "Submonoid.instSetLike", "Set", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "AddCommGroup", "AddZeroClass.toAddZero", "AddZero.toAdd", "CommRing", "CommSemiring.toCommMonoid", "Set.instMembership", "Semiring.toAddCommMonoid", "Submonoid", "Module.toDistribMulAction", "SetLike.coe", "AddCommGroup.toAddCommMonoid", "AddSubmonoid.toAddSubsemigroup", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.inter_eq_empty_of_mem_associatedPrimes_localizedModule", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (S : Submonoid R) {p : Ideal R}, p ∈ associatedPrimes R (LocalizedModule S M) → p.carrier ∩ ↑S = ∅", "constCategory": "Theorem"}, {"references": ["LE"], "name": "LE.mk", "constType": "{α : Type u} → (α → α → Prop) → LE α", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "Iff", "PartialOrder", "LE.le", "And", "Preorder.toLT", "Preorder.toLE", "HarderNarasimhan.StrictIntvl.instPartialOrder"], "name": "HarderNarasimhan.StrictIntvl.le_def", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I J : HarderNarasimhan.StrictIntvl ℒ},\n I ≤ J ↔ J.left ≤ I.left ∧ I.right ≤ J.right", "constCategory": "Theorem"}, {"references": ["LT"], "name": "HarderNarasimhan.PayoffFunction", "constType": "(ℒ : Type u_1) → [LT ℒ] → Type u_2 → Type (max u_1 u_2)", "constCategory": "Other"}, {"references": ["PrimeSpectrum", "CommRing.toCommSemiring", "LinearExtension", "Ideal.IsPrime", "CommSemiring.toSemiring", "PrimeSpectrum.asIdeal", "CommRing"], "name": "HarderNarasimhan.instIsPrimeAsIdeal_harderNarasimhan", "constType": "∀ {R : Type u_1} [inst : CommRing R] (p : LinearExtension (PrimeSpectrum R)), p.asIdeal.IsPrime", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "SetLike.instMembership", "Subtype", "Submodule.setLike", "Module", "Membership.mem", "AddSubmonoidClass.toAddCommMonoid", "Submodule.addSubmonoidClass", "Submodule", "Semiring"], "name": "Submodule.addCommMonoid", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] →\n [inst_1 : AddCommMonoid M] → {module_M : _root_.Module R M} → (p : Submodule R M) → AddCommMonoid ↥p", "constCategory": "Definition"}, {"references": ["RingHom", "Submodule.toAddSubmonoid", "LinearMap.instFunLike", "RingHomSurjective", "Module", "AddCommMonoid.toAddMonoid", "DFunLike.coe", "Submodule.map._proof_4", "Submodule", "AddSubsemigroup.mk", "AddCommMonoid", "Semiring.toNonAssocSemiring", "Submodule.map._proof_3", "Submodule.map._proof_1", "AddSubmonoid.mk", "LinearMap", "AddZero.toAdd", "AddZeroClass.toAddZero", "Submodule.map._proof_2", "Set.image", "SetLike.coe", "AddSubmonoid", "Submodule.mk", "Submodule.setLike", "AddSubmonoid.map", "AddMonoid.toAddZeroClass", "Semiring"], "name": "Submodule.map", "constType": "{R : Type u_1} →\n {R₂ : Type u_3} →\n {M : Type u_5} →\n {M₂ : Type u_7} →\n [inst : Semiring R] →\n [inst_1 : Semiring R₂] →\n [inst_2 : AddCommMonoid M] →\n [inst_3 : AddCommMonoid M₂] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : _root_.Module R₂ M₂] →\n {σ₁₂ : R →+* R₂} → [RingHomSurjective σ₁₂] → (M →ₛₗ[σ₁₂] M₂) → Submodule R M → Submodule R₂ M₂", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.left", "Set.Icc", "Set", "HarderNarasimhan.StrictIntvl.right", "Iff", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE", "Set.instMembership"], "name": "HarderNarasimhan.StrictIntvl.mem_iff_mem_Icc", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ}, x ∈ I ↔ x ∈ Set.Icc I.left I.right", "constCategory": "Theorem"}, {"references": ["MulOneClass.toMulOne", "MonoidHom.mk", "RingHom", "MulOne.toOne", "MulZeroOneClass.toMulOneClass", "RingHom.id._proof_2", "RingHom.id._proof_1", "OneHom.mk", "RingHom.id._proof_3", "RingHom.id._proof_4", "NonAssocSemiring", "RingHom.mk", "NonAssocSemiring.toMulZeroOneClass"], "name": "RingHom.id", "constType": "(α : Type u_5) → [inst : NonAssocSemiring α] → α →+* α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Set.ofPred", "Set", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.breakpoints", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.StrictIntvl ℒ → Set ℒ", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "GeneralizedCoheytingAlgebra.toLattice", "PartialOrder.toPreorder", "CoheytingAlgebra.toGeneralizedCoheytingAlgebra", "SemilatticeSup.toPartialOrder", "OrderTop", "Preorder.toLE", "CoheytingAlgebra"], "name": "CoheytingAlgebra.toOrderTop", "constType": "{α : Type u_4} → [self : CoheytingAlgebra α] → OrderTop α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "SemilatticeInf.toPartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.mk", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "ConditionallyCompleteLattice.toLattice", "CompleteLattice.toBoundedOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_1", "Module", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.piecewise_isCoprimary", "Submodule", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_6", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_8", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "HarderNarasimhan.CoprimaryFiltration", "HarderNarasimhan.Coprimary.payoff", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_7", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_9", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Submodule.completeLattice", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration", "constType": "(R : Type u_1) →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n (M : Type u_2) →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] → [inst_5 : Module.Finite R M] → HarderNarasimhan.CoprimaryFiltration R M", "constCategory": "Definition"}, {"references": ["Nat.lt", "Nat", "LT.mk", "LT"], "name": "instLTNat", "constType": "LT ℕ", "constCategory": "Definition"}, {"references": ["RingHom", "AddCommMonoid", "Semiring.toNonAssocSemiring", "Submodule.comap", "Module", "Submodule.instBot", "LinearMap", "Bot.bot", "Submodule", "Semiring"], "name": "LinearMap.ker", "constType": "{R : Type u_1} →\n {R₂ : Type u_2} →\n {M : Type u_5} →\n {M₂ : Type u_7} →\n [inst : Semiring R] →\n [inst_1 : Semiring R₂] →\n [inst_2 : AddCommMonoid M] →\n [inst_3 : AddCommMonoid M₂] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : _root_.Module R₂ M₂] → {τ₁₂ : R →+* R₂} → (M →ₛₗ[τ₁₂] M₂) → Submodule R M", "constCategory": "Definition"}, {"references": ["Subtype.instLT", "HarderNarasimhan.StrictIntvl", "Subtype", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "WellFoundedGT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instWellFoundedGTSubtypeMem", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} [hw : WellFoundedGT ℒ],\n WellFoundedGT { x // x ∈ I }", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Bot.bot", "OfNat.ofNat", "Submodule", "Nat", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration.toFun", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "Submodule.instBot", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Eq", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.head_eq_bot", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M), self.toFun 0 = ⊥", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "CompleteLattice.toLattice", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ ⊤ ≠ ⊤ → μ.FiniteTotalPayoff", "constCategory": "Other"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsAttained", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "And", "Relation.SymmGen", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_eq_or_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y z : ℒ},\n x ∈ I →\n y ∈ I →\n z ∈ I →\n ∀ (h₁ : x < y) (h₂ : y < z),\n Relation.SymmGen (fun x1 x2 => x1 ≤ x2) (μ.A { left := x, right := y, lt := h₁ })\n (μ.A { left := y, right := z, lt := h₂ }) ∨\n μ.IsAttained { left := x, right := z, lt := ⋯ } →\n μ.A { left := y, right := z, lt := h₂ } = μ.A { left := x, right := z, lt := ⋯ } ∨\n μ.A { left := x, right := y, lt := h₁ } ≤ μ.A { left := x, right := z, lt := ⋯ } ∧\n μ.A { left := x, right := z, lt := ⋯ } < μ.A { left := y, right := z, lt := h₂ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.rec", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.JordanHolderFiltration → Sort u} →\n (t : μ.JordanHolderFiltration) →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (antitone : Antitone toFun) →\n (head_eq_top : toFun 0 = ⊤) →\n (length_eq_bot : toFun length = ⊥) →\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length)) →\n (step_payoff_eq :\n ∀ (i : ℕ) (hi : i < length),\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤) →\n (payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, antitone := antitone,\n head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq,\n payoff_lt_of_between := payoff_lt_of_between }) →\n motive t", "constCategory": "Definition"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.slopelike", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsSlopeLike] (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "Set.ofPred", "Set", "LE.le", "Preorder.toLT", "And", "Preorder.toLE"], "name": "Set.Ioc", "constType": "{α : Type u_1} → [Preorder α] → α → α → Set α", "constCategory": "Definition"}, {"references": [], "name": "Iff", "constType": "Prop → Prop → Prop", "constCategory": "Other"}, {"references": ["IsLeftCancelAdd", "PartialOrder.toPreorder", "Add", "PartialOrder", "Preorder.toLE", "AddLeftReflectLE"], "name": "instIsLeftCancelAddOfAddLeftReflectLE", "constType": "∀ {α : Type u_1} [inst : Add α] [inst_1 : PartialOrder α] [AddLeftReflectLE α], IsLeftCancelAdd α", "constCategory": "Theorem"}, {"references": ["Preorder", "AddCommMonoid", "AddCommMonoid.toAddMonoid", "Preorder.toLE", "AddZeroClass.toAddZero", "AddZero.toAdd", "AddLeftReflectLE", "AddMonoid.toAddZeroClass", "IsOrderedCancelAddMonoid"], "name": "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "constType": "∀ {α : Type u_2} [inst : AddCommMonoid α] [inst_1 : Preorder α] [IsOrderedCancelAddMonoid α], AddLeftReflectLE α", "constCategory": "Theorem"}, {"references": ["Top"], "name": "Top.top", "constType": "{α : Type u_1} → [self : Top α] → α", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.CoprimaryFiltration.rec", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.casesOn", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n {motive : HarderNarasimhan.CoprimaryFiltration R M → Sort u} →\n (t : HarderNarasimhan.CoprimaryFiltration R M) →\n ((toFun : ℕ → Submodule R M) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isCoprimary :\n ∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))) →\n (associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q) →\n motive\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt }) →\n motive t", "constCategory": "Definition"}, {"references": ["Set", "Finite", "Set.Elem"], "name": "Set.Finite", "constType": "{α : Type u} → Set α → Prop", "constCategory": "Definition"}, {"references": ["LT.lt", "Preorder", "Set", "Membership.mem", "Preorder.toLT", "Set.instMembership"], "name": "StrictAntiOn", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Set α → Prop", "constCategory": "Definition"}, {"references": ["LT.lt", "Preorder", "Preorder.toLT"], "name": "StrictMono", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Prop", "constCategory": "Definition"}, {"references": ["Equiv.refl", "OrderDual", "Equiv"], "name": "OrderDual.toDual", "constType": "{α : Type u_1} → α ≃ αᵒᵈ", "constCategory": "Definition"}, {"references": ["Preorder", "Set.ofPred", "Set", "LE.le", "And", "Preorder.toLE"], "name": "Set.Icc", "constType": "{α : Type u_1} → [Preorder α] → α → α → Set α", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Semiring"], "name": "Module", "constType": "(R : Type u) → (M : Type v) → [Semiring R] → [AddCommMonoid M] → Type (max u v)", "constCategory": "Other"}, {"references": ["Submonoid", "SetLike.instMembership", "Subtype", "CommMonoid.toMonoid", "Submonoid.instSetLike", "Membership.mem", "Monoid.toMulOneClass", "OreLocalization.OreSet.mk", "OreLocalization.oreSetComm._proof_2", "OreLocalization.oreSetComm._proof_3", "CommMonoid", "OreLocalization.OreSet"], "name": "OreLocalization.oreSetComm", "constType": "{R : Type u_2} → [inst : CommMonoid R] → (S : Submonoid R) → OreLocalization.OreSet S", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "PartialOrder.toPreorder", "Module", "StrictMonoOn", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.CoprimaryFiltration.length", "AddCommGroup", "CommRing", "Set.Iic", "Nat.instPreorder", "Submodule", "HarderNarasimhan.CoprimaryFiltration.toFun", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.strictMonoOn", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M), StrictMonoOn self.toFun (Set.Iic self.length)", "constCategory": "Theorem"}, {"references": ["Nat", "CommRing.toCommSemiring", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Module", "Nontrivial", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.length", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] → [inst_5 : Module.Finite R M] → HarderNarasimhan.CoprimaryFiltration R M → ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Nontrivial", "Top.top", "Preorder.toLE", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.mem_top", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ] (x : ℒ), x ∈ ⊤", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "AddMonoid"], "name": "AddCommMonoid.toAddMonoid", "constType": "{M : Type u} → [self : AddCommMonoid M] → AddMonoid M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "Module", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "Nat.instPreorder", "Submodule", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "instDistribLatticeOfLinearOrder", "Monotone", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "Nat", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.completeLattice", "ConditionallyCompleteLattice.toLattice", "CompleteLattice.toBoundedOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_6", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n Monotone (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.toFun", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (mem : x ∈ I)\n (ne_left : I.left ≠ x),\n (∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } < μ.A { left := I.left, right := y, lt := ⋯ }) →\n (∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } = μ.A { left := I.left, right := x, lt := ⋯ } → y ≤ x) →\n μ.IsBreakpoint I x", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.StrictIntvl ℒ → ℒ → Prop", "constCategory": "Other"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "LT.lt", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "Lattice", "HarderNarasimhan.PayoffFunction.A", "Eq", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_eq_of_ge", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y z : ℒ},\n x ∈ I →\n y ∈ I →\n z ∈ I →\n ∀ (h₁ : x < y) (h₂ : y < z),\n μ.A { left := y, right := z, lt := h₂ } ≤ μ.A { left := x, right := y, lt := h₁ } →\n μ.A { left := y, right := z, lt := h₂ } = μ.A { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.ADCC.mk", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.ADCC → Sort u} →\n ((dcc :\n ∀ (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f →\n ∃ N, ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }) →\n motive ⋯) →\n (t : μ.ADCC) → motive t", "constCategory": "Other"}, {"references": ["ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "ConditionallyCompletePartialOrderInf.mk", "ConditionallyCompletePartialOrder.isGLB_csInf_of_directed", "ConditionallyCompletePartialOrder.toInfSet", "ConditionallyCompletePartialOrderInf", "ConditionallyCompletePartialOrder"], "name": "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf", "constType": "{α : Type u_3} → [self : ConditionallyCompletePartialOrder α] → ConditionallyCompletePartialOrderInf α", "constCategory": "Definition"}, {"references": [], "name": "Unique", "constType": "Sort u → Sort (max 1 u)", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.PayoffFunction.WeakACC.mk", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }) →\n μ.WeakACC", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Membership.mk", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.right", "LE.le", "And", "LE", "Membership", "LT"], "name": "HarderNarasimhan.StrictIntvl.instMembership", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → [LE ℒ] → Membership ℒ (HarderNarasimhan.StrictIntvl ℒ)", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "PartialOrder", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "instLENat", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length_le_of_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.HarderNarasimhanFiltration} {m : ℕ}, F m = ⊤ → F.length ≤ m", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop"], "name": "HarderNarasimhan.PayoffFunction.A_top_eq_min_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [h₁ : μ.WeakACC] [h₂ : μ.WeakSlopeLikeAtTop],\n μ.A ⊤ = μ.min ⊤", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Module", "Nontrivial", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "Nonempty", "AddCommGroup", "CommRing", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instNonemptyCoprimaryFiltration", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n Nonempty (HarderNarasimhan.CoprimaryFiltration R M)", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "lt_of_lt_of_le", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a < b → b ≤ c → a < c", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "OrderTop", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.instOrderTop._proof_1", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "Top.top", "Top.mk", "OrderBot.toBot", "Preorder.toLE", "OrderTop.toTop", "bot_lt_top", "OrderTop.mk"], "name": "HarderNarasimhan.StrictIntvl.instOrderTop", "constType": "{ℒ : Type u_1} → [Nontrivial ℒ] → [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → OrderTop (HarderNarasimhan.StrictIntvl ℒ)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsBreakpoint.left_lt", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.isBreakpoint_left", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (hx : μ.IsBreakpoint I x),\n μ.IsBreakpoint { left := I.left, right := x, lt := ⋯ } x", "constCategory": "Theorem"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "LT.lt", "Nat", "PartialOrder", "Nontrivial", "LE.le", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "instLENat", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.apply_lt_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {i j : ℕ},\n i < j → j ≤ F.length → F j < F i", "constCategory": "Theorem"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.ctorIdx", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : Nontrivial ℒ} →\n {inst_1 : PartialOrder ℒ} →\n {inst_2 : BoundedOrder ℒ} →\n {inst_3 : CompleteLattice S} → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.JordanHolderFiltration → ℕ", "constCategory": "Definition"}, {"references": [], "name": "CompleteLattice", "constType": "Type u_8 → Type u_8", "constCategory": "Other"}, {"references": ["Subtype", "LE.le", "LE.mk", "LE", "Subtype.val"], "name": "Subtype.instLE", "constType": "{α : Type u} → [LE α] → {P : α → Prop} → LE (Subtype P)", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.ADCC.mk", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC.rec", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.ADCC → Sort u} →\n (t : μ.ADCC) →\n ((dcc :\n ∀ (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f →\n ∃ N,\n ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Inter"], "name": "Inter.inter", "constType": "{α : Type u} → [self : Inter α] → α → α → α", "constCategory": "Definition"}, {"references": ["Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "DFunLike.coe", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.CoprimaryFiltration", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Concept.instCompleteLattice", "Colex", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Exists", "Lattice.toSemilatticeInf", "LinearExtension", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "CompleteLattice.toBoundedOrder", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.exists_hnFiltration", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (a : HarderNarasimhan.CoprimaryFiltration R M), ∃ F, ⇑a = ⇑F", "constCategory": "Theorem"}, {"references": ["Nat", "Zero.ofOfNat0", "instOfNatNat", "One.ofOfNat1", "instLENat", "ZeroLEOneClass"], "name": "Nat.instZeroLEOneClass", "constType": "ZeroLEOneClass ℕ", "constCategory": "Theorem"}, {"references": ["LT.lt.trans", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "ConditionallyCompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.inf_le_A", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y z : ℒ},\n x ∈ I →\n y ∈ I →\n z ∈ I →\n ∀ (h₁ : x < y) (h₂ : y < z),\n μ.A { left := x, right := y, lt := h₁ } ⊓ μ.A { left := y, right := z, lt := h₂ } ≤\n μ.A { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (mem : x ∈ I)\n (ne_left : I.left ≠ x),\n (∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } < μ.A { left := I.left, right := y, lt := ⋯ }) →\n (∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } = μ.A { left := I.left, right := x, lt := ⋯ } → y ≤ x) →\n μ.IsBreakpoint I x", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HEq.refl", "eq_of_heq", "HEq", "HarderNarasimhan.PayoffFunction.casesOn", "Eq.ndrec", "Eq", "HarderNarasimhan.PayoffFunction", "LT", "HarderNarasimhan.PayoffFunction.noConfusionType"], "name": "HarderNarasimhan.PayoffFunction.noConfusion", "constType": "{P : Sort u} →\n {ℒ : Type u_1} →\n {inst : LT ℒ} →\n {S : Type u_2} →\n {t : HarderNarasimhan.PayoffFunction ℒ S} →\n {ℒ' : Type u_1} →\n {inst' : LT ℒ'} →\n {S' : Type u_2} →\n {t' : HarderNarasimhan.PayoffFunction ℒ' S'} →\n ℒ = ℒ' → inst ≍ inst' → S = S' → t ≍ t' → HarderNarasimhan.PayoffFunction.noConfusionType P t t'", "constCategory": "Definition"}, {"references": ["Semiring.natCast_succ", "Semiring.left_distrib", "NonUnitalNonAssocSemiring.mk", "Semiring.toMonoid", "Monoid.toOne", "Monoid.mul_one", "NonAssocSemiring.mk", "Semigroup.toMul", "Semiring.mul_zero", "Semiring.toAddCommMonoid", "Semiring.right_distrib", "Semiring.natCast_zero", "Semiring.toNatCast", "Semiring.zero_mul", "NonAssocSemiring", "Monoid.one_mul", "Monoid.toSemigroup", "Semiring"], "name": "Semiring.toNonAssocSemiring", "constType": "{α : Type u} → [self : Semiring α] → NonAssocSemiring α", "constCategory": "Definition"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "LT.lt", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "LE.le", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Lattice", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_le_A_of_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x y z : ℒ},\n x ∈ I →\n y ∈ I →\n z ∈ I →\n ∀ (h₁ : x < y) (h₂ : y < z),\n μ.A { left := x, right := y, lt := h₁ } < μ.A { left := y, right := z, lt := h₂ } →\n μ.A { left := x, right := y, lt := h₁ } ≤ μ.A { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": [], "name": "Or", "constType": "Prop → Prop → Prop", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "StrictMonoOn", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "HEq", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.casesOn", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.noConfusionType", "constType": "Sort u →\n {ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n μ.HarderNarasimhanFiltration →\n {ℒ' : Type u_1} →\n {S' : Type u_2} →\n [inst' : PartialOrder ℒ'] →\n [inst'_1 : BoundedOrder ℒ'] →\n [inst'_2 : CompleteLattice S'] →\n {μ' : HarderNarasimhan.PayoffFunction ℒ' S'} → μ'.HarderNarasimhanFiltration → Sort u", "constCategory": "Definition"}, {"references": ["LT"], "name": "LT.mk", "constType": "{α : Type u} → (α → α → Prop) → LT α", "constCategory": "Other"}, {"references": [], "name": "CompleteLinearOrder", "constType": "Type u_8 → Type u_8", "constCategory": "Other"}, {"references": ["SubNegMonoid", "AddGroup"], "name": "AddGroup.toSubNegMonoid", "constType": "{A : Type u} → [self : AddGroup A] → SubNegMonoid A", "constCategory": "Definition"}, {"references": ["Finset.instSetLike", "SetLike.instMembership", "Exists", "Finset", "Membership.mem"], "name": "Finset.Nonempty", "constType": "{α : Type u_1} → Finset α → Prop", "constCategory": "Definition"}, {"references": ["And"], "name": "And.left", "constType": "∀ {a b : Prop}, a ∧ b → a", "constCategory": "Theorem"}, {"references": ["SemilatticeInf", "PartialOrder"], "name": "SemilatticeInf.toPartialOrder", "constType": "{α : Type u} → [self : SemilatticeInf α] → PartialOrder α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "Subtype", "PartialOrder.toPreorder", "PartialOrder", "LE.le", 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{μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration), F 0 = ⊤", "constCategory": "Theorem"}, {"references": ["Or"], "name": "Relation.SymmGen", "constType": "{α : Type u_1} → (α → α → Prop) → α → α → Prop", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "True", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Nontrivial", "Top.top", "Preorder.toLE", "Eq", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.mem_top._simp_1", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ] (x : ℒ), (x ∈ ⊤) = True", "constCategory": "Theorem"}, {"references": ["instAddNat", "HAdd.hAdd", "LT.lt", "instLTNat", "Nat", "instOfNatNat", "instHAdd", "OfNat.ofNat"], "name": "Nat.lt_add_one", "constType": "∀ (n : ℕ), 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μ.FiniteTotalPayoff", "constCategory": "Definition"}, {"references": ["LT"], "name": "HarderNarasimhan.StrictIntvl", "constType": "(ℒ : Type u_1) → [LT ℒ] → Type u_1", "constCategory": "Other"}, {"references": ["Multiset.instSingleton", "Finset.mk", "Finset", "Singleton.singleton", "Singleton.mk", "Multiset.nodup_singleton", "Multiset", "Singleton"], "name": "Finset.instSingleton", "constType": "{α : Type u_1} → Singleton α (Finset α)", "constCategory": "Definition"}, {"references": ["Semiring.toMonoid", "Real", "Monoid", "inferInstance", "Real.semiring"], "name": "Real.instMonoid", "constType": "Monoid ℝ", "constCategory": "Definition"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "Function.Injective"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat._proof_1", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n Function.Injective HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "constCategory": "Theorem"}, {"references": ["OrderDual", "BoundedOrder.toOrderBot", "BoundedOrder.toOrderTop", "OrderDual.instOrderTopOfOrderBot", "LE", "BoundedOrder", "OrderDual.instLE", "OrderDual.instOrderBotOfOrderTop", "BoundedOrder.mk"], "name": "OrderDual.instBoundedOrder", "constType": "(α : Type u) → [inst : LE α] → [BoundedOrder α] → BoundedOrder αᵒᵈ", "constCategory": "Definition"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.isSlopeLike_iff_seesaw", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike ↔\n ∀ (x y z : ℒ) (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := y, lt := h₁ } < μ { left := x, right := z, lt := ⋯ } ∧\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := h₂ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := h₁ } ∧\n μ { left := y, right := z, lt := h₂ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := x, right := y, lt := h₁ } = μ { left := x, right := z, lt := ⋯ } ∧\n μ { left := x, right := z, lt := ⋯ } = μ { left := y, right := z, lt := h₂ }", "constCategory": "Theorem"}, {"references": [], "name": "OrderDual", "constType": "Type u_2 → Type u_2", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "PartialOrder", "LE.le", "And", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_3", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] (x x_1 x_2 : HarderNarasimhan.StrictIntvl ℒ),\n x_1.left ≤ x.left ∧ x.right ≤ x_1.right →\n x_2.left ≤ x_1.left ∧ x_1.right ≤ x_2.right → x_2.left ≤ x.left ∧ x.right ≤ x_2.right", "constCategory": "Theorem"}, {"references": ["Nat", "NatCast.natCast", "NatCast"], "name": "Nat.cast", "constType": "{R : Type u} → [NatCast R] → ℕ → R", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.rec", "HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.recOn", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {S : Type u_2} →\n {motive : HarderNarasimhan.PayoffFunction ℒ S → Sort u} →\n (t : HarderNarasimhan.PayoffFunction ℒ S) →\n ((toFun : HarderNarasimhan.StrictIntvl ℒ → S) → motive { toFun := toFun }) → motive t", "constCategory": "Definition"}, {"references": ["Subtype", "SetLike.instMembership", "Module", "Distrib.toMul", "instDistribOfSemiring", "Submodule.module._proof_1", "Membership.mem", "instSMulOfMul", "Submodule", "Submodule.module'", "AddCommMonoid", "Submodule.setLike", "Submodule.addCommMonoid", "Semiring"], "name": "Submodule.module", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] →\n [inst_1 : AddCommMonoid M] → {module_M : _root_.Module R M} → (p : Submodule R M) → _root_.Module R ↥p", "constCategory": "Definition"}, {"references": ["instAddNat", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "HarderNarasimhan.CoprimaryFiltration.length", "Submodule", "Submodule.Quotient.addCommGroup", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "instLTNat", "CommRing.toCommSemiring", "SetLike.instMembership", "instHAdd", "Submodule.addCommGroup", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "CommRing.toRing", "Nat", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration.toFun", "HarderNarasimhan.IsCoprimary", "Submodule.setLike", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.piecewise_isCoprimary", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M),\n ∀ i < self.length,\n HarderNarasimhan.IsCoprimary R (↥(self.toFun (i + 1)) ⧸ (self.toFun i).submoduleOf (self.toFun (i + 1)))", "constCategory": "Theorem"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "Bot.bot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "LT.lt", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "Nontrivial", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.bot_lt_of_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n m < F.length → ⊥ < F m", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk._flat_ctor", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n (toFun : ℕ → ℒ) →\n (length : ℕ) →\n Monotone toFun →\n toFun 0 = ⊥ →\n toFun length = ⊤ →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable) →\n (∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }) →\n μ.HarderNarasimhanFiltration", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsAffine", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.IsConvex", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.toIsConvex", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [haff : μ.IsAffine], μ.IsConvex", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.PayoffFunction.WeakACC.mk", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakACC → Sort u} →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.WeakACC) → motive t", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "PartialOrder", "LE.le", "And", "Preorder.toLT", "Eq", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_4", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] (x x_1 : HarderNarasimhan.StrictIntvl ℒ),\n x_1.left ≤ x.left ∧ x.right ≤ x_1.right → x.left ≤ x_1.left ∧ x_1.right ≤ x.right → x = x_1", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["OrderDual", "PartialOrder.toPreorder", "Equiv.instEquivLike", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Equiv", "OrderDual.ofDual", "OrderDual.instNontrivial", "PartialOrder", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderDual.instCompleteLattice", "OrderDual.instPartialOrder", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.dual", "BoundedOrder", "OrderDual.instPreorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "OrderDual.instBoundedOrder", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_top_dual", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, OrderDual.ofDual (μ.dual.A ⊤) = μ.B ⊤", "constCategory": "Theorem"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.EventuallyTopDCC → Sort u} →\n ((exists_eq_top :\n ∀ (x : ℕ → ℒ) (hx : StrictAnti x), ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤) →\n motive ⋯) →\n (t : μ.EventuallyTopDCC) → motive t", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "Equiv.instEquivLike", "PartialOrder.toPreorder", "Finset", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Preorder.toLT", "DFunLike.coe", "Equiv", "Submodule", "instDistribLatticeOfLinearOrder", "EquivLike.toFunLike", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Colex", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "toColex", "Lattice.toSemilatticeInf", "LinearExtension", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Set.toFinset", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "DedekindCut.principal", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.payoff", "constType": "(R : Type u_1) →\n [inst : CommRing R] →\n [IsNoetherianRing R] →\n (M : Type u_2) →\n [inst_2 : AddCommGroup M] →\n [inst_3 : _root_.Module R M] →\n [Module.Finite R M] →\n HarderNarasimhan.PayoffFunction (Submodule R M)\n (DedekindCut (Colex (Finset (LinearExtension (PrimeSpectrum R)))))", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsConvex.mk", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.IsConvex.rec", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsConvex → Sort u} →\n (t : μ.IsConvex) →\n ((le :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PrimeSpectrum", "Ideal", "CommSemiring.toSemiring", "CommSemiring"], "name": "PrimeSpectrum.asIdeal", "constType": "{R : Type u_1} → [inst : CommSemiring R] → PrimeSpectrum R → Ideal R", "constCategory": "Definition"}, {"references": ["List"], "name": "List.nil", "constType": "{α : Type u} → List α", "constCategory": "Other"}, {"references": ["False"], "name": "Not", "constType": "Prop → Prop", "constCategory": "Definition"}, {"references": ["Submodule.Quotient.mk", "Submodule.hasQuotient", "HasQuotient.Quotient", "Module", "LinearMap.mk", "Submodule.Quotient.module", "AddHom.mk", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "LinearMap", "Submodule.mkQ._proof_1", "Ring.toSemiring", "Submodule", "Submodule.mkQ._proof_2", "Semiring.toNonAssocSemiring", "AddCommMonoid.toAddCommSemigroup", "RingHom.id", "AddCommGroup.toAddCommMonoid", "AddCommSemigroup.toAddCommMagma", "AddCommMagma.toAdd", "Ring"], "name": "Submodule.mkQ", "constType": "{R : Type u_1} →\n {M : Type u_2} →\n [inst : Ring R] → [inst_1 : AddCommGroup M] → [inst_2 : _root_.Module R M] → (p : Submodule R M) → M →ₗ[R] M ⧸ p", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.mk._flat_ctor", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] → {S : Type u_2} → (HarderNarasimhan.StrictIntvl ℒ → S) → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.IsCoprimary.mk", "CommRing.toCommSemiring", "associatedPrimes", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "HarderNarasimhan.IsCoprimary.rec", "CommRing", "Set.instMembership", "HarderNarasimhan.IsCoprimary", "Ideal", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.casesOn", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n {M : Type u_2} →\n [inst_1 : AddCommGroup M] →\n [inst_2 : _root_.Module R M] →\n {motive : HarderNarasimhan.IsCoprimary R M → Sort u} →\n (t : HarderNarasimhan.IsCoprimary R M) →\n ((existsUnique_associatedPrime : ∃! p, p ∈ associatedPrimes R M) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["Submonoid", "AddCommMonoid", "OreLocalization", "Monoid.toMulOneClass", "OreLocalization.add_comm", "DistribMulAction.toMulAction", "AddCommMonoid.mk", "Monoid", "AddCommMonoid.toAddMonoid", "OreLocalization.instAddMonoid", "DistribMulAction", "OreLocalization.OreSet"], "name": "OreLocalization.instAddCommMonoidOreLocalization", "constType": "{R : Type u_1} →\n [inst : Monoid R] →\n {S : Submonoid R} →\n [inst_1 : OreLocalization.OreSet S] →\n {X : Type u_2} →\n [inst_2 : AddCommMonoid X] → [inst_3 : DistribMulAction R X] → AddCommMonoid (OreLocalization S X)", "constCategory": "Definition"}, {"references": ["Add.add", "HAdd.mk", "HAdd", "Add"], "name": "instHAdd", "constType": "{α : Type u_1} → [Add α] → HAdd α α α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.CoprimaryFiltration.noConfusion", "instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "heq_of_eq", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Eq.refl", "Submodule.instBot", "Nontrivial", "id", "HEq", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Submodule", "Nat.instPreorder", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "HEq.refl", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk.noConfusion", "constType": "{R : Type u_1} →\n {inst : CommRing R} →\n {inst_1 : IsNoetherianRing R} →\n {M : Type u_2} →\n {inst_2 : Nontrivial M} →\n {inst_3 : AddCommGroup M} →\n {inst_4 : _root_.Module R M} →\n {inst_5 : Module.Finite R M} →\n {P : Sort u} →\n {toFun : ℕ → Submodule R M} →\n {length : ℕ} →\n {monotone : Monotone toFun} →\n {head_eq_bot : toFun 0 = ⊥} →\n {length_eq_top : toFun length = ⊤} →\n {strictMonoOn : StrictMonoOn toFun (Set.Iic length)} →\n {piecewise_isCoprimary :\n ∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))} →\n {associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q} →\n {toFun' : ℕ → Submodule R M} →\n {length' : ℕ} →\n {monotone' : Monotone toFun'} →\n {head_eq_bot' : toFun' 0 = ⊥} →\n {length_eq_top' : toFun' length' = ⊤} →\n {strictMonoOn' : StrictMonoOn toFun' (Set.Iic length')} →\n {piecewise_isCoprimary' :\n ∀ i < length',\n HarderNarasimhan.IsCoprimary R\n (↥(toFun' (i + 1)) ⧸ (toFun' i).submoduleOf (toFun' (i + 1)))} →\n {associatedPrime_succ_lt' :\n ∀ (i : ℕ),\n i + 1 < length' →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun' (i + 2)) ⧸\n (toFun' (i + 1)).submoduleOf (toFun' (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun' (i + 1)) ⧸\n (toFun' i).submoduleOf (toFun' (i + 1))) →\n toLinearExtension p < toLinearExtension q} →\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn,\n piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt } =\n { toFun := toFun', length := length', monotone := monotone',\n head_eq_bot := head_eq_bot', length_eq_top := length_eq_top',\n strictMonoOn := strictMonoOn',\n piecewise_isCoprimary := piecewise_isCoprimary',\n associatedPrime_succ_lt := associatedPrime_succ_lt' } →\n (toFun ≍ toFun' → length = length' → P) → P", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n (toFun : ℕ → ℒ) →\n (length : ℕ) →\n Antitone toFun →\n toFun 0 = ⊤ →\n toFun length = ⊥ →\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length)) →\n (∀ (i : ℕ) (hi : i < length), μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤) →\n (∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }) →\n μ.JordanHolderFiltration", "constCategory": "Other"}, {"references": ["Real.instCommSemiring", "Real", "CommSemiring.toSemiring", "inferInstance", "Semiring"], "name": "Real.semiring", "constType": "Semiring ℝ", "constCategory": "Definition"}, {"references": ["Lattice"], "name": "IsModularLattice", "constType": "(α : Type u_2) → [Lattice α] → Prop", "constCategory": "Other"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_anti_left", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ.A { left := x, right := z, lt := ⋯ } ≤ μ.A { left := y, right := z, lt := h₂ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "instHAdd", "SizeOf", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "SizeOf.sizeOf", "HarderNarasimhan.StrictIntvl", "instOfNatNat", "instSizeOfDefault", "Eq", "LT", "HarderNarasimhan.StrictIntvl._sizeOf_inst"], "name": "HarderNarasimhan.StrictIntvl.mk.sizeOf_spec", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] [inst_1 : SizeOf ℒ] (left right : ℒ) (lt : left < right),\n sizeOf { left := left, right := right, lt := lt } = 1 + sizeOf left + sizeOf right + sizeOf lt", "constCategory": "Theorem"}, {"references": [], "name": "Finset", "constType": "Type u_4 → Type u_4", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) → μ.IsSemistable", "constCategory": "Definition"}, {"references": ["CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.Admissible", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Lattice", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "CompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Unique.0.HarderNarasimhan.PayoffFunction.instUniqueHarderNarasimhanFiltration._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {S : Type u_2} [inst_1 : CompleteLinearOrder S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.Admissible", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.left_mem", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), I.left ∈ I", "constCategory": "Theorem"}, {"references": ["Subtype", "Set", "Membership.mem", "Set.instMembership"], "name": "Set.Elem", "constType": "{α : Type u} → Set α → Type u", "constCategory": "Definition"}, {"references": ["upperBounds", "Set", "Membership.mem", "And", "LE", "Set.instMembership"], "name": "IsGreatest", "constType": "{α : Type u_1} → [LE α] → Set α → α → Prop", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Iff", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.ext_iff", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n {F G : HarderNarasimhan.CoprimaryFiltration R M}, F = G ↔ ∀ (n : ℕ), F n = G n", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.restrict", "Subtype.partialOrder", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.instEventuallyTopDCCSubtypeMemStrictIntvlRestrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [h : μ.EventuallyTopDCC] {I : HarderNarasimhan.StrictIntvl ℒ},\n (μ.restrict I).EventuallyTopDCC", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Nat", "HarderNarasimhan.PayoffFunction.rec", "instOfNatNat", "HarderNarasimhan.PayoffFunction", "SizeOf", "LT", "OfNat.ofNat"], "name": "HarderNarasimhan.PayoffFunction._sizeOf_1", "constType": "{ℒ : Type u_1} → {inst : LT ℒ} → {S : Type u_2} → [SizeOf ℒ] → [SizeOf S] → HarderNarasimhan.PayoffFunction ℒ S → ℕ", "constCategory": "Definition"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_right_lt_total_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := y, right := z, lt := h₂ } < μ { left := x, right := z, lt := ⋯ } ↔\n μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := h₁ }", "constCategory": "Theorem"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "LT.lt", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "PartialOrder", "Iff", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.ne_top_iff_lt_length", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.HarderNarasimhanFiltration} {m : ℕ}, F m ≠ ⊤ ↔ m < F.length", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "Nat.instPreorder", "Nat", "PartialOrder", "Antitone", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.antitone", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration),\n Antitone self.toFun", "constCategory": "Theorem"}, {"references": ["OrderDual", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.dual", "Preorder.toLT", "BoundedOrder", "PartialOrder", "OrderDual.instCompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "OrderDual.instPartialOrder", "OrderDual.instBoundedOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop"], "name": "HarderNarasimhan.PayoffFunction.instWeakSlopeLikeAtTopOrderDualDualOfWeakSlopeLikeAtBot", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [h₂ : μ.WeakSlopeLikeAtBot], μ.dual.WeakSlopeLikeAtTop", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "Preorder.toLT", "And", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.left", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "LE.le", "Top.top", "OrderBot.toBot", "Preorder.toLE", "OrderTop.toTop", "bot_lt_top"], "name": "HarderNarasimhan.StrictIntvl.instOrderTop._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n (x : HarderNarasimhan.StrictIntvl ℒ),\n { left := ⊥, right := ⊤, lt := ⋯ }.left ≤ x.left ∧ x.right ≤ { left := ⊥, right := ⊤, lt := ⋯ }.right", "constCategory": "Theorem"}, {"references": [], "name": "Prod", "constType": "Type u → Type v → Type (max u v)", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Set", "PartialOrder", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.breakpoints", "Eq", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "Set.instMembership"], "name": "HarderNarasimhan.PayoffFunction.mem_breakpoints._simp_1", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ},\n (x ∈ μ.breakpoints I) = μ.IsBreakpoint I x", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.StrictIntvl ℒ → Prop", "constCategory": "Other"}, {"references": ["CommRing.toCommSemiring", "associatedPrimes", "HarderNarasimhan.IsCoprimary.mk", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "CommRing", "Set.instMembership", "HarderNarasimhan.IsCoprimary", "Ideal", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.mk._flat_ctor", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M],\n (∃! p, p ∈ associatedPrimes R M) → HarderNarasimhan.IsCoprimary R M", "constCategory": "Definition"}, {"references": ["instAddNat", "RelSeries.toFun", "PartialOrder.toPreorder", "Preorder.toLT", "Fin", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "RelSeries.length", "CompleteLinearOrder", "SemilatticeInf.toPartialOrder", "HarderNarasimhan.PayoffFunction.relSeries_step_lt", "instLTNat", "And", "BoundedOrder", "instNeZeroNatHAdd_1", "Bot.bot", "RelSeries.last", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "RelSeries.head", "Nontrivial", "Lattice", "Top.top", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "HarderNarasimhan.PayoffFunction.semistableRel", "Nat.cast", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "RelSeries", "CompleteLinearOrder.toCompletelyDistribLattice", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "Nat.instNeZeroSucc", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Fin.NatCast.instNatCast", "BoundedOrder.toOrderTop", "instOfNatNat", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "WellFoundedGT", "OrderBot.toBot", "Eq", "Preorder.toLE", "Not", "Lattice.toSemilatticeInf", "instHAdd", "HarderNarasimhan.PayoffFunction.relSeries_succ_step_lt", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "CompletelyDistribLattice.toCompleteLattice", "LT.lt", "HAdd.hAdd", "LE.le", "ExistsUnique", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.existsUnique_relSeries_semistableRel", "constType": "∀ {ℒ : Type u_1} [Nontrivial ℒ] [inst : Lattice ℒ] [inst_1 : BoundedOrder ℒ] [WellFoundedGT ℒ] {S : Type u_2}\n [inst_3 : CompleteLinearOrder S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [μ.ADCC] [μ.IsConvex],\n ∃! s,\n s.head = ⊥ ∧\n s.last = ⊤ ∧\n ∀ (i : ℕ) (hi : i + 1 < s.length),\n ¬μ.A { left := s.toFun ↑i, right := s.toFun ↑(i + 1), lt := ⋯ } ≤\n μ.A { left := s.toFun ↑(i + 1), right := s.toFun ↑(i + 2), lt := ⋯ }", "constCategory": "Theorem"}, {"references": [], "name": "HEq", "constType": "{α : Sort u} → α → {β : Sort u} → β → Prop", "constCategory": "Other"}, {"references": ["HarderNarasimhan.Coprimary.coprimaryFiltration", "CommRing.toCommSemiring", "Inhabited.mk", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Inhabited", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instInhabitedCoprimaryFiltration", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] → Inhabited (HarderNarasimhan.CoprimaryFiltration R M)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn", "LE.le", "Lattice", "Preorder.toLT", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.StrictIntvl.instPartialOrder", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.mono", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I₁ I₂ : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I₁ → I₂ ≤ I₁ → μ.IsConvexOn I₂", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "Module", "Submodule.instTop", "CommSemiring.toSemiring", "HarderNarasimhan.CoprimaryFiltration.length", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "LE.le", "Nontrivial", "Top.top", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "instLENat", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.length_le_of_eq_top", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n {F : HarderNarasimhan.CoprimaryFiltration R M} {m : ℕ}, F m = ⊤ → F.length ≤ m", "constCategory": "Theorem"}, {"references": ["Ne"], "name": "Ne.symm", "constType": "∀ {α : Sort u} {a b : α}, a ≠ b → b ≠ a", "constCategory": "Theorem"}, {"references": ["Set"], "name": "Set.ofPred", "constType": "{α : Type u} → (α → Prop) → Set α", "constCategory": "Definition"}, {"references": ["Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "DedekindCut", "AddCommGroup", "CommRing", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.ADCC", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M], (HarderNarasimhan.Coprimary.payoff R M).ADCC", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "GeneralizedCoheytingAlgebra.mk", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "SemilatticeSup.toPartialOrder", "Order.Coframe.toSDiff", "Order.Coframe.toHNot", "Order.Coframe.toCompleteLattice", "BoundedOrder.toOrderBot", "CoheytingAlgebra.mk", "BoundedOrder.toOrderTop", "Order.Coframe", "Order.Coframe.top_sdiff", "Order.Coframe.sdiff_le_iff", "Preorder.toLE", "CompleteLattice.toBoundedOrder", "CoheytingAlgebra"], "name": "Order.Coframe.toCoheytingAlgebra", "constType": "{α : Type u_1} → [self : Order.Coframe α] → CoheytingAlgebra α", "constCategory": "Definition"}, {"references": ["EmptyCollection"], "name": "EmptyCollection.emptyCollection", "constType": "{α : Type u} → [self : EmptyCollection α] → α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mk", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsBreakpoint.rec", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n {x : ℒ} →\n {motive : μ.IsBreakpoint I x → Sort u} →\n (t : μ.IsBreakpoint I x) →\n ((mem : x ∈ I) →\n (ne_left : I.left ≠ x) →\n (not_lt :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } <\n μ.A { left := I.left, right := y, lt := ⋯ }) →\n (le_of_eq :\n ∀ (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } =\n μ.A { left := I.left, right := x, lt := ⋯ } →\n y ≤ x) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.noConfusion", "HEq.refl", "Eq.refl", "id", "HEq", "heq_of_eq", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk.noConfusion", "constType": "{ℒ : Type u_1} →\n {inst : LT ℒ} →\n {P : Sort u} →\n {left right : ℒ} →\n {lt : left < right} →\n {left' right' : ℒ} →\n {lt' : left' < right'} →\n { left := left, right := right, lt := lt } = { left := left', right := right', lt := lt' } →\n (left ≍ left' → right ≍ right' → P) → P", "constCategory": "Definition"}, {"references": ["OrderDual", "OrderDual.instCompleteLattice._proof_1", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "instCompleteSemilatticeInfOrderDualOfCompleteSemilatticeSup", "CompleteLattice.mk", "OrderDual.instCompleteLattice._proof_2", "OrderDual.instLattice", "CompleteLattice.toLattice", "CompleteSemilatticeSup.toSupSet", "CompleteLattice.toCompleteSemilatticeSup", "instCompleteSemilatticeSupOrderDualOfCompleteSemilatticeInf", "CompleteSemilatticeInf.toInfSet", "OrderDual.instBoundedOrder", "Preorder.toLE", "CompleteLattice.toCompleteSemilatticeInf", "CompleteLattice.toBoundedOrder", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "OrderDual.instCompleteLattice", "constType": "{α : Type u_1} → [CompleteLattice α] → CompleteLattice αᵒᵈ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.toFun", "FunLike", "HarderNarasimhan.StrictIntvl", "DFunLike.mk", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl._proof_1", "Eq", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} → [inst : LT ℒ] → FunLike (HarderNarasimhan.PayoffFunction ℒ S) (HarderNarasimhan.StrictIntvl ℒ) S", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "SizeOf", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.rec", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instSizeOfNat", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "SizeOf.sizeOf", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "instSizeOfDefault", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration._sizeOf_1", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : Nontrivial ℒ} →\n {inst_1 : PartialOrder ℒ} →\n {inst_2 : BoundedOrder ℒ} →\n {inst_3 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} → [SizeOf ℒ] → [SizeOf S] → μ.JordanHolderFiltration → ℕ", "constCategory": "Definition"}, {"references": ["Submodule.instNontrivial", "Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Preorder.toLT", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "HarderNarasimhan.StrictIntvl.instOrderTop", "DistribLattice.toLattice", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "Top.top", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ConditionallyCompleteLattice.toLattice", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instIsConvexOnSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoffTopStrictIntvl", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] [inst_5 : Nontrivial M],\n (HarderNarasimhan.Coprimary.payoff R M).IsConvexOn ⊤", "constCategory": "Theorem"}, {"references": ["CompleteLattice.isLUB_sSup", "Lattice.toSemilatticeSup", "Set.Nonempty", "PartialOrder.toPreorder", "Set", "CompleteLattice.toLattice", "SemilatticeSup.toPartialOrder", "BddBelow", "CompleteSemilatticeSup.toSupSet", "CompleteLattice.toCompleteSemilatticeSup", "BddAbove", "CompleteSemilatticeInf.toInfSet", "ConditionallyCompleteLattice", "CompleteLattice.isGLB_sInf", "CompleteLattice.toCompleteSemilatticeInf", "Preorder.toLE", "CompleteLattice", "ConditionallyCompleteLattice.mk"], "name": "CompleteLattice.toConditionallyCompleteLattice", "constType": "{α : Type u_1} → [CompleteLattice α] → ConditionallyCompleteLattice α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "iInf", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf", "ConditionallyCompletePartialOrderInf.toInfSet", "Eq", "Set.Ico", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "HarderNarasimhan.PayoffFunction.min", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) (I : HarderNarasimhan.StrictIntvl ℒ),\n μ.min I = ⨅ u, ⨅ (hu : u ∈ Set.Ico I.left I.right), μ { left := u, right := I.right, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_monotone", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_not_A_le_succ", "HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.hnFiltration._proof_1", "Preorder.toLT", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_piecewise_isSemistable", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_strictMonoOn", "HarderNarasimhan.PayoffFunction.IsConvex", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_length_eq_top", "Nontrivial", "Lattice", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.hnFiltration", "constType": "{ℒ : Type u_1} →\n [Nontrivial ℒ] →\n [inst : Lattice ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [hwf : WellFoundedGT ℒ] →\n {S : Type u_2} →\n [inst_2 : CompleteLattice S] →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) →\n [μ.ADCC] → [μ.IsConvex] → [hadm : μ.Admissible] → μ.HarderNarasimhanFiltration", "constCategory": "Definition"}, {"references": ["Concept", "Preorder", "LE.le", "Preorder.toLE"], "name": "DedekindCut", "constType": "(α : Type u_1) → [Preorder α] → Type u_1", "constCategory": "Definition"}, {"references": [], "name": "AddCommGroup", "constType": "Type u → Type u", "constCategory": "Other"}, {"references": [], "name": "Inhabited", "constType": "Sort u → Sort (max 1 u)", "constCategory": "Other"}, {"references": ["CommSemiring"], "name": "PrimeSpectrum", "constType": "(R : Type u_1) → [CommSemiring R] → Type u_1", "constCategory": "Other"}, {"references": ["Max"], "name": "Max.max", "constType": "{α : Type u} → [self : Max α] → α → α → α", "constCategory": "Definition"}, {"references": [], "name": "Preorder", "constType": "Type u_2 → Type u_2", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "StrictMonoOn", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "eq_of_heq", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.noConfusionType", "Nat.instIsOrderedAddMonoid", "Eq.ndrec", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Bot.bot", "Set.Iic", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Eq.refl", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "HEq", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.casesOn", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "HEq.refl", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.noConfusion", "constType": "{P : Sort u} →\n {ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : PartialOrder ℒ} →\n {inst_1 : BoundedOrder ℒ} →\n {inst_2 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {t : μ.HarderNarasimhanFiltration} →\n {ℒ' : Type u_1} →\n {S' : Type u_2} →\n {inst' : PartialOrder ℒ'} →\n {inst'_1 : BoundedOrder ℒ'} →\n {inst'_2 : CompleteLattice S'} →\n {μ' : HarderNarasimhan.PayoffFunction ℒ' S'} →\n {t' : μ'.HarderNarasimhanFiltration} →\n ℒ = ℒ' →\n S = S' →\n inst ≍ inst' →\n inst_1 ≍ inst'_1 →\n inst_2 ≍ inst'_2 →\n μ ≍ μ' →\n t ≍ t' →\n HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.noConfusionType P\n t t'", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "HarderNarasimhan.PayoffFunction.max", "Lattice", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, μ.IsConvexOn I → μ.max.A I = μ.A I", "constCategory": "Theorem"}, {"references": ["Exists", "And", "Eq"], "name": "ExistsUnique", "constType": "{α : Sort u_1} → (α → Prop) → Prop", "constCategory": "Definition"}, {"references": ["Not", "LT.lt", "Max.max", "PartialOrder.toPreorder", "Iff", "LE.le", "SemilatticeSup.toMax", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "SemilatticeSup", "Preorder.toLE"], "name": "right_lt_sup", "constType": "∀ {α : Type u} [inst : SemilatticeSup α] {a b : α}, b < a ⊔ b ↔ ¬a ≤ b", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "LT.lt", "BoundedOrder.toOrderBot", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "Top.top", "OrderBot.toBot", "Preorder.toLE", "OrderTop.toTop"], "name": "bot_lt_top", "constType": "∀ {α : Type u} [inst : PartialOrder α] [inst_1 : BoundedOrder α] [Nontrivial α], ⊥ < ⊤", "constCategory": "Theorem"}, {"references": ["Nat.find", "HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Classical.propDecidable", "Preorder.toLT", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "HarderNarasimhan.PayoffFunction.IsConvex", "Nat", "BoundedOrder.toOrderTop", "Nontrivial", "Lattice", "Top.top", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_exists_eq_top", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen", "constType": "{ℒ : Type u_1} →\n [Nontrivial ℒ] →\n [inst : Lattice ℒ] →\n [BoundedOrder ℒ] →\n [hwf : WellFoundedGT ℒ] →\n {S : Type u_2} →\n [inst_2 : CompleteLattice S] →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) → [μ.ADCC] → [μ.IsConvex] → [hadm : μ.Admissible] → ℕ", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Bot.bot", "Set.Iic", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HEq", "Top.top", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.casesOn", "Submodule.submoduleOf", "OrderHom.instFunLike", "associatedPrimes", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.noConfusionType", "constType": "Sort u →\n {R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n HarderNarasimhan.CoprimaryFiltration R M →\n {R' : Type u_1} →\n [inst' : CommRing R'] →\n [inst'_1 : IsNoetherianRing R'] →\n {M' : Type u_2} →\n [inst'_2 : Nontrivial M'] →\n [inst'_3 : AddCommGroup M'] →\n [inst'_4 : _root_.Module R' M'] →\n [inst'_5 : Module.Finite R' M'] → HarderNarasimhan.CoprimaryFiltration R' M' → Sort u", "constCategory": "Definition"}, {"references": ["Lattice", "SemilatticeSup"], "name": "Lattice.toSemilatticeSup", "constType": "{α : Type u} → [self : Lattice α] → SemilatticeSup α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "PartialOrder", "LE.le", "And", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_2", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] (x : HarderNarasimhan.StrictIntvl ℒ), x.left ≤ x.left ∧ x.right ≤ x.right", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "lt_of_le_of_lt'", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "Max.max", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "Lattice", "LE.le", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction.max", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.max_inf_le_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x w t : ℒ},\n x ∈ I →\n w ∈ I →\n ∀ (hxw : ¬x ≤ w) (hxwt : x ⊔ w ≤ t),\n μ.max { left := x ⊓ w, right := x, lt := ⋯ } ≤ μ.max { left := w, right := t, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "Preorder.toLT", "Ne", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.ne_left", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ},\n μ.IsBreakpoint I x → I.left ≠ x", "constCategory": "Theorem"}, {"references": ["Nat", "IsOrderedAddMonoid", "Nat.instAddCommMonoid", "Nat.instPreorder"], "name": "Nat.instIsOrderedAddMonoid", "constType": "IsOrderedAddMonoid ℕ", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.IsStable", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.toIsSemistable", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.IsStable], μ.IsSemistable", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSlopeLike → Sort u} →\n ((slopelike :\n ∀ (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })) →\n motive ⋯) →\n (t : μ.IsSlopeLike) → motive t", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsBreakpoint.ne_left", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Not", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mem", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.not_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (self : μ.IsBreakpoint I x)\n (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n ¬μ.A { left := I.left, right := x, lt := ⋯ } < μ.A { left := I.left, right := y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["AddLeftMono", "Preorder", "AddCommMonoid", "IsOrderedAddMonoid", "AddCommMonoid.toAddMonoid", "Preorder.toLE", "AddZeroClass.toAddZero", "AddZero.toAdd", "AddMonoid.toAddZeroClass"], "name": "IsOrderedAddMonoid.toAddLeftMono", "constType": "∀ {α : Type u_1} [inst : AddCommMonoid α] [inst_1 : Preorder α] [IsOrderedAddMonoid α], AddLeftMono α", "constCategory": "Theorem"}, {"references": [], "name": "Fintype", "constType": "Type u_4 → Type u_4", "constCategory": "Other"}, {"references": ["CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.Admissible", "Preorder", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "CompleteLinearOrder", "HarderNarasimhan.PayoffFunction"], "name": "HarderNarasimhan.PayoffFunction.instAdmissible", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLinearOrder S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S), μ.Admissible", "constCategory": "Theorem"}, {"references": [], "name": "IsWellOrder", "constType": "(α : Type u) → (α → α → Prop) → Prop", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Submodule", "Nat.instPreorder", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk._flat_ctor", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n (toFun : ℕ → Submodule R M) →\n (length : ℕ) →\n Monotone toFun →\n toFun 0 = ⊥ →\n toFun length = ⊤ →\n StrictMonoOn toFun (Set.Iic length) →\n (∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))) →\n (∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q) →\n HarderNarasimhan.CoprimaryFiltration R M", "constCategory": "Definition"}, {"references": ["Iff"], "name": "Iff.mpr", "constType": "∀ {a b : Prop}, (a ↔ b) → b → a", "constCategory": "Theorem"}, {"references": ["outParam", "HSMul"], "name": "HSMul.hSMul", "constType": "{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HSMul α β γ] → α → β → γ", "constCategory": "Definition"}, {"references": ["Preorder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC", "constType": "{ℒ : Type u_1} → {S : Type u_2} → [inst : Preorder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["Equiv.left_inv", "Equiv.invFun", "Equiv.right_inv", "Equiv.instEquivLike._proof_1", "EquivLike.mk", "Equiv.toFun", "Eq", "EquivLike", "Equiv"], "name": "Equiv.instEquivLike", "constType": "{α : Sort u} → {β : Sort v} → EquivLike (α ≃ β) α β", "constCategory": "Definition"}, {"references": ["Lattice", "GeneralizedCoheytingAlgebra"], "name": "GeneralizedCoheytingAlgebra.toLattice", "constType": "{α : Type u_4} → [self : GeneralizedCoheytingAlgebra α] → Lattice α", "constCategory": "Definition"}, {"references": ["Submodule.hasQuotient", "HasQuotient.Quotient", "IsScalarTower", "Module", "AddCommGroup.toAddGroup", "SMulZeroClass.toSMul", "Submodule", "SMul", "AddGroup.toSubNegMonoid", "DistribSMul.toSMulZeroClass", "Submodule.Quotient.module'", "Semiring.toMonoid", "DistribMulAction.toDistribSMul", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "AddZeroClass.toAddZero", "Ring.toSemiring", "Module.toDistribMulAction", "SubNegMonoid.toAddMonoid", "AddCommGroup.toAddCommMonoid", "AddZero.toZero", "AddMonoid.toAddZeroClass", "Ring", "Semiring", "Module.Finite"], "name": "Module.Finite.quotient", "constType": "∀ (R : Type u_6) {A : Type u_7} {M : Type u_8} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Ring A]\n [inst_3 : _root_.Module A M] [inst_4 : _root_.Module R M] [inst_5 : SMul R A] [inst_6 : IsScalarTower R A M]\n [Module.Finite R M] (N : Submodule A M), Module.Finite R (M ⧸ N)", "constCategory": "Theorem"}, {"references": ["PartialOrder", "SemilatticeSup"], "name": "SemilatticeSup.toPartialOrder", "constType": "{α : Type u} → [self : SemilatticeSup α] → PartialOrder α", "constCategory": "Definition"}, {"references": ["PartialOrder.mk", "PartialOrder.toPreorder", "Subtype", "PartialOrder", "Subtype.preorder", "Subtype.partialOrder._proof_1"], "name": "Subtype.partialOrder", "constType": "{α : Type u_2} → [PartialOrder α] → (p : α → Prop) → PartialOrder (Subtype p)", "constCategory": "Definition"}, {"references": ["Not", "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_3", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_2", "Preorder.toLT", "And", "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_4", "PartialOrder.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder", "LE.mk", "LE.le", "HarderNarasimhan.StrictIntvl.instPartialOrder._proof_1", "Preorder.mk", "LT.mk", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instPartialOrder", "constType": "{ℒ : Type u_1} → [inst : PartialOrder ℒ] → PartialOrder (HarderNarasimhan.StrictIntvl ℒ)", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Module", "Semiring"], "name": "Submodule", "constType": "(R : Type u) → (M : Type v) → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [_root_.Module R M] → Type v", "constCategory": "Other"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.PayoffFunction.restrict_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] (μ : HarderNarasimhan.PayoffFunction ℒ S)\n (I : HarderNarasimhan.StrictIntvl ℒ) (J : HarderNarasimhan.StrictIntvl { x // x ∈ I }),\n (μ.restrict I) J = μ (HarderNarasimhan.StrictIntvl.ofSub J)", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "AddGroup.mk", "HasQuotient.Quotient", "Submodule.Quotient.addCommGroup._proof_10", "Module", "AddCommGroup.toAddGroup", "Submodule.Quotient.addCommGroup._aux_1", "Submodule.Quotient.addCommGroup._proof_6", "Submodule", "ZSMul.toSMul", "Submodule.Quotient.addCommGroup._proof_8", "AddCommGroup.mk", "Ring.toAddGroupWithOne", "Sub.mk", "Submodule.Quotient.addCommGroup._proof_7", "AddGroup.toSubNegMonoid", "SubNegMonoid.mk", "Submodule.Quotient.addCommGroup._aux_3", "Submodule.Quotient.addCommGroup._proof_9", "Submodule.Quotient.instSMul'", "AddCommGroup", "Submodule.Quotient.addMonoid", "Ring.toSemiring", "ZSMul.ofSMul", "Int", "AddCommGroup.intIsScalarTower", "AddGroupWithOne.toAddGroup", "AddCommGroup.toAddCommMonoid", "SubNegMonoid.toZSMul", "Neg.mk", "Submodule.Quotient.addCommGroup._proof_5", "Ring"], "name": "Submodule.Quotient.addCommGroup", "constType": "{R : Type u_1} →\n {M : Type u_2} →\n [inst : Ring R] →\n [inst_1 : AddCommGroup M] → [inst_2 : _root_.Module R M] → (p : Submodule R M) → AddCommGroup (M ⧸ p)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "Submodule", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "Eq", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "Top.top", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ConditionallyCompleteLattice.toLattice", "OrderTop.toTop", "CompleteLattice.toBoundedOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_8", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.toFun\n (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.length =\n ⊤", "constCategory": "Theorem"}, {"references": ["LT.lt", "OrderDual", "LT.mk", "LT"], "name": "OrderDual.instLT", "constType": "(α : Type u_2) → [h : LT α] → LT αᵒᵈ", "constCategory": "Definition"}, {"references": ["NatCast.mk", "NeZero", "Nat", "Zero.ofOfNat0", "instOfNatNat", "NatCast", "Fin", "Fin.ofNat"], "name": "Fin.NatCast.instNatCast", "constType": "(n : ℕ) → [NeZero n] → NatCast (Fin n)", "constCategory": "Definition"}, {"references": ["Nat", "SizeOf", "SizeOf.mk"], "name": "instSizeOfNat", "constType": "SizeOf ℕ", "constCategory": "Definition"}, {"references": ["Nat", "Nat.instAddCommMonoid", "Nat.instPreorder", "IsOrderedCancelAddMonoid"], "name": "Nat.instIsOrderedCancelAddMonoid", "constType": "IsOrderedCancelAddMonoid ℕ", "constCategory": "Theorem"}, {"references": ["SMul", "HSMul", "SMul.smul", "HSMul.mk"], "name": "instHSMul", "constType": "{α : Type u_1} → {β : Type u_2} → [SMul α β] → HSMul α β β", "constCategory": "Definition"}, {"references": ["OrderDual.instPreorder", "OrderDual", "PartialOrder.mk", "PartialOrder.toPreorder", "PartialOrder", "LE.le", "Preorder.toLE", "le_antisymm"], "name": "OrderDual.instPartialOrder", "constType": "(α : Type u_2) → [PartialOrder α] → PartialOrder αᵒᵈ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun_eq_coe", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.HarderNarasimhanFiltration), F.toFun = ⇑F", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y : ℒ) (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsAffine", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.Admissible.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.PayoffFunction.Admissible.rec", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.Admissible → Sort u} →\n (t : μ.Admissible) →\n ((total_or_attained :\n (Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.ext", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n {F G : HarderNarasimhan.CoprimaryFiltration R M}, (∀ (n : ℕ), F n = G n) → F = G", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "AddCommSemigroup.mk", "AddMonoid.toAddSemigroup", "AddCommMonoid.add_comm", "AddCommMonoid.toAddMonoid", "AddCommSemigroup"], "name": "AddCommMonoid.toAddCommSemigroup", "constType": "{M : Type u} → [self : AddCommMonoid M] → AddCommSemigroup M", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.dcc", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Preorder ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.ADCC] (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f → ∃ N, ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["NNReal.instSemiring._proof_26", "Mul.mk", "NNReal.instSemiring._proof_14", "NNReal.instSemiring._proof_9", "NPow.mk", "NNReal.instSemiring._proof_15", "NNReal.instSemiring._aux_11", "NNReal.instSemiring._proof_23", "NNReal.instSemiring._aux_24", "NNReal.instSemiring._proof_5", "NNReal.instZero", "NNReal.instSemiring._aux_1", "Semigroup.mk", "NNReal.instOne", "NNReal.instSemiring._proof_18", "AddSemigroup.mk", "NNReal.instSemiring._proof_27", "NNReal.instSemiring._aux_16", "NNReal.instSemiring._proof_3", "NNReal.instSemiring._proof_20", "NNReal", "NNReal.instSemiring._proof_8", "AddCommMonoid.mk", "Semiring.mk", "NNReal.instSemiring._proof_21", "NNReal.instSemiring._proof_4", "AddMonoid.mk", "NNReal.instSemiring._proof_10", "NatCast.mk", "NSMul.mk", "NNReal.instSemiring._proof_13", "NNReal.instSemiring._proof_19", "NNReal.instSemiring._aux_6", "Add.mk", "Monoid.mk", "NNReal.instSemiring._proof_22", "Semiring"], "name": "NNReal.instSemiring", "constType": "Semiring NNReal", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.hnFiltration", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "Nontrivial", "Lattice", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.hnFiltration.congr_simp", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible], μ.hnFiltration = μ.hnFiltration", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_4", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_2", "PartialOrder.toPreorder", "Subtype", "Bot.mk", "Subtype.instLE", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "BoundedOrder", "BoundedOrder.mk", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_1", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_3", "PartialOrder", "Top.mk", "Subtype.mk", "Preorder.toLE", "OrderBot.mk", "OrderTop.mk"], "name": "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "constType": "{ℒ : Type u_1} → [inst : PartialOrder ℒ] → {I : HarderNarasimhan.StrictIntvl ℒ} → BoundedOrder { x // x ∈ I }", "constCategory": "Definition"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.right", "LT"], "name": "HarderNarasimhan.StrictIntvl.lt", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] (self : HarderNarasimhan.StrictIntvl ℒ), self.left < self.right", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mem", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "LE.le", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsBreakpoint.ne_left", "HarderNarasimhan.PayoffFunction.A", "Ne", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.le_of_eq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (self : μ.IsBreakpoint I x)\n (y : ℒ) (hyI : y ∈ I) (hy : I.left ≠ y),\n μ.A { left := I.left, right := y, lt := ⋯ } = μ.A { left := I.left, right := x, lt := ⋯ } → y ≤ x", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "Submodule.Quotient.module'", "HasQuotient.Quotient", "Module", "Submodule.Quotient.instSMul._proof_1", "Distrib.toMul", "instDistribOfSemiring", "Submodule.Quotient.addCommMonoid", "instSMulOfMul", "AddCommGroup", "Ring.toSemiring", "Submodule", "AddCommGroup.toAddCommMonoid", "Ring"], "name": "Submodule.Quotient.module", "constType": "{R : Type u_1} →\n {M : Type u_2} →\n [inst : Ring R] →\n [inst_1 : AddCommGroup M] → [inst_2 : _root_.Module R M] → (P : Submodule R M) → _root_.Module R (M ⧸ P)", "constCategory": "Definition"}, {"references": ["AddGroup", "AddCommGroup"], "name": "AddCommGroup.toAddGroup", "constType": "{G : Type u} → [self : AddCommGroup G] → AddGroup G", "constCategory": "Definition"}, {"references": ["Unique", "Eq", "Inhabited", "Inhabited.default"], "name": "Unique.mk", "constType": "{α : Sort u} → (toInhabited : Inhabited α) → (∀ (a : α), a = default) → Unique α", "constCategory": "Other"}, {"references": ["NNReal.instLinearOrder._proof_10", "SemilatticeInf.toMin", "NNReal.instLinearOrder._aux_1", "NNReal", "LinearOrder", "NNReal.instLinearOrder._proof_11", "NNReal.instPartialOrder", "LinearOrder.mk", "NNReal.instLinearOrder._aux_4", "NNReal.instSemilatticeSup", "NNReal.instSemilatticeInf", "NNReal.instLinearOrder._proof_3", "NNReal.instLinearOrder._proof_12", "Ord.mk", "SemilatticeSup.toMax", "NNReal.instLinearOrder._aux_6", "NNReal.instLinearOrder._aux_8"], "name": "NNReal.instLinearOrder", "constType": "LinearOrder NNReal", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.semistableRel", "instAddNat", "RelSeries.toFun", "PartialOrder.toPreorder", "Fin.instPartialOrder", "instHAdd", "Preorder.toLT", "RelSeries", "Fin", "OfNat.ofNat", "HAdd.hAdd", "Nat", "instOfNatNat", "RelSeries.length", "PartialOrder", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.relSeries_strictMono", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (s : RelSeries μ.semistableRel), StrictMono s.toFun", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "SizeOf", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration._sizeOf_inst", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instSizeOfNat", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "SizeOf.sizeOf", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "instSizeOfDefault", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk.sizeOf_spec", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_3 : SizeOf ℒ] [inst_4 : SizeOf S] (toFun : ℕ → ℒ) (length : ℕ)\n (monotone : Monotone toFun) (head_eq_bot : toFun 0 = ⊥) (length_eq_top : toFun length = ⊤)\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length))\n (piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length), (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable)\n (not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }),\n sizeOf\n { toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot,\n length_eq_top := length_eq_top, strictMonoOn := strictMonoOn, piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ } =\n 1 + sizeOf length + sizeOf head_eq_bot + sizeOf length_eq_top", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HEq.refl", "Eq.refl", "id", "HEq", "heq_of_eq", "HarderNarasimhan.PayoffFunction", "Eq", "HarderNarasimhan.PayoffFunction.noConfusion", "LT"], "name": "HarderNarasimhan.PayoffFunction.mk.noConfusion", "constType": "{ℒ : Type u_1} →\n {inst : LT ℒ} →\n {S : Type u_2} →\n {P : Sort u} →\n {toFun toFun' : HarderNarasimhan.StrictIntvl ℒ → S} →\n { toFun := toFun } = { toFun := toFun' } → (toFun ≍ toFun' → P) → P", "constCategory": "Definition"}, {"references": ["InfSet", "ConditionallyCompletePartialOrderInf"], "name": "ConditionallyCompletePartialOrderInf.toInfSet", "constType": "{α : Type u_3} → [self : ConditionallyCompletePartialOrderInf α] → InfSet α", "constCategory": "Definition"}, {"references": [], "name": "Exists", "constType": "{α : Sort u} → (α → Prop) → Prop", "constCategory": "Other"}, {"references": ["Semiring.toMonoid", "Module", "OreLocalization.oreSetComm", "CommSemiring", "CommSemiring.toSemiring", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "CommSemiring.toCommMonoid", "Submonoid", "Module.toDistribMulAction", "AddCommMonoid", "OreLocalization", "instMulZeroOneClassOfSemiring", "DistribMulAction.toMulAction"], "name": "LocalizedModule", "constType": "{R : Type u} →\n [inst : CommSemiring R] →\n Submonoid R → (M : Type v) → [inst_1 : AddCommMonoid M] → [_root_.Module R M] → Type (max u v)", "constCategory": "Definition"}, {"references": ["Submodule.Quotient.addCommMonoid._proof_1", "AddCommMonoid", "Submodule.hasQuotient", "HasQuotient.Quotient", "Module", "AddCommMonoid.mk", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "Submodule.Quotient.addMonoid", "Submodule", "Ring.toSemiring", "Ring"], "name": "Submodule.Quotient.addCommMonoid", "constType": "{R : Type u_1} →\n {M : Type u_2} →\n [inst : Ring R] →\n [inst_1 : AddCommGroup M] → [inst_2 : _root_.Module R M] → (p : Submodule R M) → AddCommMonoid (M ⧸ p)", "constCategory": "Definition"}, {"references": ["LE"], "name": "BoundedOrder", "constType": "(α : Type u) → [LE α] → Type u", "constCategory": "Other"}, {"references": ["Preorder", "Set.ofPred", "Set", "LE.le", "Preorder.toLE"], "name": "Set.Iic", "constType": "{α : Type u_1} → [Preorder α] → α → Set α", "constCategory": "Definition"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "HEq", "HarderNarasimhan.StrictIntvl.casesOn", "LT"], "name": "HarderNarasimhan.StrictIntvl.noConfusionType", "constType": "Sort u →\n {ℒ : Type u_1} →\n [inst : LT ℒ] →\n HarderNarasimhan.StrictIntvl ℒ → {ℒ' : Type u_1} → [inst' : LT ℒ'] → HarderNarasimhan.StrictIntvl ℒ' → Sort u", "constCategory": "Definition"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "OfNat.ofNat", "LT.lt", "Nat", "instOfNatNat", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length_pos", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration), 0 < F.length", "constCategory": "Theorem"}, {"references": ["Preorder", "LE.le", "Preorder.toLE"], "name": "Antitone", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Prop", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Type u_1", "constCategory": "Other"}, {"references": ["Finset", "PartialOrder.toPreorder", "Preorder.toLT", "Finset.Colex.instDecidableLT", "compareOfLessAndEq", "Min.mk", "instDistribLatticeOfLinearOrder", "Finset.Colex.instLinearOrder._proof_7", "Ord.mk", "Colex", "Preorder.toLE", "Finset.Colex.instLinearOrder._proof_4", "SemilatticeInf.toPartialOrder", "Finset.Colex.instLinearOrder._proof_6", "Lattice.toSemilatticeInf", "ite", "LinearOrder", "LinearOrder.mk", "decidableEqOfDecidableLE", "Finset.Colex.instLinearOrder._proof_5", "Max.mk", "LinearOrder.toDecidableEq", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LinearOrder.toDecidableLE", "LE.le", "Finset.Colex.instDecidableLE"], "name": "Finset.Colex.instLinearOrder", "constType": "{α : Type u_1} → [LinearOrder α] → LinearOrder (Colex (Finset α))", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Iff", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "OrderTop.toTop", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.isConvexOn_top_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_2 : Nontrivial ℒ] [inst_3 : BoundedOrder ℒ],\n μ.IsConvexOn ⊤ ↔ μ.IsConvex", "constCategory": "Theorem"}, {"references": ["outParam", "HasQuotient"], "name": "HasQuotient.Quotient", "constType": "(A : outParam (Type u)) → {B : Type v} → [self : HasQuotient A B] → B → Type (max u v)", "constCategory": "Definition"}, {"references": ["ConditionallyCompleteLattice.isGLB_csInf", "ConditionallyCompletePartialOrderSup.mk", "ConditionallyCompleteLattice.toInfSet", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Set", "GE.ge", "ConditionallyCompletePartialOrder", "DirectedOn", "ConditionallyCompleteLattice", "LE.le", "ConditionallyCompleteLattice.toLattice", "ConditionallyCompleteLattice.toSupSet", "Preorder.toLE", "ConditionallyCompleteLattice.isLUB_csSup", "ConditionallyCompletePartialOrder.mk", "SemilatticeInf.toPartialOrder"], "name": "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "constType": "{α : Type u_1} → [ConditionallyCompleteLattice α] → ConditionallyCompletePartialOrder α", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsAffine", "constType": "{ℒ : Type u_1} → {S : Type u_2} → [inst : Lattice ℒ] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["MulOneClass", "MulZeroOneClass"], "name": "MulZeroOneClass.toMulOneClass", "constType": "{M₀ : Type u} → [self : MulZeroOneClass M₀] → MulOneClass M₀", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "StrictMonoOn", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.rec", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "SizeOf", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "instSizeOfNat", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "SizeOf.sizeOf", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "instSizeOfDefault", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration._sizeOf_1", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : PartialOrder ℒ} →\n {inst_1 : BoundedOrder ℒ} →\n {inst_2 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} → [SizeOf ℒ] → [SizeOf S] → μ.HarderNarasimhanFiltration → ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.restrict", "Subtype.partialOrder", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.instIsSlopeLikeSubtypeMemStrictIntvlRestrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} [hsl : μ.IsSlopeLike],\n (μ.restrict I).IsSlopeLike", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "PartialOrder", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "instLENat", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.eq_top_of_length_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.HarderNarasimhanFiltration} {m : ℕ}, F.length ≤ m → F m = ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) → μ.IsSemistable", "constCategory": "Other"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "And", "HarderNarasimhan.StrictIntvl", "Max.max", "HarderNarasimhan.StrictIntvl.left", "SemilatticeSup.toMax", "Lattice", "LE.le", "Preorder.toLE", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} (x x_1 : ℒ),\n x ∈ I → x_1 ∈ I → I.left ≤ x ⊔ x_1 ∧ x ⊔ x_1 ≤ I.right", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.eq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.HasNashEquilibrium], μ.A ⊤ = μ.B ⊤", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "LE.le.trans_lt", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b < c → a < c", "constCategory": "Theorem"}, {"references": ["Preorder", "PartialOrder", "LE.le", "Eq", "Preorder.toLE"], "name": "PartialOrder.mk", "constType": "{α : Type u_2} → [toPreorder : Preorder α] → (∀ (a b : α), a ≤ b → b ≤ a → a = b) → PartialOrder α", "constCategory": "Other"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "LT.lt", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "BoundedOrder.toOrderTop", "PartialOrder", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.ne_top_of_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.HarderNarasimhanFiltration} {m : ℕ}, m < F.length → F m ≠ ⊤", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "Subtype.instLE", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderTop", "PartialOrder", "Top.top", "Eq", "Preorder.toLE", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.val_top", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, ↑⊤ = I.right", "constCategory": "Theorem"}, {"references": ["ConditionallyCompleteLinearOrderBot", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot._proof_3", "CompleteLinearOrder.toDecidableLE", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot._proof_1", "Lattice.toSemilatticeSup", "CompleteLinearOrder.toDecidableEq", "PartialOrder.toPreorder", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot._proof_2", "ConditionallyCompleteLinearOrder.mk", "CompleteLattice.toLattice", "SemilatticeSup.toPartialOrder", "CompleteLinearOrder.toDecidableLT", "CompleteLinearOrder.compare_eq_compareOfLessAndEq", "ConditionallyCompleteLinearOrderBot.mk", "CompleteLinearOrder.le_total", "BoundedOrder.toOrderBot", "ConditionallyCompleteLattice", "CompleteLinearOrder.toCompleteLattice", "CompleteLinearOrder", "Preorder.toLE", "CompleteLinearOrder.toOrd", "CompleteLattice.toBoundedOrder", "CompleteLattice.toConditionallyCompleteLattice"], "name": "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "constType": "{α : Type u_5} → [h : CompleteLinearOrder α] → ConditionallyCompleteLinearOrderBot α", "constCategory": "Definition"}, {"references": ["Semiring.toNonAssocSemiring", "MulZeroOneClass", "inferInstance", "NonAssocSemiring.toMulZeroOneClass", "Semiring"], "name": "instMulZeroOneClassOfSemiring", "constType": "{α : Type u} → [Semiring α] → MulZeroOneClass α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Subtype.preorder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_restrict_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}\n {J : HarderNarasimhan.StrictIntvl { x // x ∈ I }}, (μ.restrict I).B J = μ.B (HarderNarasimhan.StrictIntvl.ofSub J)", "constCategory": "Theorem"}, {"references": [], "name": "Eq", "constType": "{α : Sort u_1} → α → α → Prop", "constCategory": "Other"}, {"references": ["ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "iInf", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf", "ConditionallyCompletePartialOrderInf.toInfSet", "Eq", "Set.Ico", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "Set.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.PayoffFunction.max", "LE.le", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) (I : HarderNarasimhan.StrictIntvl ℒ),\n μ.A I = ⨅ a, ⨅ (ha : a ∈ Set.Ico I.left I.right), μ.max { left := a, right := I.right, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Fin.instLinearOrder", "instDistribLatticeOfLinearOrder", "Nat", "DistribLattice.toLattice", "Lattice.toSemilatticeInf", "PartialOrder", "Fin", "inferInstance", "SemilatticeInf.toPartialOrder"], "name": "Fin.instPartialOrder", "constType": "{n : ℕ} → PartialOrder (Fin n)", "constCategory": "Definition"}, {"references": ["OrderDual", "OrderDual.toDual", "HarderNarasimhan.StrictIntvl.lt", "Equiv.instEquivLike", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.PayoffFunction.dual", "DFunLike.coe", "Equiv", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "OrderDual.instLT", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction", "Eq", "LT"], "name": "HarderNarasimhan.PayoffFunction.dual_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : LT ℒ] (μ : HarderNarasimhan.PayoffFunction ℒ S)\n (p : HarderNarasimhan.StrictIntvl ℒᵒᵈ), μ.dual p = OrderDual.toDual (μ { left := p.right, right := p.left, lt := ⋯ })", "constCategory": "Theorem"}, {"references": ["LE"], "name": "OrderTop", "constType": "(α : Type u) → [LE α] → Type u", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n ((Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I) → μ.Admissible", "constCategory": "Other"}, {"references": ["Add", "AddZero"], "name": "AddZero.toAdd", "constType": "{M : Type u_2} → [self : AddZero M] → Add M", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "HarderNarasimhan.PayoffFunction.IsAffine.rec", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsAffine.mk", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsAffine → Sort u} →\n (t : μ.IsAffine) →\n ((eq :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Nat", "CommRing.toCommSemiring", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Module", "Nontrivial", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing", "Submodule", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.toFun", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] → HarderNarasimhan.CoprimaryFiltration R M → ℕ → Submodule R M", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Semiring.toMonoid", "Module", "AddCommMonoid.toAddMonoid", "DistribMulAction", "Semiring"], "name": "Module.toDistribMulAction", "constType": "{R : Type u} →\n {M : Type v} → {inst : Semiring R} → {inst_1 : AddCommMonoid M} → [self : _root_.Module R M] → DistribMulAction R M", "constCategory": "Definition"}, {"references": ["_private.Mathlib.Data.Real.Basic.0.Real.zero", "Real", "Zero.mk", "Zero"], "name": "Real.instZero", "constType": "Zero ℝ", "constCategory": "Definition"}, {"references": ["outParam"], "name": "CoeSort", "constType": "Sort u → outParam (Sort v) → Sort (max (max 1 u) v)", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Preorder.toLT", "bot_le", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "LE.le.trans_lt", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "WellFoundedGT", "OrderBot.toBot", "Preorder.toLE", "Eq", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.hnFiltration", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Lattice", "Nontrivial", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.hnFiltration_A_bot_eq_A", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible] {n : ℕ} {y : ℒ} (hy : μ.hnFiltration n < y),\n μ.A { left := ⊥, right := y, lt := ⋯ } = μ.A { left := μ.hnFiltration n, right := y, lt := hy }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "Top.top", "Preorder.toLE", "Eq", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.right_top", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ], ⊤.right = ⊤", "constCategory": "Theorem"}, {"references": ["Not", "Eq"], "name": "Ne", "constType": "{α : Sort u} → α → α → Prop", "constCategory": "Definition"}, {"references": ["AddLeftMono", "PartialOrder.toPreorder", "IsLeftCancelAdd", "Add", "AddLeftStrictMono", "PartialOrder", "Preorder.toLT", "Preorder.toLE"], "name": "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "constType": "∀ (N : Type u_2) [inst : Add N] [IsLeftCancelAdd N] [inst_2 : PartialOrder N] [AddLeftMono N], AddLeftStrictMono N", "constCategory": "Theorem"}, {"references": ["Eq"], "name": "Function.Injective", "constType": "{α : Sort u_1} → {β : Sort u_2} → (α → β) → Prop", "constCategory": "Definition"}, {"references": ["instLinearOrderLinearExtensionOfPartialOrder._proof_4", "instLinearOrderLinearExtensionOfPartialOrder._proof_9", "PartialOrder.toPreorder", "Pi.hasLe", "instLinearOrderLinearExtensionOfPartialOrder._proof_7", "compareOfLessAndEq", "Min.mk", "PartialOrder.mk", "decidableLTOfDecidableLE", "IsLinearOrder", "Ord.mk", "PartialOrder", "Prop.le", "Classical.decRel", "LT.mk", "Preorder.toLE", "instLinearOrderLinearExtensionOfPartialOrder._proof_1", "Not", "LinearExtension", "instLinearOrderLinearExtensionOfPartialOrder._proof_8", "ite", "LinearOrder", "And", "decidableEqOfDecidableLE", "LinearOrder.mk", "Exists.choose", "Max.mk", "instLinearOrderLinearExtensionOfPartialOrder._proof_3", "instLinearOrderLinearExtensionOfPartialOrder._proof_5", "instLinearOrderLinearExtensionOfPartialOrder._proof_2", "LE.le", "LE.mk", "instLinearOrderLinearExtensionOfPartialOrder._proof_6", "Preorder.mk"], "name": "instLinearOrderLinearExtensionOfPartialOrder", "constType": "{α : Type u} → [PartialOrder α] → LinearOrder (LinearExtension α)", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n (∀ (x y : ℒ),\n x ∈ I →\n y ∈ I → ∀ (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsConvexOn I", "constCategory": "Other"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder", "HarderNarasimhan.StrictIntvl.ofSub", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.StrictIntvl.ofSub_left", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}\n (J : HarderNarasimhan.StrictIntvl { x // x ∈ I }), (HarderNarasimhan.StrictIntvl.ofSub J).left = ↑J.left", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE"], "name": "HarderNarasimhan.PayoffFunction.restrict", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n HarderNarasimhan.PayoffFunction ℒ S →\n (I : HarderNarasimhan.StrictIntvl ℒ) → HarderNarasimhan.PayoffFunction { x // x ∈ I } S", "constCategory": "Definition"}, {"references": ["Nat"], "name": "Fin", "constType": "ℕ → Type", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "heq_of_eq", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Eq.refl", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "id", "HEq", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "HEq.refl", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.noConfusion", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk.noConfusion", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : Nontrivial ℒ} →\n {inst_1 : PartialOrder ℒ} →\n {inst_2 : BoundedOrder ℒ} →\n {inst_3 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {P : Sort u} →\n {toFun : ℕ → ℒ} →\n {length : ℕ} →\n {antitone : Antitone toFun} →\n {head_eq_top : toFun 0 = ⊤} →\n {length_eq_bot : toFun length = ⊥} →\n {strictAntiOn : StrictAntiOn toFun (Set.Iic length)} →\n {step_payoff_eq :\n ∀ (i : ℕ) (hi : i < length),\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤} →\n {payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }} →\n {toFun' : ℕ → ℒ} →\n {length' : ℕ} →\n {antitone' : Antitone toFun'} →\n {head_eq_top' : toFun' 0 = ⊤} →\n {length_eq_bot' : toFun' length' = ⊥} →\n {strictAntiOn' : StrictAntiOn toFun' (Set.Iic length')} →\n {step_payoff_eq' :\n ∀ (i : ℕ) (hi : i < length'),\n μ { left := toFun' (i + 1), right := toFun' i, lt := ⋯ } = μ ⊤} →\n {payoff_lt_of_between' :\n ∀ (i : ℕ) (hi : i < length') (z : ℒ) (h' : toFun' (i + 1) < z),\n z < toFun' i →\n μ { left := toFun' (i + 1), right := z, lt := h' } <\n μ { left := toFun' (i + 1), right := toFun' i, lt := ⋯ }} →\n { toFun := toFun, length := length, antitone := antitone,\n head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq,\n payoff_lt_of_between := payoff_lt_of_between } =\n { toFun := toFun', length := length', antitone := antitone',\n head_eq_top := head_eq_top', length_eq_bot := length_eq_bot',\n strictAntiOn := strictAntiOn', step_payoff_eq := step_payoff_eq',\n payoff_lt_of_between := payoff_lt_of_between' } →\n (toFun ≍ toFun' → length = length' → P) → P", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "PartialOrder.toPreorder", "LT.lt.trans_le", "LinearExtension", "Set", "Module", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Preorder.toLT", "AddCommGroup", "CommRing", "Submodule", "HarderNarasimhan.StrictIntvl.mk", "PrimeSpectrum", "LT.lt", "LE.le", "AddCommGroup.toAddCommMonoid", "Preorder.toLE", "Set.instLE"], "name": "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_mono_right", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n {N₁ u N₃ : Submodule R M} (h₁ : N₁ < u) (h₂ : u ≤ N₃),\n HarderNarasimhan.Coprimary.subquotientAssociatedPrimes { left := N₁, right := u, lt := h₁ } ⊆\n HarderNarasimhan.Coprimary.subquotientAssociatedPrimes { left := N₁, right := N₃, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Nat", "AddCommMonoid", "inferInstance", "Nat.instAddCancelCommMonoid", "AddCancelCommMonoid.toAddCommMonoid"], "name": "Nat.instAddCommMonoid", "constType": "AddCommMonoid ℕ", "constCategory": "Definition"}, {"references": ["Subtype", "SetLike.instMembership", "Module", "LinearMap.mk", "Submodule.module", "AddHom.mk", "Membership.mem", "LinearMap", "Subtype.val", "Submodule.subtype._proof_1", "Submodule", "Submodule.subtype._proof_2", "AddCommMonoid", "Semiring.toNonAssocSemiring", "AddCommMonoid.toAddCommSemigroup", "RingHom.id", "Submodule.setLike", "AddCommSemigroup.toAddCommMagma", "AddCommMagma.toAdd", "Submodule.addCommMonoid", "Semiring"], "name": "Submodule.subtype", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : _root_.Module R M} → (p : Submodule R M) → ↥p →ₗ[R] M", "constCategory": "Definition"}, {"references": ["Real.instZero", "Real", "Zero.mk", "NNReal", "NNReal.mk", "Zero.toOfNat0", "Zero", "NNReal.instZero._proof_1", "OfNat.ofNat"], "name": "NNReal.instZero", "constType": "Zero NNReal", "constCategory": "Definition"}, {"references": ["Preorder", "LE.le", "Preorder.toLE"], "name": "Monotone", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Prop", "constCategory": "Definition"}, {"references": ["MulAction", "Monoid", "AddMonoid", "DistribMulAction"], "name": "DistribMulAction.toMulAction", "constType": "{M : Type u_12} →\n {A : Type u_13} → {inst : Monoid M} → {inst_1 : AddMonoid A} → [self : DistribMulAction M A] → MulAction M A", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.right", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.ext", "constType": "∀ {ℒ : Type u_1} {inst : LT ℒ} {x y : HarderNarasimhan.StrictIntvl ℒ}, x.left = y.left → x.right = y.right → x = y", "constCategory": "Theorem"}, {"references": ["Iff", "Membership.mem", "List.instMembership", "List"], "name": "List.TFAE", "constType": "List Prop → Prop", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Subtype.instLE", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "lt_of_le_of_ne", "Set.Elem", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction.breakpoints", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "And.left", "Lattice.toSemilatticeInf", "Set", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "LE.le", "Lattice", "Ne", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.breakpoints_total", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ((Std.Total fun x1 x2 => x1 ≤ x2) ∨\n ∀ (z : ℒ) (hzI : z ∈ I) (hz : I.left ≠ z), μ.IsAttained { left := I.left, right := z, lt := ⋯ }) →\n Std.Total fun x1 x2 => x1 ≤ x2", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.semistableRel", "instAddNat", "instLTNat", "Nat.cast", "RelSeries.toFun", "PartialOrder.toPreorder", "instHAdd", "Preorder.toLT", "Fin", "RelSeries", "instNeZeroNatHAdd_1", "OfNat.ofNat", "Nat.instNeZeroSucc", "LT.lt", "HAdd.hAdd", "Nat", "Fin.NatCast.instNatCast", "instOfNatNat", "RelSeries.length", "PartialOrder", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.relSeries_step_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (s : RelSeries μ.semistableRel) {i : ℕ},\n i + 1 < s.length → s.toFun ↑i < s.toFun ↑(i + 1)", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "Set", "HarderNarasimhan.StrictIntvl.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "Preorder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "iSup", "Set.Ioc", "ConditionallyCompletePartialOrderSup.toSupSet", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.max._proof_1", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.PayoffFunction ℒ S", "constCategory": "Definition"}, {"references": ["LT.lt", "Subtype", "LT.mk", "Subtype.val", "LT"], "name": "Subtype.instLT", "constType": "{α : Type u} → [LT α] → {P : α → Prop} → LT (Subtype P)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.mem", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ}, μ.IsBreakpoint I x → x ∈ I", "constCategory": "Theorem"}, {"references": ["And"], "name": "And.right", "constType": "∀ {a b : Prop}, a ∧ b → b", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "lt_of_le_of_lt", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b < c → a < c", "constCategory": "Theorem"}, {"references": ["Prod", "Set"], "name": "SetRel", "constType": "Type u_6 → Type u_7 → Type (max u_7 u_6)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.partialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Top.top", "HarderNarasimhan.StrictIntvl.ofSub", "Eq", "Preorder.toLE", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.ofSub_top", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, HarderNarasimhan.StrictIntvl.ofSub ⊤ = I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "FunLike", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.mk", "Nat", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat._proof_1", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} → FunLike μ.HarderNarasimhanFiltration ℕ ℒ", "constCategory": "Definition"}, {"references": ["IsNoetherian", "AddCommMonoid", "SetLike.instMembership", "Subtype", "Submodule.setLike", "Module", "Submodule.module", "Membership.mem", "Submodule.addCommMonoid", "Submodule", "Semiring"], "name": "isNoetherian_submodule'", "constType": "∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : _root_.Module R M]\n [IsNoetherian R M] (N : Submodule R M), IsNoetherian R ↥N", "constCategory": "Theorem"}, {"references": ["Fin.last", "RelSeries.toFun", "RelSeries.length", "RelSeries", "SetRel"], "name": "RelSeries.last", "constType": "{α : Type u_1} → {r : SetRel α α} → RelSeries r → α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "HarderNarasimhan.PayoffFunction.IsConvex", "BoundedOrder.toOrderTop", "Nontrivial", "Lattice", "Top.top", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_length_eq_top", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible],\n HarderNarasimhan.PayoffFunction.HNFil✝ μ (HarderNarasimhan.PayoffFunction.HNlen✝ μ) = ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsAffine", "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem", "HarderNarasimhan.StrictIntvl", "Lattice", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instIsAffineSubtypeMemStrictIntvlRestrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] {μ : HarderNarasimhan.PayoffFunction ℒ S}\n {I : HarderNarasimhan.StrictIntvl ℒ} [haff : μ.IsAffine], (μ.restrict I).IsAffine", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.WeakACC.rec", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.PayoffFunction.WeakACC.mk", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakACC → Sort u} →\n (t : μ.WeakACC) →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "LT.lt", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "Iff", "Nontrivial", "Ne", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.ne_bot_iff_lt_length", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n F m ≠ ⊥ ↔ m < F.length", "constCategory": "Theorem"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.StrictIntvl.ofSub", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.StrictIntvl.ofSub_right", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}\n (J : HarderNarasimhan.StrictIntvl { x // x ∈ I }), (HarderNarasimhan.StrictIntvl.ofSub J).right = ↑J.right", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.PayoffFunction.IsSemistable", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "PartialOrder", "Iff", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.isBreakpoint_right_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsBreakpoint I I.right ↔ (μ.restrict I).IsSemistable", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "And", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Submodule", "Nat.instPreorder", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "HarderNarasimhan.CoprimaryFiltration", "instOfNatNat", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.mk.injEq", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M] (toFun : ℕ → Submodule R M)\n (length : ℕ) (monotone : Monotone toFun) (head_eq_bot : toFun 0 = ⊥) (length_eq_top : toFun length = ⊤)\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length))\n (piecewise_isCoprimary :\n ∀ i < length, HarderNarasimhan.IsCoprimary R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))))\n (associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q)\n (toFun_1 : ℕ → Submodule R M) (length_1 : ℕ) (monotone_1 : Monotone toFun_1) (head_eq_bot_1 : toFun_1 0 = ⊥)\n (length_eq_top_1 : toFun_1 length_1 = ⊤) (strictMonoOn_1 : StrictMonoOn toFun_1 (Set.Iic length_1))\n (piecewise_isCoprimary_1 :\n ∀ i < length_1, HarderNarasimhan.IsCoprimary R (↥(toFun_1 (i + 1)) ⧸ (toFun_1 i).submoduleOf (toFun_1 (i + 1))))\n (associatedPrime_succ_lt_1 :\n ∀ (i : ℕ),\n i + 1 < length_1 →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(toFun_1 (i + 2)) ⧸ (toFun_1 (i + 1)).submoduleOf (toFun_1 (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(toFun_1 (i + 1)) ⧸ (toFun_1 i).submoduleOf (toFun_1 (i + 1))) →\n toLinearExtension p < toLinearExtension q),\n ({ toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt } =\n { toFun := toFun_1, length := length_1, monotone := monotone_1, head_eq_bot := head_eq_bot_1,\n length_eq_top := length_eq_top_1, strictMonoOn := strictMonoOn_1,\n piecewise_isCoprimary := piecewise_isCoprimary_1, associatedPrime_succ_lt := associatedPrime_succ_lt_1 }) =\n (toFun = toFun_1 ∧ length = length_1)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Set.Ico", "CompleteLattice.toConditionallyCompleteLattice", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "LE.le", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {a : ℒ}\n (ha : a ∈ Set.Ico I.left I.right), μ.A I ≤ μ.max { left := a, right := I.right, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.strictAntiOn", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Nat.instIsOrderedAddMonoid", "Preorder.toLE", "Eq", "LT.lt.le", "instLTNat", "Nat.instAddMonoid", "Nat.instPartialOrder", "instHAdd", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "Nat", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.step_payoff_eq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration) (i : ℕ)\n (hi : i < self.length), μ { left := self.toFun (i + 1), right := self.toFun i, lt := ⋯ } = μ ⊤", "constCategory": "Theorem"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "Preorder.toLT", "DFunLike.coe", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_left_eq_right_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := y, lt := h₁ } = μ { left := y, right := z, lt := h₂ } ↔\n μ { left := x, right := y, lt := h₁ } = μ { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.lattice", "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem._proof_1", "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem._proof_2", "HarderNarasimhan.StrictIntvl", "Lattice", "Preorder.toLE", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem", "constType": "{ℒ : Type u_1} → [inst : Lattice ℒ] → {I : HarderNarasimhan.StrictIntvl ℒ} → Lattice { x // x ∈ I }", "constCategory": "Definition"}, {"references": ["LinearOrder", "instDistribLatticeOfLinearOrder._proof_4", "DistribLattice.mk", "DistribLattice", "LinearOrder.toLattice"], "name": "instDistribLatticeOfLinearOrder", "constType": "{α : Type u} → [LinearOrder α] → DistribLattice α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.Admissible.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.Admissible → Sort u} →\n ((total_or_attained :\n (Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I) →\n motive ⋯) →\n (t : μ.Admissible) → motive t", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Subtype", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "HarderNarasimhan.PayoffFunction.IsBreakpoint.left_lt", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Lattice", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.isSemistable_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (hx : μ.IsBreakpoint I x),\n (μ.restrict { left := I.left, right := x, lt := ⋯ }).IsSemistable", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.Admissible.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.PayoffFunction.Admissible.rec", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.Admissible → Sort u} →\n (t : μ.Admissible) →\n ((total_or_attained :\n (Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.HarderNarasimhanFiltration → ℕ", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.ne_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.FiniteTotalPayoff], μ ⊤ ≠ ⊤", "constCategory": "Theorem"}, {"references": ["LE", "OrderTop", "BoundedOrder"], "name": "BoundedOrder.toOrderTop", "constType": "{α : Type u} → {inst : LE α} → [self : BoundedOrder α] → OrderTop α", "constCategory": "Definition"}, {"references": ["Inv"], "name": "Inv.inv", "constType": "{α : Type u} → [self : Inv α] → α → α", "constCategory": "Definition"}, {"references": ["LE.le", "Top.top", "LE", "OrderTop", "OrderTop.toTop"], "name": "le_top", "constType": "∀ {α : Type u} [inst : LE α] [inst_1 : OrderTop α] {a : α}, a ≤ ⊤", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "Nontrivial", "Lattice", "CompleteLinearOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.hasNashEquilibrium", "constType": "∀ {ℒ : Type u_3} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] {S : Type u_4}\n [inst_3 : CompleteLinearOrder S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSemistable → ∀ [μ.WeakACC] [μ.WeakSlopeLikeAtTop], μ.HasNashEquilibrium", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.right", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → HarderNarasimhan.StrictIntvl ℒ → ℒ", "constCategory": "Definition"}, {"references": ["LT"], "name": "LT.lt", "constType": "{α : Type u} → [self : LT α] → α → α → Prop", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} → [inst : PartialOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.ctorIdx", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : PartialOrder ℒ} →\n {inst_1 : BoundedOrder ℒ} →\n {inst_2 : CompleteLattice S} → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.HarderNarasimhanFiltration → ℕ", "constCategory": "Definition"}, {"references": ["CompletelyDistribLattice.toCompleteLattice", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "PartialOrder", "Preorder.toLT", "BoundedOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "CompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop"], "name": "HarderNarasimhan.PayoffFunction.instWeakSlopeLikeAtTopOfIsSlopeLike", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] {S : Type u_3} [inst_2 : CompleteLinearOrder S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike], μ.WeakSlopeLikeAtTop", "constCategory": "Theorem"}, {"references": ["Set.Ici", "Preorder", "LE.le", "DedekindCut", "DedekindCut.principal._proof_3", "DedekindCut.principal._proof_2", "Preorder.toLE", "Set.Iic", "Concept.ofObject", "Concept.copy"], "name": "DedekindCut.principal", "constType": "{α : Type u_1} → [inst : Preorder α] → α → DedekindCut α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsStable", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "HarderNarasimhan.PayoffFunction.IsStable.mk", "CompleteLattice", "HarderNarasimhan.PayoffFunction.IsStable.rec"], "name": "HarderNarasimhan.PayoffFunction.IsStable.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsStable → Sort u} →\n (t : μ.IsStable) →\n ([toIsSemistable : μ.IsSemistable] →\n (ne : ∀ (x : ℒ) (hx : ⊥ < x), x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["default.sizeOf", "SizeOf", "SizeOf.mk"], "name": "instSizeOfDefault", "constType": "(α : Sort u) → SizeOf α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.IsConvex", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "BoundedOrder.toOrderTop", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLinearOrder", "OrderBot.toBot", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "CompletelyDistribLattice.toCompleteLattice", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Lattice", "Top.top", "HarderNarasimhan.PayoffFunction.A", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.A_right_eq_of_A_left_gt", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {S : Type u_3} [inst_1 : CompleteLinearOrder S] [inst_2 : Nontrivial ℒ]\n [inst_3 : BoundedOrder ℒ] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsConvex →\n ∀ {x : ℒ} (h₁ : ⊥ < x) (h₂ : x < ⊤),\n μ.A ⊤ < μ.A { left := ⊥, right := x, lt := h₁ } → μ.A { left := x, right := ⊤, lt := h₂ } = μ.A ⊤", "constCategory": "Theorem"}, {"references": [], "name": "LT", "constType": "Type u → Type u", "constCategory": "Other"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.HarderNarasimhanFiltration → ℕ → ℒ", "constCategory": "Definition"}, {"references": ["SubtractionMonoid", "SubtractionMonoid.toSubNegMonoid", "SubNegZeroMonoid.mk", "SubNegMonoid", "SubtractionMonoid.toSubNegZeroMonoid._proof_1", "SubNegZeroMonoid"], "name": "SubtractionMonoid.toSubNegZeroMonoid", "constType": "{α : Type u_1} → [SubtractionMonoid α] → SubNegZeroMonoid α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "SizeOf.sizeOf", "Nat", "HarderNarasimhan.StrictIntvl", "instOfNatNat", "HarderNarasimhan.PayoffFunction", "Eq", "SizeOf", "LT", "OfNat.ofNat", "HarderNarasimhan.PayoffFunction._sizeOf_inst"], "name": "HarderNarasimhan.PayoffFunction.mk.sizeOf_spec", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] {S : Type u_2} [inst_1 : SizeOf ℒ] [inst_2 : SizeOf S]\n (toFun : HarderNarasimhan.StrictIntvl ℒ → S), sizeOf { toFun := toFun } = 1", "constCategory": "Theorem"}, {"references": ["outParam", "Singleton"], "name": "Singleton.singleton", "constType": "{α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "Eq", "LT"], "name": "HarderNarasimhan.PayoffFunction.mk.inj", "constType": "∀ {ℒ : Type u_1} {inst : LT ℒ} {S : Type u_2} {toFun toFun_1 : HarderNarasimhan.StrictIntvl ℒ → S},\n { toFun := toFun } = { toFun := toFun_1 } → toFun = toFun_1", "constCategory": "Theorem"}, {"references": [], "name": "LE", "constType": "Type u → Type u", "constCategory": "Other"}, {"references": ["outParam"], "name": "Membership", "constType": "outParam (Type u) → Type v → Type (max u v)", "constCategory": "Other"}, {"references": ["Subtype"], "name": "Subtype.val", "constType": "{α : Sort u} → {p : α → Prop} → Subtype p → α", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toInfSet", "DedekindCut.instCompleteLinearOrder._proof_3", "BiheytingAlgebra.toHNot", "Lattice.mk", "Lattice.inf", "DedekindCut.instCompleteLinearOrder._proof_15", "CompleteLinearOrder", "BiheytingAlgebra", "SemilatticeInf.toPartialOrder", "DedekindCut.instCompleteLinearOrder._proof_17", "CompleteLattice.toLattice", "SemilatticeSup.sup", "LinearOrder", "DedekindCut.instCompleteLinearOrder._proof_14", "LinearOrder.toOrd", "LinearOrder.toDecidableLT", "DedekindCut.instCompleteLinearOrder._proof_5", "HeytingAlgebra.toCompl", "CompleteLattice.toBoundedOrder", "DedekindCut.instCompleteLinearOrder._proof_7", "CompleteLattice.toSupSet", "Concept.instBoundedOrderConcept", "DedekindCut.instCompleteLinearOrder._proof_8", "GeneralizedHeytingAlgebra.toHImp", "CompleteLinearOrder.mk", "instDistribLatticeOfLinearOrder", "DedekindCut.instCompleteLinearOrder._proof_2", "DedekindCut.instCompleteLinearOrder._proof_6", "Concept.instCompleteLattice", "Preorder.toLE", "HeytingAlgebra.toGeneralizedHeytingAlgebra", "Lattice.toSemilatticeInf", "BiheytingAlgebra.toHeytingAlgebra", "CompleteLattice.mk", "BiheytingAlgebra.toSDiff", "DedekindCut", "DedekindCut.instCompleteLinearOrder._proof_13", "LinearOrder.toPartialOrder", "LinearOrder.toDecidableEq", "DedekindCut.instCompleteLinearOrder._proof_12", "DedekindCut.instCompleteLinearOrder._proof_16", "DistribLattice.toLattice", "LinearOrder.toDecidableLE", "LE.le", "inferInstance", "DedekindCut.instLinearOrder", "SemilatticeSup.mk", "DedekindCut.instCompleteLinearOrder._proof_1", "LinearOrder.toBiheytingAlgebra", "CompleteLattice", "DedekindCut.instCompleteLinearOrder._proof_4"], "name": "DedekindCut.instCompleteLinearOrder", "constType": "{α : Type u_1} → [inst : LinearOrder α] → CompleteLinearOrder (DedekindCut α)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Preorder", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.right_mem", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), I.right ∈ I", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "DFunLike.coe", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.A", "CompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "Eq"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.min_eq_A", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {S : Type u_3} [inst_1 : CompleteLinearOrder S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [WellFoundedGT ℒ],\n μ.IsSlopeLike → ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.min I = μ.A I", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsAffine.mk", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y : ℒ) (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsAffine", "constCategory": "Definition"}, {"references": ["FunLike", "CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Submodule", "HarderNarasimhan.CoprimaryFiltration.toFun", "IsNoetherianRing", "DFunLike.mk", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule._proof_1", "AddCommGroup.toAddCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] → FunLike (HarderNarasimhan.CoprimaryFiltration R M) ℕ (Submodule R M)", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.instMembership", "Nat", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "associatedPrimes", "Submodule.hasQuotient", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "HarderNarasimhan.CoprimaryFiltration.length", "DFunLike.coe", "Submodule", "instDistribLatticeOfLinearOrder", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "Set", "instHAdd", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "PrimeSpectrum", "HAdd.hAdd", "LT.lt", "CommRing.toRing", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration.toFun", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.associatedPrime_succ_lt", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (self : HarderNarasimhan.CoprimaryFiltration R M) (i : ℕ),\n i + 1 < self.length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈ associatedPrimes R (↥(self.toFun (i + 2)) ⧸ (self.toFun (i + 1)).submoduleOf (self.toFun (i + 2))) →\n q.asIdeal ∈ associatedPrimes R (↥(self.toFun (i + 1)) ⧸ (self.toFun i).submoduleOf (self.toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q", "constCategory": "Theorem"}, {"references": ["SetLike.instMembership", "Subtype", "Submodule.setLike", "Module", "Membership.mem", "Submodule.addSubgroupClass", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "AddSubgroupClass.toAddCommGroup", "Submodule", "Ring.toSemiring", "Ring"], "name": "Submodule.addCommGroup", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Ring R] → [inst_1 : AddCommGroup M] → {module_M : _root_.Module R M} → (p : Submodule R M) → AddCommGroup ↥p", "constCategory": "Definition"}, {"references": ["instAddNat", "HAdd.hAdd", "NeZero", "Nat", "Zero.ofOfNat0", "instHAdd", "instOfNatNat"], "name": "instNeZeroNatHAdd_1", "constType": "∀ {n m : ℕ} [h : NeZero m], NeZero (n + m)", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Eq.ndrec", "Eq", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk.congr_simp", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] (left left_1 : ℒ) (e_left : left = left_1) (right right_1 : ℒ)\n (e_right : right = right_1) (lt : left < right),\n { left := left, right := right, lt := lt } = { left := left_1, right := right_1, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Membership.mk", "Set", "Set.Mem", "Membership"], "name": "Set.instMembership", "constType": "{α : Type u} → Membership α (Set α)", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Set.Ico", "CompleteLattice.toConditionallyCompleteLattice", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "LE.le", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.le_A", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {s : S},\n (∀ (a : ℒ) (ha : a ∈ Set.Ico I.left I.right), s ≤ μ.max { left := a, right := I.right, lt := ⋯ }) → s ≤ μ.A I", "constCategory": "Theorem"}, {"references": ["LinearOrder.toPartialOrder", "PartialOrder.toPreorder", "DecidableLT", "Preorder.toLT", "LinearOrder"], "name": "LinearOrder.toDecidableLT", "constType": "{α : Type u_2} → [self : LinearOrder α] → DecidableLT α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HasNashEquilibrium.rec", "HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HasNashEquilibrium → Sort u} →\n (t : μ.HasNashEquilibrium) → ((eq : μ.A ⊤ = μ.B ⊤) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.CoprimaryFiltration.rec", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.recOn", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n {motive : HarderNarasimhan.CoprimaryFiltration R M → Sort u} →\n (t : HarderNarasimhan.CoprimaryFiltration R M) →\n ((toFun : ℕ → Submodule R M) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isCoprimary :\n ∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))) →\n (associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q) →\n motive\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt }) →\n motive t", "constCategory": "Definition"}, {"references": ["Eq"], "name": "Eq.refl", "constType": "∀ {α : Sort u_1} (a : α), a = a", "constCategory": "Other"}, {"references": ["AddCommMonoid", "SubNegMonoid.toAddMonoid", "AddCommGroup.add_comm", "AddCommGroup.toAddGroup", "AddCommMonoid.mk", "AddCommGroup", "AddGroup.toSubNegMonoid"], "name": "AddCommGroup.toAddCommMonoid", "constType": "{G : Type u} → [self : AddCommGroup G] → AddCommMonoid G", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.IsCoprimary.mk", "CommRing.toCommSemiring", "associatedPrimes", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "HarderNarasimhan.IsCoprimary.rec", "CommRing", "Set.instMembership", "HarderNarasimhan.IsCoprimary", "Ideal", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.recOn", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n {M : Type u_2} →\n [inst_1 : AddCommGroup M] →\n [inst_2 : _root_.Module R M] →\n {motive : HarderNarasimhan.IsCoprimary R M → Sort u} →\n (t : HarderNarasimhan.IsCoprimary R M) →\n ((existsUnique_associatedPrime : ∃! p, p ∈ associatedPrimes R M) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsStable", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "HarderNarasimhan.PayoffFunction.IsStable.mk", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsStable → Sort u} →\n ([toIsSemistable : μ.IsSemistable] →\n (ne : ∀ (x : ℒ) (hx : ⊥ < x), x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤) → motive ⋯) →\n (t : μ.IsStable) → motive t", "constCategory": "Other"}, {"references": ["Set", "Set.Finite.nonempty_fintype", "Fintype", "Set.Finite", "Set.Elem", "Nonempty.some"], "name": "Set.Finite.fintype", "constType": "{α : Type u} → {s : Set α} → s.Finite → Fintype ↑s", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "DFunLike.coe", "Submodule", "HarderNarasimhan.CoprimaryFiltration.toFun", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule", "Eq", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.toFun_eq_coe", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (F : HarderNarasimhan.CoprimaryFiltration R M), F.toFun = ⇑F", "constCategory": "Theorem"}, {"references": ["Finset", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Submodule.map", "Membership.mem", "Preorder.toLT", "Submodule.subtype", "Semiring.toNonAssocSemiring", "RingHom.id", "Module.IsNoetherian.finite", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "SetLike.instMembership", "Submodule.addCommGroup", "Bot.bot", "Ring.toSemiring", "isNoetherian_submodule'", "HarderNarasimhan.StrictIntvl", "Submodule.comap", "Iff.mpr", "Submodule.instBot", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.A", "Submodule.submoduleOf", "Submodule.hasQuotient", "Ne.symm", "Subtype", "HasQuotient.Quotient", "Submodule.Quotient.instSMul._proof_1", "Module.Finite.quotient", "Module", "Submodule.module", "lt_of_le_of_ne", "DFunLike.coe", "Submodule", "Submodule.Quotient.addCommGroup", "instDistribLatticeOfLinearOrder", "isNoetherian_of_isNoetherianRing_of_finite", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.Coprimary.payoff", "OrderBot.toBot", "Colex", "Concept.instCompleteLattice", "Submodule.instOrderBot", "Eq", "Preorder.toLE", "Submodule.mkQ", "bot_lt_iff_ne_bot", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instDistribOfSemiring", "Distrib.toMul", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "instSMulOfMul", "DedekindCut", "AddCommGroup", "CommRing", "RingHomSurjective.ids", "LT.lt", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl.mk", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Submodule.setLike", "LE.le", "Ne", "HarderNarasimhan.PayoffFunction", "Submodule.addCommMonoid", "HarderNarasimhan.Coprimary.map_comap_ne_bot", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.A_restrict_eq_quotient", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] {N₁ N₂ W : Submodule R M} (h₁ : N₁ ≤ W) (h₂ : W ≤ N₂)\n (h₃ : W ≠ N₁),\n (HarderNarasimhan.Coprimary.payoff R M).A { left := N₁, right := W, lt := ⋯ } =\n (HarderNarasimhan.Coprimary.payoff R (↥N₂ ⧸ N₁.submoduleOf N₂)).A\n { left := ⊥, right := Submodule.map (N₁.submoduleOf N₂).mkQ (Submodule.comap N₂.subtype W), lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.ADCC.mk", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "GT.gt", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLattice.toConditionallyCompleteLattice", "Not", "Exists", "instHAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "Nat", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.ADCC.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (a : ℒ) (f : ℕ → ℒ) (h₁ : ∀ (n : ℕ), f n > a),\n StrictAnti f → ∃ N, ¬μ.A { left := a, right := f N, lt := ⋯ } < μ.A { left := a, right := f (N + 1), lt := ⋯ }) →\n μ.ADCC", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Set.Ico", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.le_min", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {s : S},\n (∀ (u : ℒ) (hu : u ∈ Set.Ico I.left I.right), s ≤ μ { left := u, right := I.right, lt := ⋯ }) → s ≤ μ.min I", "constCategory": "Theorem"}, {"references": ["LinearOrder.toPartialOrder", "Nat", "Preorder", "PartialOrder.toPreorder", "inferInstance", "Nat.instLinearOrder"], "name": "Nat.instPreorder", "constType": "Preorder ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder", "ConditionallyCompletePartialOrderSup"], "name": "ConditionallyCompletePartialOrderSup.toPartialOrder", "constType": "{α : Type u_3} → [self : ConditionallyCompletePartialOrderSup α] → PartialOrder α", "constCategory": "Definition"}, {"references": ["SMul.mk", "SMul", "Real", "HSMul.hSMul", "NNReal", "NNReal.toReal", "instHSMul"], "name": "NNReal.instSMulOfReal", "constType": "{M : Type u_1} → [SMul ℝ M] → SMul NNReal M", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.IsBreakpoint.left_lt", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.not_A_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (hx : μ.IsBreakpoint I x),\n μ.IsConvexOn I →\n ∀ {y : ℒ},\n y ∈ I → ∀ (hy : x < y), ¬μ.A { left := I.left, right := x, lt := ⋯ } ≤ μ.A { left := x, right := y, lt := hy }", "constCategory": "Theorem"}, {"references": ["Nat", "OfNat", "OfNat.mk"], "name": "instOfNatNat", "constType": "(n : ℕ) → OfNat ℕ n", "constCategory": "Definition"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.lt", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }) →\n μ.WeakSlopeLikeAtBot", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.strictAntiOn", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Nat.instOne", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Nat.instIsOrderedAddMonoid", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "instLTNat", "Nat.instAddMonoid", "Nat.instPartialOrder", "instHAdd", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "HAdd.hAdd", "lt_add_one", "Nat", "HarderNarasimhan.StrictIntvl", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.payoff_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration) {i : ℕ}\n (hi : i < F.length) {z : ℒ} (h' : F (i + 1) < z),\n z < F i → μ { left := F (i + 1), right := z, lt := h' } < μ { left := F (i + 1), right := F i, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HasNashEquilibrium → Sort u} →\n ((eq : μ.A ⊤ = μ.B ⊤) → motive ⋯) → (t : μ.HasNashEquilibrium) → motive t", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.IsConvex", "ConditionallyCompletePartialOrderSup.toPartialOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_lt_succ", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "WellFoundedGT", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Not", "instLTNat", "Lattice.toSemilatticeInf", "instHAdd", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_ne_top_iff", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HAdd.hAdd", "Nat.lt_of_succ_lt", "HarderNarasimhan.StrictIntvl", "Nat", "Iff.mpr", "Lattice", "Nontrivial", "LE.le", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction.ADCC", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_not_A_le_succ", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible] (i : ℕ) (hi : i + 1 < HarderNarasimhan.PayoffFunction.HNlen✝ μ),\n ¬μ.A\n { left := HarderNarasimhan.PayoffFunction.HNFil✝ μ i, right := HarderNarasimhan.PayoffFunction.HNFil✝ μ (i + 1),\n lt := ⋯ } ≤\n μ.A\n { left := HarderNarasimhan.PayoffFunction.HNFil✝ μ (i + 1),\n right := HarderNarasimhan.PayoffFunction.HNFil✝ μ (i + 2), lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "RelSeries", "CompleteLinearOrder.toCompletelyDistribLattice", "BoundedOrder.toOrderTop", "WellFoundedGT", "OrderBot.toBot", "CompleteLinearOrder", "Preorder.toLE", "Eq", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "Exists", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "HarderNarasimhan.PayoffFunction.jordanHolderRel", "And", "BoundedOrder", "Bot.bot", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "RelSeries.last", "BoundedOrder.toOrderBot", "RelSeries.head", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.exists_relSeries_jordanHolderRel", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hacc : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLinearOrder S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [hsl : μ.IsSlopeLike]\n [hftp : μ.FiniteTotalPayoff] [hdc : μ.EventuallyTopDCC] [hst : μ.IsSemistable], ∃ s, s.head = ⊤ ∧ s.last = ⊥", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "HarderNarasimhan.PayoffFunction.B", "Set", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.PayoffFunction.min", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "Set.Ioc", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.le_B", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {b : ℒ}\n (hb : b ∈ Set.Ioc I.left I.right), μ.min { left := I.left, right := b, lt := ⋯ } ≤ μ.B I", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} → [inst : PartialOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.max_top_eq_apply_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike],\n μ.max ⊤ = μ ⊤ ↔ μ.min ⊤ = μ.max ⊤", "constCategory": "Theorem"}, {"references": ["CommRing", "Ring"], "name": "CommRing.toRing", "constType": "{α : Type u} → [self : CommRing α] → Ring α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.A ⊤ = μ.B ⊤ → μ.HasNashEquilibrium", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.rec", "HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.casesOn", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {S : Type u_2} →\n {motive : HarderNarasimhan.PayoffFunction ℒ S → Sort u} →\n (t : HarderNarasimhan.PayoffFunction ℒ S) →\n ((toFun : HarderNarasimhan.StrictIntvl ℒ → S) → motive { toFun := toFun }) → motive t", "constCategory": "Definition"}, {"references": ["instAddNat", "instHAdd", "SizeOf", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl", "Nat", "SizeOf.sizeOf", "instOfNatNat", "HarderNarasimhan.StrictIntvl.rec", "instSizeOfDefault", "LT"], "name": "HarderNarasimhan.StrictIntvl._sizeOf_1", "constType": "{ℒ : Type u_1} → {inst : LT ℒ} → [SizeOf ℒ] → HarderNarasimhan.StrictIntvl ℒ → ℕ", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Module", "Semiring"], "name": "Module.Finite", "constType": "(R : Type u_1) → (M : Type u_4) → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [_root_.Module R M] → Prop", "constCategory": "Other"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "LT.lt.trans_le", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a < b → b ≤ c → a < c", "constCategory": "Theorem"}, {"references": ["Ideal", "Semiring"], "name": "Ideal.IsPrime", "constType": "{α : Type u} → [inst : Semiring α] → Ideal α → Prop", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "Subtype.instLE", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "Bot.bot", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "BoundedOrder.toOrderBot", "PartialOrder", "OrderBot.toBot", "Eq", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.val_bot", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, ↑⊥ = I.left", "constCategory": "Theorem"}, {"references": ["Subtype.instLT", "Preorder", "Subtype", "Subtype.instLE", "Preorder.toLT", "Subtype.preorder._proof_2", "Subtype.preorder._proof_1", "Preorder.mk", "Subtype.preorder._proof_3", "Preorder.toLE"], "name": "Subtype.preorder", "constType": "{α : Type u_2} → [Preorder α] → (p : α → Prop) → Preorder (Subtype p)", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "Nat.brecOn", "HarderNarasimhan.PayoffFunction.IsConvex", "Nat", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil._f", "Nontrivial", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "constType": "{ℒ : Type u_1} →\n [Nontrivial ℒ] →\n [inst : Lattice ℒ] →\n [BoundedOrder ℒ] →\n [hwf : WellFoundedGT ℒ] →\n {S : Type u_2} →\n [inst_2 : CompleteLattice S] →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) → [μ.ADCC] → [μ.IsConvex] → [hadm : μ.Admissible] → ℕ → ℒ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsStable", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "HarderNarasimhan.PayoffFunction.IsStable.mk", "CompleteLattice", "HarderNarasimhan.PayoffFunction.IsStable.rec"], "name": "HarderNarasimhan.PayoffFunction.IsStable.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsStable → Sort u} →\n (t : μ.IsStable) →\n ([toIsSemistable : μ.IsSemistable] →\n (ne : ∀ (x : ℒ) (hx : ⊥ < x), x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.IsSlopeLike.rec", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSlopeLike → Sort u} →\n (t : μ.IsSlopeLike) →\n ((slopelike :\n ∀ (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": [], "name": "Std.Total", "constType": "{α : Sort u} → (α → α → Prop) → Prop", "constCategory": "Other"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsConvex.mk", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y : ℒ) (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsConvex", "constCategory": "Definition"}, {"references": ["outParam", "Membership"], "name": "Membership.mk", "constType": "{α : outParam (Type u)} → {γ : Type v} → (γ → α → Prop) → Membership α γ", "constCategory": "Other"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Subtype.preorder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.max_restrict_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}\n {J : HarderNarasimhan.StrictIntvl { x // x ∈ I }}, (μ.restrict I).max J = μ.max (HarderNarasimhan.StrictIntvl.ofSub J)", "constCategory": "Theorem"}, {"references": ["Prod.casesOn", "Prod", "Prod.mk"], "name": "HarderNarasimhan.PayoffFunction.jordanHolderRel.match_1", "constType": "{ℒ : Type u_1} → (motive : ℒ × ℒ → Sort u_2) → (x : ℒ × ℒ) → ((x y : ℒ) → motive (x, y)) → motive x", "constCategory": "Definition"}, {"references": ["LT.lt", "Preorder", "Set.ofPred", "Set", "LE.le", "Preorder.toLT", "And", "Preorder.toLE"], "name": "Set.Ico", "constType": "{α : Type u_1} → [Preorder α] → α → α → Set α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "HarderNarasimhan.PayoffFunction.IsConvex", "Nat.instPreorder", "Nat", "Monotone", "Nontrivial", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_monotone", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible], Monotone (HarderNarasimhan.PayoffFunction.HNFil✝ μ)", "constCategory": "Theorem"}, {"references": ["Membership.mk", "SetLike.coe", "Set", "Membership.mem", "SetLike", "Membership", "Set.instMembership"], "name": "SetLike.instMembership", "constType": "{A : Type u_1} → {B : Type u_2} → [i : SetLike A B] → Membership B A", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.min", "BoundedOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "HarderNarasimhan.PayoffFunction.max", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_top_le_A_top_of_min_eq_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.min ⊤ = μ.max ⊤ → μ.B ⊤ ≤ μ.A ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_top_eq_apply_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hμ : μ.IsSlopeLike],\n μ.min ⊤ = μ ⊤ ↔ μ.min ⊤ = μ.max ⊤", "constCategory": "Theorem"}, {"references": ["MulOneClass.toMulOne", "Submonoid", "Submonoid.instSetLike._proof_1", "Subsemigroup.carrier", "SetLike.mk", "MulOne.toMul", "MulOneClass", "SetLike", "Submonoid.toSubsemigroup"], "name": "Submonoid.instSetLike", "constType": "{M : Type u_1} → [inst : MulOneClass M] → SetLike (Submonoid M) M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.total_or_attained", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Preorder ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.Admissible],\n (Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instHAdd", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "OfNat.ofNat", "HAdd.hAdd", "LT.lt", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "instOfNatNat", "BoundedOrder.toOrderTop", "PartialOrder", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.lt_succ_of_ne_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.HarderNarasimhanFiltration} {m : ℕ}, F m ≠ ⊤ → F m < F (m + 1)", "constCategory": "Theorem"}, {"references": ["Lattice", "CompleteLattice"], "name": "CompleteLattice.toLattice", "constType": "{α : Type u_8} → [self : CompleteLattice α] → Lattice α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "Subtype", "CoeSort", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "LE", "CoeSort.mk", "LT"], "name": "HarderNarasimhan.StrictIntvl.instCoeSortTypeOfLE", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → [LE ℒ] → CoeSort (HarderNarasimhan.StrictIntvl ℒ) (Type u_1)", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "And", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk.inj", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Nontrivial ℒ} {inst_1 : PartialOrder ℒ} {inst_2 : BoundedOrder ℒ}\n {inst_3 : CompleteLattice S} {μ : HarderNarasimhan.PayoffFunction ℒ S} {toFun : ℕ → ℒ} {length : ℕ}\n {antitone : Antitone toFun} {head_eq_top : toFun 0 = ⊤} {length_eq_bot : toFun length = ⊥}\n {strictAntiOn : StrictAntiOn toFun (Set.Iic length)}\n {step_payoff_eq : ∀ (i : ℕ) (hi : i < length), μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤}\n {payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } < μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }}\n {toFun_1 : ℕ → ℒ} {length_1 : ℕ} {antitone_1 : Antitone toFun_1} {head_eq_top_1 : toFun_1 0 = ⊤}\n {length_eq_bot_1 : toFun_1 length_1 = ⊥} {strictAntiOn_1 : StrictAntiOn toFun_1 (Set.Iic length_1)}\n {step_payoff_eq_1 : ∀ (i : ℕ) (hi : i < length_1), μ { left := toFun_1 (i + 1), right := toFun_1 i, lt := ⋯ } = μ ⊤}\n {payoff_lt_of_between_1 :\n ∀ (i : ℕ) (hi : i < length_1) (z : ℒ) (h' : toFun_1 (i + 1) < z),\n z < toFun_1 i →\n μ { left := toFun_1 (i + 1), right := z, lt := h' } <\n μ { left := toFun_1 (i + 1), right := toFun_1 i, lt := ⋯ }},\n { toFun := toFun, length := length, antitone := antitone, head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq, payoff_lt_of_between := payoff_lt_of_between } =\n { toFun := toFun_1, length := length_1, antitone := antitone_1, head_eq_top := head_eq_top_1,\n length_eq_bot := length_eq_bot_1, strictAntiOn := strictAntiOn_1, step_payoff_eq := step_payoff_eq_1,\n payoff_lt_of_between := payoff_lt_of_between_1 } →\n toFun = toFun_1 ∧ length = length_1", "constCategory": "Theorem"}, {"references": [], "name": "Nontrivial", "constType": "Type u_3 → Prop", "constCategory": "Other"}, {"references": ["Zero", "NegZeroClass"], "name": "NegZeroClass.toZero", "constType": "{G : Type u_2} → [self : NegZeroClass G] → Zero G", "constCategory": "Definition"}, {"references": ["CompleteDistribLattice.toHNot", "CompleteDistribLattice.toFrame", "Order.Frame.toCompleteLattice", "Order.Coframe", "CompleteDistribLattice.toSDiff", "CompleteDistribLattice.sdiff_le_iff", "CompleteDistribLattice.top_sdiff", "Order.Coframe.mk", "CompleteDistribLattice"], "name": "CompleteDistribLattice.toCoframe", "constType": "{α : Type u_1} → [self : CompleteDistribLattice α] → Order.Coframe α", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.mk", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.rec", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.FiniteTotalPayoff → Sort u} →\n (t : μ.FiniteTotalPayoff) → ((ne_top : μ ⊤ ≠ ⊤) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakACC", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": [], "name": "Subtype", "constType": "{α : Sort u} → (α → Prop) → Sort (max 1 u)", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.ext", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F G : μ.JordanHolderFiltration},\n (∀ (n : ℕ), F n = G n) → F = G", "constCategory": "Theorem"}, {"references": ["SetRel"], "name": "RelSeries", "constType": "{α : Type u_1} → SetRel α α → Type u_1", "constCategory": "Other"}, {"references": ["EmptyCollection.mk", "Set", "False", "EmptyCollection"], "name": "Set.instEmptyCollection", "constType": "{α : Type u} → EmptyCollection (Set α)", "constCategory": "Definition"}, {"references": ["outParam", "CoeSort"], "name": "CoeSort.mk", "constType": "{α : Sort u} → {β : outParam (Sort v)} → (α → β) → CoeSort α β", "constCategory": "Other"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_left_lt_right_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := x, right := y, lt := h₁ } < μ { left := y, right := z, lt := h₂ } ↔\n μ { left := x, right := y, lt := h₁ } < μ { left := x, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.HasNashEquilibrium.rec", "HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HasNashEquilibrium → Sort u} →\n (t : μ.HasNashEquilibrium) → ((eq : μ.A ⊤ = μ.B ⊤) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.mk", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.FiniteTotalPayoff → Sort u} →\n ((ne_top : μ ⊤ ≠ ⊤) → motive ⋯) → (t : μ.FiniteTotalPayoff) → motive t", "constCategory": "Other"}, {"references": ["OfNat", "OfNat.mk", "Zero", "Zero.zero"], "name": "Zero.toOfNat0", "constType": "{α : Type u_1} → [Zero α] → OfNat α 0", "constCategory": "Definition"}, {"references": [], "name": "Colex", "constType": "Type u_2 → Type u_2", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "WellFoundedGT", "CompleteLinearOrder", "OrderBot.toBot", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "HarderNarasimhan.StrictIntvl.instMembership", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl.mk", "CompletelyDistribLattice.toCompleteLattice", "LT.lt", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Lattice", "Nontrivial", "HarderNarasimhan.PayoffFunction"], "name": "HarderNarasimhan.PayoffFunction.instFiniteTotalPayoffSubtypeMemStrictIntvlMkBotRestrictOfIsSlopeLikeOfIsSemistableOfEventuallyTopDCC", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [WellFoundedGT ℒ]\n [inst_4 : CompleteLinearOrder S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [hftp : μ.FiniteTotalPayoff]\n [μ.IsSlopeLike] [hst : μ.IsSemistable] [μ.EventuallyTopDCC] {x : ℒ} {hx : ⊥ < x},\n (μ.restrict { left := ⊥, right := x, lt := hx }).FiniteTotalPayoff", "constCategory": "Theorem"}, {"references": ["CommRing.toRing", "CommSemiring", "CommRing.mul_comm", "CommSemiring.mk", "CommRing", "Ring.toSemiring"], "name": "CommRing.toCommSemiring", "constType": "{α : Type u} → [s : CommRing α] → CommSemiring α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "lt_of_le_of_ne", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.IsAttained", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsBreakpoint.mem", "HarderNarasimhan.StrictIntvl.right", "lt_of_le_of_lt", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "LE.le", "Lattice", "HarderNarasimhan.PayoffFunction.A", "Ne", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.A_eq_A_of_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ} (hx : μ.IsBreakpoint I x),\n μ.IsConvexOn I →\n ((Std.Total fun x1 x2 => x1 ≤ x2) ∨\n ∀ (z : ℒ) (hzI : z ∈ I) (hz : I.left ≠ z), μ.IsAttained { left := I.left, right := z, lt := ⋯ }) →\n ∀ {y : ℒ},\n y ∈ I → ∀ (hxy : x < y), μ.A { left := I.left, right := y, lt := ⋯ } = μ.A { left := x, right := y, lt := hxy }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.dual._proof_1", "HarderNarasimhan.PayoffFunction.mk", "OrderDual", "OrderDual.toDual", "Equiv.instEquivLike", "HarderNarasimhan.StrictIntvl.right", "DFunLike.coe", "Equiv", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "OrderDual.instLT", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.dual", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} → [inst : LT ℒ] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.PayoffFunction ℒᵒᵈ Sᵒᵈ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_top_eq_max_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.StrongDCC] [μ.WeakSlopeLikeAtBot],\n μ.B ⊤ = μ.max ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Finset", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Preorder.toLT", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "Eq", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "HarderNarasimhan.PayoffFunction.max", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.max_payoff", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M],\n (HarderNarasimhan.Coprimary.payoff R M).max = HarderNarasimhan.Coprimary.payoff R M", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsAffine", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "HarderNarasimhan.PayoffFunction.IsAffine.rec", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsAffine.mk", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "HarderNarasimhan.PayoffFunction", "right_lt_sup"], "name": "HarderNarasimhan.PayoffFunction.IsAffine.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsAffine → Sort u} →\n (t : μ.IsAffine) →\n ((eq :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } = μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.PayoffFunction.IsStable", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "HarderNarasimhan.PayoffFunction.IsStable.mk", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) →\n (∀ (x : ℒ) (hx : ⊥ < x), x < ⊤ → μ.A { left := ⊥, right := x, lt := hx } ≠ μ.A ⊤) → μ.IsStable", "constCategory": "Definition"}, {"references": ["Semiring.toAddCommMonoid", "IsNoetherian", "Semiring.toModule", "Semiring"], "name": "IsNoetherianRing", "constType": "(R : Type u_1) → [Semiring R] → Prop", "constCategory": "Definition"}, {"references": ["LE"], "name": "LE.le", "constType": "{α : Type u} → [self : LE α] → α → α → Prop", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "StrictMonoOn", "Preorder.toLT", "BoundedOrder", "Set.Iic", "Nat.instPreorder", "Nat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.strictMonoOn", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration),\n StrictMonoOn self.toFun (Set.Iic self.length)", "constCategory": "Theorem"}, {"references": ["LT.lt.trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.seesaw_right_lt_left_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.IsSlopeLike →\n ∀ {x y z : ℒ} (h₁ : x < y) (h₂ : y < z),\n μ { left := y, right := z, lt := h₂ } < μ { left := x, right := y, lt := h₁ } ↔\n μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := h₁ }", "constCategory": "Theorem"}, {"references": ["Nat", "LE.mk", "LE", "Nat.le"], "name": "instLENat", "constType": "LE ℕ", "constCategory": "Definition"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtBot → Sort u} →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }) →\n motive ⋯) →\n (t : μ.WeakSlopeLikeAtBot) → motive t", "constCategory": "Other"}, {"references": ["AddCommMonoid", "AddSubmonoid", "Module", "AddCommMonoid.toAddMonoid", "AddMonoid.toAddZeroClass", "Submodule", "Semiring"], "name": "Submodule.toAddSubmonoid", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → Submodule R M → AddSubmonoid M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.IsCoprimary.mk", "CommRing.toCommSemiring", "associatedPrimes", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "CommRing", "Set.instMembership", "HarderNarasimhan.IsCoprimary", "Ideal", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.rec", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n {M : Type u_2} →\n [inst_1 : AddCommGroup M] →\n [inst_2 : _root_.Module R M] →\n {motive : HarderNarasimhan.IsCoprimary R M → Sort u} →\n ((existsUnique_associatedPrime : ∃! p, p ∈ associatedPrimes R M) → motive ⋯) →\n (t : HarderNarasimhan.IsCoprimary R M) → motive t", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "Monotone", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.Iic", "Bot.bot", "Set.instMembership", "Nat", "HarderNarasimhan.IsCoprimary", "Submodule.instBot", "Nontrivial", "HarderNarasimhan.CoprimaryFiltration.mk", "Top.top", "AddCommGroup.toAddCommMonoid", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "DFunLike.coe", "Nat.instPreorder", "Submodule", "instDistribLatticeOfLinearOrder", "Submodule.Quotient.addCommGroup", "Ideal", "instOfNatNat", "HarderNarasimhan.CoprimaryFiltration", "PrimeSpectrum.asIdeal", "Eq", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instHAdd", "Set", "Submodule.instTop", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "toLinearExtension", "AddCommGroup", "CommRing", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Submodule.setLike", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration.rec", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n [inst_1 : IsNoetherianRing R] →\n {M : Type u_2} →\n [inst_2 : Nontrivial M] →\n [inst_3 : AddCommGroup M] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : Module.Finite R M] →\n {motive : HarderNarasimhan.CoprimaryFiltration R M → Sort u} →\n ((toFun : ℕ → Submodule R M) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isCoprimary :\n ∀ i < length,\n HarderNarasimhan.IsCoprimary R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1)))) →\n (associatedPrime_succ_lt :\n ∀ (i : ℕ),\n i + 1 < length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 2)) ⧸ (toFun (i + 1)).submoduleOf (toFun (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥(toFun (i + 1)) ⧸ (toFun i).submoduleOf (toFun (i + 1))) →\n toLinearExtension p < toLinearExtension q) →\n motive\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isCoprimary := piecewise_isCoprimary,\n associatedPrime_succ_lt := associatedPrime_succ_lt }) →\n (t : HarderNarasimhan.CoprimaryFiltration R M) → motive t", "constCategory": "Other"}, {"references": ["instDistribLatticeOfLinearOrder", "DistribLattice.toLattice", "Lattice.toSemilatticeInf", "Finset", "id", "LinearOrder", "Finset.Nonempty", "Finset.inf'"], "name": "Finset.min'", "constType": "{α : Type u_2} → [LinearOrder α] → (s : Finset α) → s.Nonempty → α", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.exists_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : BoundedOrder ℒ} {inst_2 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.WeakACC] (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["associatedPrimes", "Submodule.hasQuotient", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Submodule", "Ideal", "Preorder.toLE", "Set.instLE", "CommRing.toCommSemiring", "SetLike.instMembership", "Set", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "CommRing", "CommRing.toRing", "Submodule.setLike", "LE.le", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf"], "name": "HarderNarasimhan.associatedPrimes_subset_of_submoduleOf_le", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (N A B : Submodule R M), A ≤ B → associatedPrimes R (↥A ⧸ N.submoduleOf A) ⊆ associatedPrimes R (↥B ⧸ N.submoduleOf B)", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "And", "HarderNarasimhan.PayoffFunction.IsSlopeLike.mk", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.IsSlopeLike.rec", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "And.left"], "name": "HarderNarasimhan.PayoffFunction.IsSlopeLike.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSlopeLike → Sort u} →\n (t : μ.IsSlopeLike) →\n ((slopelike :\n ∀ (x y z : ℒ) (h : x < y ∧ y < z),\n (μ { left := x, right := y, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := y, lt := ⋯ } < μ { left := x, right := z, lt := ⋯ } ∨\n μ { left := y, right := z, lt := ⋯ } ≤ μ { left := x, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } < μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } ≤ μ { left := y, right := z, lt := ⋯ }) ∧\n (μ { left := x, right := z, lt := ⋯ } ≤ μ { left := x, right := y, lt := ⋯ } ∨\n μ { left := x, right := z, lt := ⋯ } < μ { left := y, right := z, lt := ⋯ })) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.preorder", "HarderNarasimhan.PayoffFunction.min", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction", "Eq", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, (μ.restrict I).min = μ.min.restrict I", "constCategory": "Theorem"}, {"references": [], "name": "LinearOrder", "constType": "Type u_2 → Type u_2", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "BoundedOrder.toOrderTop", "PartialOrder", "Nontrivial", "Top.top", "OrderBot.toBot", "Eq", "Preorder.toLE", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.mk_bot_top", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ] (h : ⊥ < ⊤),\n { left := ⊥, right := ⊤, lt := h } = ⊤", "constCategory": "Theorem"}, {"references": ["Semiring", "Ring"], "name": "Ring.toSemiring", "constType": "{R : Type u} → [self : Ring R] → Semiring R", "constCategory": "Definition"}, {"references": ["Set", "outParam", "SetLike"], "name": "SetLike.coe", "constType": "{A : Type u_1} → {B : outParam (Type u_2)} → [self : SetLike A B] → A → Set B", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "OrderDual", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "HarderNarasimhan.PayoffFunction.dual", "Preorder.toLT", "BoundedOrder", "PartialOrder", "OrderDual.instCompleteLattice", "OrderDual.instPartialOrder", "OrderDual.instBoundedOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.instWeakACCOrderDualDualOfStrongDCC", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [h₁ : μ.StrongDCC], μ.dual.WeakACC", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "bot_lt_iff_ne_bot", "HarderNarasimhan.PayoffFunction.min", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Iff.mpr", "Iff", "Nontrivial", "LE.le", "Top.top", "Ne", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.hasNashEquilibrium_iff_min_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.WeakACC] [μ.WeakSlopeLikeAtTop],\n μ.HasNashEquilibrium ↔ ∀ (y : ℒ) (hy : y ≠ ⊥), μ.min { left := ⊥, right := y, lt := ⋯ } ≤ μ.min ⊤", "constCategory": "Theorem"}, {"references": ["Real.instInv", "Inv.inv", "Inv", "Real", "Inv.mk", "NNReal", "NNReal.toReal", "NNReal.mk", "NNReal.instInv._proof_1"], "name": "NNReal.instInv", "constType": "Inv NNReal", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Type u_1", "constCategory": "Other"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "And", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk.inj", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : BoundedOrder ℒ} {inst_2 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {toFun : ℕ → ℒ} {length : ℕ} {monotone : Monotone toFun}\n {head_eq_bot : toFun 0 = ⊥} {length_eq_top : toFun length = ⊤} {strictMonoOn : StrictMonoOn toFun (Set.Iic length)}\n {piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length), (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable}\n {not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }}\n {toFun_1 : ℕ → ℒ} {length_1 : ℕ} {monotone_1 : Monotone toFun_1} {head_eq_bot_1 : toFun_1 0 = ⊥}\n {length_eq_top_1 : toFun_1 length_1 = ⊤} {strictMonoOn_1 : StrictMonoOn toFun_1 (Set.Iic length_1)}\n {piecewise_isSemistable_1 :\n ∀ (i : ℕ) (hi : i < length_1), (μ.restrict { left := toFun_1 i, right := toFun_1 (i + 1), lt := ⋯ }).IsSemistable}\n {not_A_le_succ_1 :\n ∀ (i : ℕ) (hi : i + 1 < length_1),\n ¬μ.A { left := toFun_1 i, right := toFun_1 (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun_1 (i + 1), right := toFun_1 (i + 2), lt := ⋯ }},\n { toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ } =\n { toFun := toFun_1, length := length_1, monotone := monotone_1, head_eq_bot := head_eq_bot_1,\n length_eq_top := length_eq_top_1, strictMonoOn := strictMonoOn_1,\n piecewise_isSemistable := piecewise_isSemistable_1, not_A_le_succ := not_A_le_succ_1 } →\n toFun = toFun_1 ∧ length = length_1", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "Eq", "LT"], "name": "HarderNarasimhan.PayoffFunction.mk.injEq", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] {S : Type u_2} (toFun toFun_1 : HarderNarasimhan.StrictIntvl ℒ → S),\n ({ toFun := toFun } = { toFun := toFun_1 }) = (toFun = toFun_1)", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.mk", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderTop", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.rec", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "Ne", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.FiniteTotalPayoff → Sort u} →\n (t : μ.FiniteTotalPayoff) → ((ne_top : μ ⊤ ≠ ⊤) → motive ⋯) → motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsSemistable.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Not", "BoundedOrder", "Bot.bot", "HarderNarasimhan.PayoffFunction.IsSemistable.rec", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "BoundedOrder.toOrderBot", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsSemistable → Sort u} →\n (t : μ.IsSemistable) →\n ((not_lt : ∀ (x : ℒ) (hx : ⊥ < x), ¬μ.A ⊤ < μ.A { left := ⊥, right := x, lt := hx }) → motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.rec", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "StrictAnti", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.EventuallyTopDCC → Sort u} →\n (t : μ.EventuallyTopDCC) →\n ((exists_eq_top :\n ∀ (x : ℕ → ℒ) (hx : StrictAnti x), ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "lt_of_le_of_lt", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.A", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_le_A_sup", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x w u : ℒ},\n x ∈ I →\n w ∈ I →\n ∀ (hxw : ¬x ≤ w) (huxw : u ≤ x ⊓ w),\n μ.A { left := u, right := x, lt := ⋯ } ≤ μ.A { left := w, right := x ⊔ w, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Subtype", "Module", "OreLocalization.oreSetComm", "LinearMap.mk", "Membership.mem", "CommSemiring", "AddHom.mk", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "LocalizedModule.mk", "Algebra.id", "LocalizedModule.mkLinearMap._proof_1", "AddCommMonoid", "Semiring.toNonAssocSemiring", "LocalizedModule.mkLinearMap._proof_3", "RingHom.id", "instMulZeroOneClassOfSemiring", "DistribMulAction.toMulAction", "IsScalarTower.right", "AddCommMagma.toAdd", "AddCommSemigroup.toAddCommMagma", "Semiring.toModule", "SetLike.instMembership", "IsScalarTower.left", "Semiring.toMonoid", "OreLocalization.instAddCommMonoidOreLocalization", "LocalizedModule", "Submonoid.instSetLike", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "LinearMap", "LocalizedModule.mkLinearMap._proof_2", "Submonoid.one", "OfNat.ofNat", "CommSemiring.toCommMonoid", "Submonoid", "Module.toDistribMulAction", "One.toOfNat1", "AddCommMonoid.toAddCommSemigroup", "LocalizedModule.algebra'._proof_4"], "name": "LocalizedModule.mkLinearMap", "constType": "{R : Type u} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n (M : Type v) → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → M →ₗ[R] LocalizedModule S M", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "lt_of_le_of_ne", "IsGreatest", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction.breakpoints", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "SemilatticeInf.toPartialOrder", "Exists", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "LE.le", "HarderNarasimhan.PayoffFunction.ADCC", "Ne", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.exists_isGreatest_breakpoints", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} [hwf : WellFoundedGT ℒ] [μ.ADCC],\n μ.IsConvexOn I →\n ((Std.Total fun x1 x2 => x1 ≤ x2) ∨\n ∀ (z : ℒ) (hzI : z ∈ I) (hz : I.left ≠ z), μ.IsAttained { left := I.left, right := z, lt := ⋯ }) →\n ∃ s, IsGreatest (μ.breakpoints I) s", "constCategory": "Theorem"}, {"references": ["SubtractionMonoid.toSubNegZeroMonoid", "Real.instPreorder", "LT.lt.trans", "PartialOrder.toPreorder", "AddCommGroup.toAddGroup", "Preorder.toLT", "SMulZeroClass.toSMul", "NNReal.instZero", "PartialOrder", "IsOrderedAddMonoid", "AddGroup.toSubNegMonoid", "DistribSMul.toSMulZeroClass", "SemilatticeInf.toPartialOrder", "Real", "NNReal", "DistribMulAction.toDistribSMul", "LinearOrder", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl", "Real.instMonoid", "Nontrivial", "AddCommGroup.toAddCommMonoid", "NegZeroClass.toZero", "AddZero.toZero", "AddMonoid.toAddZeroClass", "Module", "SubtractionCommMonoid.toSubtractionMonoid", "SubNegZeroMonoid.toNegZeroClass", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.PayoffFunction.slope", "AddCommMagma.toAdd", "AddCommSemigroup.toAddCommMagma", "Zero.toOfNat0", "Concept.instCompleteLattice", "Preorder.toLE", "Eq", "Distrib.toAdd", "PosSMulStrictMono", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "instHAdd", "instDistribOfSemiring", "DedekindCut", "AddCommGroup", "Real.semiring", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Module.toDistribMulAction", "DistribLattice.toLattice", "AddCommGroup.toDivisionAddCommMonoid", "SubNegMonoid.toAddMonoid", "Real.instZero", "AddCommMonoid.toAddCommSemigroup", "LE.le", "NNReal.instSemiring"], "name": "HarderNarasimhan.PayoffFunction.isSlopeLike_slope", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {V : Type u_2} [inst_1 : AddCommGroup V] [inst_2 : _root_.Module ℝ V]\n [inst_3 : LinearOrder V] [IsOrderedAddMonoid V] [PosSMulStrictMono ℝ V] [Nontrivial V]\n (r : HarderNarasimhan.StrictIntvl ℒ → NNReal) (d : HarderNarasimhan.StrictIntvl ℒ → V),\n (∀ (x y z : ℒ) (h₁ : x < y) (h₂ : y < z),\n d { left := x, right := z, lt := ⋯ } =\n d { left := x, right := y, lt := h₁ } + d { left := y, right := z, lt := h₂ }) →\n (∀ (x y z : ℒ) (h₁ : x < y) (h₂ : y < z),\n r { left := x, right := z, lt := ⋯ } =\n r { left := x, right := y, lt := h₁ } + r { left := y, right := z, lt := h₂ }) →\n (∀ (x y : ℒ) (h : x < y), r { left := x, right := y, lt := h } = 0 → 0 < d { left := x, right := y, lt := h }) →\n (HarderNarasimhan.PayoffFunction.slope r d).IsSlopeLike", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "Nontrivial", "Top.top", "OrderBot.toBot", "Preorder.toLE", "Eq", "OrderTop.toTop"], "name": "HarderNarasimhan.StrictIntvl.left_top", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ], ⊤.left = ⊥", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "LE.le", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "instLENat", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length_le_of_eq_bot", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n F m = ⊥ → F.length ≤ m", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.rec", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] →\n {motive : HarderNarasimhan.StrictIntvl ℒ → Sort u} →\n ((left right : ℒ) → (lt : left < right) → motive { left := left, right := right, lt := lt }) →\n (t : HarderNarasimhan.StrictIntvl ℒ) → motive t", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "Preorder", "True", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE", "Eq"], "name": "HarderNarasimhan.StrictIntvl.right_mem._simp_1", "constType": "∀ {ℒ : Type u_1} [inst : Preorder ℒ] (I : HarderNarasimhan.StrictIntvl ℒ), (I.right ∈ I) = True", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk", "Nontrivial", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HasNashEquilibrium.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.A ⊤ = μ.B ⊤ → μ.HasNashEquilibrium", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "Eq", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "DFunLike.coe", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.ext", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F G : μ.HarderNarasimhanFiltration}, (∀ (n : ℕ), F n = G n) → F = G", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "OrderTop.toTop", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instIsConvexOnTopStrictIntvlOfIsConvex", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_2 : Nontrivial ℒ] [inst_3 : BoundedOrder ℒ] [μ.IsConvex],\n μ.IsConvexOn ⊤", "constCategory": "Theorem"}, {"references": [], "name": "LinearExtension", "constType": "Type u → Type u", "constCategory": "Definition"}, {"references": [], "name": "Set", "constType": "Type u → Type u", "constCategory": "Definition"}, {"references": ["Semiring.toNonAssocSemiring", "NonUnitalNonAssocSemiring.toDistrib", "NonAssocSemiring.toNonUnitalNonAssocSemiring", "Distrib", "inferInstance", "Semiring"], "name": "instDistribOfSemiring", "constType": "{α : Type u} → [Semiring α] → Distrib α", "constCategory": "Definition"}, {"references": ["IsScalarTower", "Module", "SMulZeroClass.toSMul", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "AddMonoidWithOne.toAddMonoid", "OreLocalization.OreSet", "Module.mk", "AddCommMonoid", "OreLocalization", "Semiring.toNonAssocSemiring", "instMulZeroOneClassOfSemiring", "DistribMulAction.toMulAction", "AddCommMonoidWithOne.toAddMonoidWithOne", "NonAssocSemiring.toAddCommMonoidWithOne", "DistribSMul.toSMulZeroClass", "OreLocalization.instAddCommMonoidOreLocalization", "Semiring.toMonoid", "Distrib.toMul", "instDistribOfSemiring", "OreLocalization.instDistribMulActionOfIsScalarTower", "DistribMulAction.toDistribSMul", "instSMulOfMul", "AddZeroClass.toAddZero", "Semiring.toAddCommMonoid", "Submonoid", "Module.toDistribMulAction", "OreLocalization.instModuleOfIsScalarTower._proof_4", "OreLocalization.instModuleOfIsScalarTower._proof_3", "AddZero.toZero", "Semiring", "AddMonoid.toAddZeroClass"], "name": "OreLocalization.instModuleOfIsScalarTower", "constType": "{R : Type u_1} →\n [inst : Semiring R] →\n {S : Submonoid R} →\n [inst_1 : OreLocalization.OreSet S] →\n {X : Type u_2} →\n [inst_2 : AddCommMonoid X] →\n [inst_3 : _root_.Module R X] →\n {R₀ : Type u_3} →\n [inst_4 : Semiring R₀] →\n [inst_5 : _root_.Module R₀ X] →\n [inst_6 : _root_.Module R₀ R] →\n [IsScalarTower R₀ R X] → [IsScalarTower R₀ R R] → _root_.Module R₀ (OreLocalization S X)", "constCategory": "Definition"}, {"references": ["instDistribLatticeOfLinearOrder", "DistribLattice.toLattice", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "toLinearExtension._proof_1", "LinearExtension", "instLinearOrderLinearExtensionOfPartialOrder", "OrderHom", "PartialOrder", "OrderHom.mk", "SemilatticeInf.toPartialOrder"], "name": "toLinearExtension", "constType": "{α : Type u} → [inst : PartialOrder α] → α →o LinearExtension α", "constCategory": "Definition"}, {"references": ["LT.lt", "HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → (left right : ℒ) → left < right → HarderNarasimhan.StrictIntvl ℒ", "constCategory": "Other"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "StrictAnti", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (hx : StrictAnti x), ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤) → μ.EventuallyTopDCC", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.PayoffFunction.StrongDCC.mk", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.StrongDCC → Sort u} →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.StrongDCC) → motive t", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.partialOrder", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "instOfNatNat", "PartialOrder", "Nat.instIsOrderedCancelAddMonoid", "Nat.instIsOrderedAddMonoid", "Preorder.toLE", "LT.lt.le", "instLTNat", "Nat.instAddMonoid", "Nat.instPartialOrder", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "Nat", "HarderNarasimhan.StrictIntvl", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.strictMonoOn", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.piecewise_isSemistable", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration) (i : ℕ) (hi : i < self.length),\n (μ.restrict { left := self.toFun i, right := self.toFun (i + 1), lt := ⋯ }).IsSemistable", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsConvexOn.mk", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n (∀ (x y : ℒ),\n x ∈ I →\n y ∈ I → ∀ (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsConvexOn I", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.Admissible.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Std.Total", "HarderNarasimhan.StrictIntvl", "Preorder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Or", "LE.le", "HarderNarasimhan.PayoffFunction.IsAttained", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.Admissible.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n ((Std.Total fun x1 x2 => x1 ≤ x2) ∨ ∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ.IsAttained I) → μ.Admissible", "constCategory": "Definition"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "HarderNarasimhan.PayoffFunction.min", "Subtype.preorder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.StrictIntvl.ofSub", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_restrict_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}\n {J : HarderNarasimhan.StrictIntvl { x // x ∈ I }}, (μ.restrict I).min J = μ.min (HarderNarasimhan.StrictIntvl.ofSub J)", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "And", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk.injEq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (toFun : ℕ → ℒ) (length : ℕ)\n (antitone : Antitone toFun) (head_eq_top : toFun 0 = ⊤) (length_eq_bot : toFun length = ⊥)\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length))\n (step_payoff_eq : ∀ (i : ℕ) (hi : i < length), μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤)\n (payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } < μ { left := toFun (i + 1), right := toFun i, lt := ⋯ })\n (toFun_1 : ℕ → ℒ) (length_1 : ℕ) (antitone_1 : Antitone toFun_1) (head_eq_top_1 : toFun_1 0 = ⊤)\n (length_eq_bot_1 : toFun_1 length_1 = ⊥) (strictAntiOn_1 : StrictAntiOn toFun_1 (Set.Iic length_1))\n (step_payoff_eq_1 : ∀ (i : ℕ) (hi : i < length_1), μ { left := toFun_1 (i + 1), right := toFun_1 i, lt := ⋯ } = μ ⊤)\n (payoff_lt_of_between_1 :\n ∀ (i : ℕ) (hi : i < length_1) (z : ℒ) (h' : toFun_1 (i + 1) < z),\n z < toFun_1 i →\n μ { left := toFun_1 (i + 1), right := z, lt := h' } <\n μ { left := toFun_1 (i + 1), right := toFun_1 i, lt := ⋯ }),\n ({ toFun := toFun, length := length, antitone := antitone, head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq, payoff_lt_of_between := payoff_lt_of_between } =\n { toFun := toFun_1, length := length_1, antitone := antitone_1, head_eq_top := head_eq_top_1,\n length_eq_bot := length_eq_bot_1, strictAntiOn := strictAntiOn_1, step_payoff_eq := step_payoff_eq_1,\n payoff_lt_of_between := payoff_lt_of_between_1 }) =\n (toFun = toFun_1 ∧ length = length_1)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "Set.Iic", "Nat.instPreorder", "Nat", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "StrictAntiOn"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.strictAntiOn", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration),\n StrictAntiOn self.toFun (Set.Iic self.length)", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "Preorder.toLT"], "name": "LT.lt.trans", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a < b → b < c → a < c", "constCategory": "Theorem"}, {"references": ["Preorder", "PartialOrder"], "name": "PartialOrder.toPreorder", "constType": "{α : Type u_2} → [self : PartialOrder α] → Preorder α", "constCategory": "Definition"}, {"references": ["outParam", "Membership"], "name": "Membership.mem", "constType": "{α : outParam (Type u)} → {γ : Type v} → [self : Membership α γ] → γ → α → Prop", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsStable", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["ConditionallyCompletePartialOrderSup", "ConditionallyCompletePartialOrder"], "name": "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "constType": "{α : Type u_3} → [self : ConditionallyCompletePartialOrder α] → ConditionallyCompletePartialOrderSup α", "constCategory": "Definition"}, {"references": [], "name": "PartialOrder", "constType": "Type u_2 → Type u_2", "constCategory": "Other"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.min", "BoundedOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.max", "LE.le", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_top_eq_max_top_of_B_top_le_A_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.WeakACC] [μ.WeakSlopeLikeAtTop]\n [μ.StrongDCC] [μ.WeakSlopeLikeAtBot], μ.B ⊤ ≤ μ.A ⊤ → μ.min ⊤ = μ.max ⊤", "constCategory": "Theorem"}, {"references": ["HEq", "Eq"], "name": "eq_of_heq", "constType": "∀ {α : Sort u} {a a' : α}, a ≍ a' → a = a'", "constCategory": "Theorem"}, {"references": ["List"], "name": "List.cons", "constType": "{α : Type u} → α → List α → List α", "constCategory": "Other"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "Nontrivial", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing", "Submodule"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "constType": "∀ (R : Type u_2) [inst : CommRing R] (M : Type u_1) [Nontrivial M] [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M], Nontrivial (Submodule R M)", "constCategory": "Theorem"}, {"references": ["AddZeroClass", "SMulZeroClass", "AddZeroClass.toAddZero", "AddZero.toZero", "DistribSMul"], "name": "DistribSMul.toSMulZeroClass", "constType": "{M : Type u_12} → {A : Type u_13} → {inst : AddZeroClass A} → [self : DistribSMul M A] → SMulZeroClass M A", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.instOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nontrivial", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop"], "name": "HarderNarasimhan.PayoffFunction.A_top_le_B_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} [μ.WeakACC] [μ.WeakSlopeLikeAtTop],\n μ.A ⊤ ≤ μ.B ⊤", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Subtype", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Subtype.preorder", "HarderNarasimhan.StrictIntvl", "PartialOrder", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Eq", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, (μ.restrict I).A = μ.A.restrict I", "constCategory": "Theorem"}, {"references": ["LinearOrder.toPartialOrder", "Nat", "PartialOrder", "inferInstance", "Nat.instLinearOrder"], "name": "Nat.instPartialOrder", "constType": "PartialOrder ℕ", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "Nat.instZeroLEOneClass", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_strictMonoOn", "Nat.instAddCommMonoid", "Nat.instIsOrderedAddMonoid", "SemilatticeInf.toPartialOrder", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "HarderNarasimhan.StrictIntvl", "Nat", "Nontrivial", "Lattice", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "HarderNarasimhan.PayoffFunction.ADCC", "AddMonoid.toAddZeroClass", "HarderNarasimhan.PayoffFunction.Admissible", "Subtype", "Subtype.partialOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "instOfNatNat", "Nat.instIsOrderedCancelAddMonoid", "WellFoundedGT", "Preorder.toLE", "LT.lt.le", "Nat.instAddMonoid", "Lattice.toSemilatticeInf", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNlen", "CompleteLattice"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil_piecewise_isSemistable", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible] (i : ℕ) (hi : i < HarderNarasimhan.PayoffFunction.HNlen✝ μ),\n (μ.restrict\n { left := HarderNarasimhan.PayoffFunction.HNFil✝ μ i, right := HarderNarasimhan.PayoffFunction.HNFil✝ μ (i + 1),\n lt := ⋯ }).IsSemistable", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "Subtype.lattice._proof_3", "Subtype.lattice._proof_1", "SemilatticeInf.toMin", "Lattice.toSemilatticeInf", "Subtype", "Subtype.lattice._proof_4", "Subtype.semilatticeSup", "SemilatticeInf.inf", "SemilatticeSup.sup", "Lattice.mk", "Subtype.lattice._proof_5", "Subtype.semilatticeInf", "Subtype.lattice._proof_6", "SemilatticeInf", "Max.max", "Min.min", "Lattice", "SemilatticeSup.toMax", "SemilatticeSup", "Subtype.lattice._proof_2", "SemilatticeSup.mk", "SemilatticeInf.toPartialOrder"], "name": "Subtype.lattice", "constType": "{α : Type u} →\n [inst : Lattice α] →\n {P : α → Prop} → (∀ ⦃x y : α⦄, P x → P y → P (x ⊔ y)) → (∀ ⦃x y : α⦄, P x → P y → P (x ⊓ y)) → Lattice { x // P x }", "constCategory": "Definition"}, {"references": ["AddZeroClass", "AddZero"], "name": "AddZeroClass.toAddZero", "constType": "{M : Type u} → [self : AddZeroClass M] → AddZero M", "constCategory": "Definition"}, {"references": ["semiOutParam"], "name": "CoeOut", "constType": "Sort u → semiOutParam (Sort v) → Sort (max (max 1 u) v)", "constCategory": "Other"}, {"references": ["Bot"], "name": "Bot.bot", "constType": "{α : Type u_1} → [self : Bot α] → α", "constCategory": "Definition"}, {"references": ["lt_trans", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "SemilatticeSup.toPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "Lattice.toSemilatticeInf", "CompleteLattice.toLattice", "HarderNarasimhan.StrictIntvl.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "LE.le", "Lattice", "Top.top", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice.toBoundedOrder", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.A_le_of_A_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I →\n ∀ {x z : ℒ},\n x ∈ I →\n z ∈ I →\n ∀ (h : x < z),\n μ.A { left := x, right := z, lt := h } = ⊤ →\n ∀ {a : ℒ},\n a ∈ I →\n ∀ (hax : a < x), μ.A { left := a, right := x, lt := hax } ≤ μ.A { left := a, right := z, lt := ⋯ }", "constCategory": "Theorem"}, {"references": [], "name": "Nat", "constType": "Type", "constCategory": "Other"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "inf_lt_left", "SemilatticeInf.toPartialOrder", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "Max.max", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "Lattice", "LE.le", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : Lattice ℒ} {inst_1 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} [self : μ.IsConvexOn I] (x y : ℒ),\n x ∈ I → y ∈ I → ∀ (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["AddZero", "Zero"], "name": "AddZero.toZero", "constType": "{M : Type u_2} → [self : AddZero M] → Zero M", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.StrictIntvl.right", "Iff", "LE.le", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "And", "LE", "LT"], "name": "HarderNarasimhan.StrictIntvl.mem_def", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] [inst_1 : LE ℒ] {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ},\n x ∈ I ↔ I.left ≤ x ∧ x ≤ I.right", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "DedekindCut.instCompleteLinearOrder", "wellFoundedGT", "HarderNarasimhan.PayoffFunction.instIsConvexOfIsConvexOnTopStrictIntvl", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "instLTNat", "SetLike.instMembership", "Submodule.addCommGroup", "Submodule.Quotient.addCommMonoid", "Set.instMembership", "HarderNarasimhan.PayoffFunction.instAdmissible", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "Nontrivial", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "ConditionallyCompleteLattice.toLattice", "Finset.Colex.instLinearOrder", "CompleteLattice.toBoundedOrder", "Submodule.submoduleOf", "OrderHom.instFunLike", "Submodule.hasQuotient", "associatedPrimes", "Subtype", "Submodule.instNontrivial", "HasQuotient.Quotient", "Module", "OrderHom", "Submodule.module", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "DFunLike.coe", "Submodule", "instDistribLatticeOfLinearOrder", "Ideal", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "instOfNatNat", "isNoetherian_of_isNoetherianRing_of_finite", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.asIdeal", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "HarderNarasimhan.Coprimary.instIsConvexOnSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoffTopStrictIntvl", "CompleteLattice.toConditionallyCompleteLattice", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "Set", "instHAdd", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "toLinearExtension", "DedekindCut", "AddCommGroup", "CommRing", "OfNat.ofNat", "PrimeSpectrum", "LT.lt", "HAdd.hAdd", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Submodule.setLike", "LE.le", "Submodule.completeLattice", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_1", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M] (i : ℕ),\n i + 1 < (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.length →\n ∀ (p q : PrimeSpectrum R),\n p.asIdeal ∈\n associatedPrimes R\n (↥((HarderNarasimhan.Coprimary.payoff R M).hnFiltration (i + 2)) ⧸\n ((HarderNarasimhan.Coprimary.payoff R M).hnFiltration (i + 1)).submoduleOf\n ((HarderNarasimhan.Coprimary.payoff R M).hnFiltration (i + 2))) →\n q.asIdeal ∈\n associatedPrimes R\n (↥((HarderNarasimhan.Coprimary.payoff R M).hnFiltration (i + 1)) ⧸\n ((HarderNarasimhan.Coprimary.payoff R M).hnFiltration i).submoduleOf\n ((HarderNarasimhan.Coprimary.payoff R M).hnFiltration (i + 1))) →\n toLinearExtension p < toLinearExtension q", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "Ideal", "Set.ofPred", "Set", "Module", "CommSemiring.toSemiring", "CommSemiring", "IsAssociatedPrime"], "name": "associatedPrimes", "constType": "(R : Type u_1) →\n [inst : CommSemiring R] → (M : Type u_2) → [inst_1 : AddCommMonoid M] → [_root_.Module R M] → Set (Ideal R)", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Module", "Nontrivial", "Submodule", "Semiring"], "name": "Submodule.instNontrivial", "constType": "∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : _root_.Module R M]\n [Nontrivial M], Nontrivial (Submodule R M)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "Nontrivial", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.apply_length", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration), F F.length = ⊥", "constCategory": "Theorem"}, {"references": ["Preorder"], "name": "OrderHom", "constType": "(α : Type u_6) → (β : Type u_7) → [Preorder α] → [Preorder β] → Type (max u_6 u_7)", "constCategory": "Other"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.StrictIntvl.instPartialOrder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "Eq", "OrderTop.toTop", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.isConvexOn_top_iff._simp_1", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [inst_2 : Nontrivial ℒ] [inst_3 : BoundedOrder ℒ],\n μ.IsConvexOn ⊤ = μ.IsConvex", "constCategory": "Theorem"}, {"references": ["LT.lt", "PartialOrder.toPreorder", "PartialOrder", "LE.le", "Preorder.toLT", "Ne", "Preorder.toLE"], "name": "lt_of_le_of_ne", "constType": "∀ {α : Type u_1} [inst : PartialOrder α] {a b : α}, a ≤ b → a ≠ b → a < b", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "Finset", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "instLinearOrderLinearExtensionOfPartialOrder", "Submodule.Quotient.module", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.restrict", "Module.IsNoetherian.finite", "PrimeSpectrum.instPartialOrder", "SemilatticeInf.toPartialOrder", "SetLike.instMembership", "Submodule.addCommGroup", "Ring.toSemiring", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "isNoetherian_submodule'", "HarderNarasimhan.StrictIntvl", "Iff", "AddCommGroup.toAddCommMonoid", "CompleteLattice.toBoundedOrder", "Submodule.submoduleOf", "Submodule.hasQuotient", "Submodule.instNontrivial", "Subtype", "HasQuotient.Quotient", "Submodule.Quotient.instSMul._proof_1", "Module.Finite.quotient", "Module", "Submodule.module", "Subtype.partialOrder", "Submodule", "Submodule.Quotient.addCommGroup", "instDistribLatticeOfLinearOrder", "isNoetherian_of_isNoetherianRing_of_finite", "HarderNarasimhan.Coprimary.payoff", "Colex", "Concept.instCompleteLattice", "Preorder.toLE", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "instDistribOfSemiring", "Distrib.toMul", "HarderNarasimhan.StrictIntvl.instMembership", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "instSMulOfMul", "DedekindCut", "AddCommGroup", "CommRing", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "PrimeSpectrum", "CommRing.toRing", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "Submodule.setLike", "LE.le", "Submodule.completeLattice", "HarderNarasimhan.Coprimary.nontrivial_quotient_of_lt", "Submodule.addCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.isSemistable_restrict_iff_quotient", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (N₁ N₂ : Submodule R M) (hN : N₁ < N₂),\n ((HarderNarasimhan.Coprimary.payoff R M).restrict { left := N₁, right := N₂, lt := hN }).IsSemistable ↔\n (HarderNarasimhan.Coprimary.payoff R (↥N₂ ⧸ N₁.submoduleOf N₂)).IsSemistable", "constCategory": "Theorem"}, {"references": ["DFunLike", "outParam"], "name": "DFunLike.coe", "constType": "{F : Sort u_1} → {α : outParam (Sort u_2)} → {β : outParam (α → Sort u_3)} → [self : DFunLike F α β] → F → (a : α) → β a", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "heq_of_eq", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Eq.refl", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "id", "HEq", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "HEq.refl", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.noConfusion", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk.noConfusion", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : PartialOrder ℒ} →\n {inst_1 : BoundedOrder ℒ} →\n {inst_2 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {P : Sort u} →\n {toFun : ℕ → ℒ} →\n {length : ℕ} →\n {monotone : Monotone toFun} →\n {head_eq_bot : toFun 0 = ⊥} →\n {length_eq_top : toFun length = ⊤} →\n {strictMonoOn : StrictMonoOn toFun (Set.Iic length)} →\n {piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable} →\n {not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }} →\n {toFun' : ℕ → ℒ} →\n {length' : ℕ} →\n {monotone' : Monotone toFun'} →\n {head_eq_bot' : toFun' 0 = ⊥} →\n {length_eq_top' : toFun' length' = ⊤} →\n {strictMonoOn' : StrictMonoOn toFun' (Set.Iic length')} →\n {piecewise_isSemistable' :\n ∀ (i : ℕ) (hi : i < length'),\n (μ.restrict\n { left := toFun' i, right := toFun' (i + 1),\n lt := ⋯ }).IsSemistable} →\n {not_A_le_succ' :\n ∀ (i : ℕ) (hi : i + 1 < length'),\n ¬μ.A { left := toFun' i, right := toFun' (i + 1), lt := ⋯ } ≤\n μ.A\n { left := toFun' (i + 1), right := toFun' (i + 2), lt := ⋯ }} →\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn,\n piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ } =\n { toFun := toFun', length := length', monotone := monotone',\n head_eq_bot := head_eq_bot', length_eq_top := length_eq_top',\n strictMonoOn := strictMonoOn',\n piecewise_isSemistable := piecewise_isSemistable',\n not_A_le_succ := not_A_le_succ' } →\n (toFun ≍ toFun' → length = length' → P) → P", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration._sizeOf_1", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "SizeOf", "SizeOf.mk", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration._sizeOf_inst", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : PartialOrder ℒ} →\n {inst_1 : BoundedOrder ℒ} →\n {inst_2 : CompleteLattice S} →\n (μ : HarderNarasimhan.PayoffFunction ℒ S) → [SizeOf ℒ] → [SizeOf S] → SizeOf μ.HarderNarasimhanFiltration", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "instHAdd", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "OfNat.ofNat", "HAdd.hAdd", "LT.lt", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "BoundedOrder.toOrderBot", "instOfNatNat", "PartialOrder", "Nontrivial", "Ne", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.succ_lt_of_ne_bot", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n F m ≠ ⊥ → F (m + 1) < F m", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "HarderNarasimhan.PayoffFunction", "DFunLike.coe", "LT"], "name": "HarderNarasimhan.PayoffFunction.ext", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : LT ℒ] {μ ν : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (I : HarderNarasimhan.StrictIntvl ℒ), μ I = ν I) → μ = ν", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "OfNat.ofNat", "Nat", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.head_eq_top", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration),\n self.toFun 0 = ⊤", "constCategory": "Theorem"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.lt", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }) →\n μ.WeakSlopeLikeAtTop", "constCategory": "Other"}, {"references": ["IsNoetherian", "IsNoetherianRing", "Module", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "Module.Finite", "Ring.toSemiring", "Ring"], "name": "isNoetherian_of_isNoetherianRing_of_finite", "constType": "∀ (R : Type u_1) (M : Type u_2) [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n [IsNoetherianRing R] [Module.Finite R M], IsNoetherian R M", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction._sizeOf_1", "HarderNarasimhan.PayoffFunction", "SizeOf", "LT", "SizeOf.mk"], "name": "HarderNarasimhan.PayoffFunction._sizeOf_inst", "constType": "(ℒ : Type u_1) → {inst : LT ℒ} → (S : Type u_2) → [SizeOf ℒ] → [SizeOf S] → SizeOf (HarderNarasimhan.PayoffFunction ℒ S)", "constCategory": "Definition"}, {"references": ["lt_trans", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.lt", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }) →\n μ.WeakSlopeLikeAtBot", "constCategory": "Definition"}, {"references": ["Set.ofPred", "Set", "Membership.mem", "Set.instMembership"], "name": "Set.preimage", "constType": "{α : Type u} → {β : Type v} → (α → β) → Set β → Set α", "constCategory": "Definition"}, {"references": ["Exists", "Set", "Membership.mem", "Set.instMembership"], "name": "Set.Nonempty", "constType": "{α : Type u} → Set α → Prop", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.exists_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} {inst : PartialOrder ℒ} {inst_1 : BoundedOrder ℒ} {inst_2 : CompleteLattice S}\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [self : μ.StrongDCC] (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Set.ofPred", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "HarderNarasimhan.PayoffFunction.jordanHolderRel.match_1", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "And", "BoundedOrder", "SetRel", "LT.lt", "Prod", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.jordanHolderRel", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] →\n [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → SetRel ℒ ℒ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "Finset", "PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "Submodule", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "instDistribLatticeOfLinearOrder", "instOfNatNat", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "OrderBot.toBot", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "Eq", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "DedekindCut", "AddCommGroup", "CommRing", "Bot.bot", "OfNat.ofNat", "PrimeSpectrum", "Nat", "IsNoetherianRing", "BoundedOrder.toOrderBot", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "Submodule.completeLattice", "AddCommGroup.toAddCommMonoid", "ConditionallyCompleteLattice.toLattice", "CompleteLattice.toBoundedOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_7", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.toFun 0 = ⊥", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsConvex.mk", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "HarderNarasimhan.PayoffFunction.IsConvex.rec", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsConvex → Sort u} →\n (t : μ.IsConvex) →\n ((le :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Mul", "Distrib"], "name": "Distrib.toMul", "constType": "{R : Type u_1} → [self : Distrib R] → Mul R", "constCategory": "Definition"}, {"references": ["SMul.mk", "SMul", "Mul", "Mul.mul"], "name": "instSMulOfMul", "constType": "{α : Type u} → [Mul α] → SMul α α", "constCategory": "Definition"}, {"references": ["Semiring.toNonAssocSemiring", "RingHomSurjective", "RingHom.id", "Semiring"], "name": "RingHomSurjective.ids", "constType": "∀ {R₁ : Type u_1} [inst : Semiring R₁], RingHomSurjective (RingHom.id R₁)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.PayoffFunction.B", "HarderNarasimhan.PayoffFunction.min", "BoundedOrder", "Bot.bot", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.StrictIntvl.instOrderTop", "Iff.mpr", "Iff", "HarderNarasimhan.PayoffFunction.max", "LE.le", "Nontrivial", "Top.top", "HarderNarasimhan.PayoffFunction.A", "Ne", "lt_top_iff_ne_top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_top_le_A_top_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S},\n μ.B ⊤ ≤ μ.A ⊤ ↔\n ∀ (x : ℒ) (hx : x ≠ ⊤) (y : ℒ) (hy : ⊥ < y),\n μ.min { left := ⊥, right := y, lt := hy } ≤ μ.max { left := x, right := ⊤, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Submodule.hasQuotient", "CommRing.toCommSemiring", "SetLike.instMembership", "Subtype", "PartialOrder.toPreorder", "HasQuotient.Quotient", "Module", "Submodule.module", "Submodule.addCommGroup", "Membership.mem", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "Preorder.toLT", "AddCommGroup", "CommRing", "Submodule", "LT.lt", "CommRing.toRing", "Submodule.setLike", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf"], "name": "HarderNarasimhan.Coprimary.nontrivial_quotient_of_lt", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n {N₁ N₂ : Submodule R M}, N₁ < N₂ → Nontrivial (↥N₂ ⧸ N₁.submoduleOf N₂)", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Finset", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Preorder.toLT", "Submodule", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Colex", "Eq", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.payoff.congr_simp", "constType": "∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_2) [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M],\n HarderNarasimhan.Coprimary.payoff R M = HarderNarasimhan.Coprimary.payoff R M", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "HarderNarasimhan.PayoffFunction.IsSemistable", "HarderNarasimhan.PayoffFunction.IsSlopeLike", "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "Preorder.toLT", "BoundedOrder", "Nonempty", "CompleteLinearOrder.toCompletelyDistribLattice", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "Nontrivial", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "CompleteLinearOrder", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.instNonemptyJordanHolderFiltration", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hacc : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLinearOrder S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [hsl : μ.IsSlopeLike]\n [hftp : μ.FiniteTotalPayoff] [hdc : μ.EventuallyTopDCC] [hst : μ.IsSemistable], Nonempty μ.JordanHolderFiltration", "constCategory": "Theorem"}, {"references": ["Top", "LE", "OrderTop"], "name": "OrderTop.toTop", "constType": "{α : Type u} → {inst : LE α} → [self : OrderTop α] → Top α", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.strictAntiOn", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Nat.instOne", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Nat.instIsOrderedAddMonoid", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "instLTNat", "Nat.instAddMonoid", "Nat.instPartialOrder", "instHAdd", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HAdd.hAdd", "lt_add_one", "Nat", "HarderNarasimhan.StrictIntvl", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.payoff_lt_of_between", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.JordanHolderFiltration) (i : ℕ)\n (hi : i < self.length) (z : ℒ) (h' : self.toFun (i + 1) < z),\n z < self.toFun i →\n μ { left := self.toFun (i + 1), right := z, lt := h' } <\n μ { left := self.toFun (i + 1), right := self.toFun i, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Nat", "Nat.add", "Add", "Add.mk"], "name": "instAddNat", "constType": "Add ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_nonempty", "Submodule", "Finset.min'", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "CommRing.toCommSemiring", "LinearExtension", "Set", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Set.toFinset", "AddCommGroup", "CommRing", "Set.instMembership", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "AddCommGroup.toAddCommMonoid", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.min'_mem_subquotientAssociatedPrimes", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I).toFinset.min' ⋯ ∈\n HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.Admissible", "PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "BoundedOrder", "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.HNFil", "HarderNarasimhan.PayoffFunction.IsConvex", "OfNat.ofNat", "Nat", "instOfNatNat", "Nontrivial", "Lattice", "WellFoundedGT", "HarderNarasimhan.PayoffFunction.ADCC", "Eq", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "_private.HarderNarasimhan.Filtration.Exists.0.HarderNarasimhan.PayoffFunction.hnFiltration._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] [hwf : WellFoundedGT ℒ]\n {S : Type u_2} [inst_3 : CompleteLattice S] (μ : HarderNarasimhan.PayoffFunction ℒ S) [inst_4 : μ.ADCC]\n [inst_5 : μ.IsConvex] [hadm : μ.Admissible],\n HarderNarasimhan.PayoffFunction.HNFil✝ μ 0 = HarderNarasimhan.PayoffFunction.HNFil✝ μ 0", "constCategory": "Theorem"}, {"references": ["lt_trans", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.rec", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtBot → Sort u} →\n (t : μ.WeakSlopeLikeAtBot) →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "PartialOrder.toPreorder", "LinearExtension", "Module", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Preorder.toLT", "Set.toFinset", "AddCommGroup", "CommRing", "Submodule", "PrimeSpectrum", "IsNoetherianRing", "HarderNarasimhan.StrictIntvl", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Finset.Nonempty", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_nonempty", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I).toFinset.Nonempty", "constCategory": "Theorem"}, {"references": ["LE", "BoundedOrder", "OrderTop", "OrderBot"], "name": "BoundedOrder.mk", "constType": "{α : Type u} → [inst : LE α] → [toOrderTop : OrderTop α] → [toOrderBot : OrderBot α] → BoundedOrder α", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.toFun", "constType": "{ℒ : Type u_1} →\n [inst : LT ℒ] → {S : Type u_2} → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.StrictIntvl ℒ → S", "constCategory": "Definition"}, {"references": ["instAddNat", "Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "DFunLike.coe", "Nat.instPreorder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "Nat.instIsOrderedAddMonoid", "Eq", "Nat.instAddMonoid", "Exists", "instHAdd", "Nat.instPartialOrder", "CompleteLattice.toLattice", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "AddZero.toAdd", "OfNat.ofNat", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "StrictAnti", "HarderNarasimhan.StrictIntvl", "Nat", "lt_add_one", "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk", "One.toOfNat1", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "instIsLeftCancelAddOfAddLeftReflectLE", "Top.top", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "HarderNarasimhan.PayoffFunction", "CompleteLattice.toBoundedOrder", "OrderTop.toTop", "CompleteLattice", "AddMonoid.toAddZeroClass"], "name": "HarderNarasimhan.PayoffFunction.EventuallyTopDCC.mk._flat_ctor", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (hx : StrictAnti x), ∃ N, μ { left := x (N + 1), right := x N, lt := ⋯ } = ⊤) → μ.EventuallyTopDCC", "constCategory": "Definition"}, {"references": ["SupSet.sSup", "SupSet", "Set.range"], "name": "iSup", "constType": "{α : Type u} → {ι : Sort v} → [SupSet α] → (ι → α) → α", "constCategory": "Definition"}, {"references": ["ConditionallyCompleteLattice", "ConditionallyCompleteLinearOrder"], "name": "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "constType": "{α : Type u_5} → [self : ConditionallyCompleteLinearOrder α] → ConditionallyCompleteLattice α", "constCategory": "Definition"}, {"references": ["Subtype.instLT", "HarderNarasimhan.StrictIntvl", "Subtype", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "CoeOut.mk", "Preorder.toLT", "HarderNarasimhan.StrictIntvl.ofSub", "Preorder.toLE", "CoeOut"], "name": "HarderNarasimhan.StrictIntvl.instCoeOutSubtypeMem", "constType": "{ℒ : Type u_1} →\n [inst : PartialOrder ℒ] →\n {I : HarderNarasimhan.StrictIntvl ℒ} →\n CoeOut (HarderNarasimhan.StrictIntvl { x // x ∈ I }) (HarderNarasimhan.StrictIntvl ℒ)", "constCategory": "Definition"}, {"references": ["IsScalarTower", "instDistribOfSemiring", "Distrib.toMul", "Algebra.toSMul", "CommSemiring", "instSMulOfMul", "Algebra", "Semiring"], "name": "IsScalarTower.right", "constType": "∀ {R : Type u} {A : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A], IsScalarTower R A A", "constCategory": "Theorem"}, {"references": ["Eq.rec", "Eq"], "name": "Eq.ndrec", "constType": "{α : Sort u2} → {a : α} → {motive : α → Sort u1} → motive a → {b : α} → a = b → motive b", "constCategory": "Definition"}, {"references": ["lt_trans", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.rec", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.mk", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "Bot.bot", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "LE.le", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtBot.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtBot → Sort u} →\n (t : μ.WeakSlopeLikeAtBot) →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : ⊥ < z.left),\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ z ∨\n μ { left := ⊥, right := z.right, lt := ⋯ } ≤ μ { left := ⊥, right := z.left, lt := hz }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Inter", "Set", "Set.inter", "Inter.mk"], "name": "Set.instInter", "constType": "{α : Type u} → Inter (Set α)", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice", "CompleteLinearOrder.toConditionallyCompleteLinearOrderBot", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "CompleteLinearOrder", "Preorder.toLE", "SemilatticeInf.toPartialOrder", "HarderNarasimhan.PayoffFunction.B", "Lattice.toSemilatticeInf", "BoundedOrder", "CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.instOrderTop", "Lattice", "Nontrivial", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction.A", "ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.B_top_le_A_top", "constType": "∀ {ℒ : Type u_3} [inst : Nontrivial ℒ] [inst_1 : Lattice ℒ] [inst_2 : BoundedOrder ℒ] {S : Type u_4}\n [inst_3 : CompleteLinearOrder S] {μ : HarderNarasimhan.PayoffFunction ℒ S}, μ.IsSemistable → μ.B ⊤ ≤ μ.A ⊤", "constCategory": "Theorem"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "HarderNarasimhan.StrictIntvl.instPartialOrder", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk._flat_ctor", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n (toFun : ℕ → ℒ) →\n (length : ℕ) →\n Antitone toFun →\n toFun 0 = ⊤ →\n toFun length = ⊥ →\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length)) →\n (∀ (i : ℕ) (hi : i < length), μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤) →\n (∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }) →\n μ.JordanHolderFiltration", "constCategory": "Definition"}, {"references": ["Semiring.toMonoid", "CommSemiring.toSemiring", "CommMonoid.mk", "CommSemiring", "CommSemiring.mul_comm", "CommMonoid"], "name": "CommSemiring.toCommMonoid", "constType": "{R : Type u} → [self : CommSemiring R] → CommMonoid R", "constCategory": "Definition"}, {"references": ["MulOneClass"], "name": "Submonoid", "constType": "(M : Type u_3) → [MulOneClass M] → Type u_3", "constCategory": "Other"}, {"references": ["RingHom", "Submodule.toAddSubmonoid", "LinearMap.instFunLike", "Module", "Membership.mem", "Submodule.comap._proof_2", "AddCommMonoid.toAddMonoid", "DFunLike.coe", "Submodule", "AddSubsemigroup.mk", "AddCommMonoid", "Semiring.toNonAssocSemiring", "Submodule.comap._proof_3", "Set.preimage", "Set", "AddSubmonoid.mk", "LinearMap", "AddZero.toAdd", "AddZeroClass.toAddZero", "AddSubmonoid.comap", "Set.instMembership", "Submodule.map._proof_2", "Submodule.comap._proof_1", "SetLike.coe", "AddSubmonoid", "Submodule.mk", "Submodule.setLike", "AddMonoid.toAddZeroClass", "Semiring"], "name": "Submodule.comap", "constType": "{R : Type u_1} →\n {R₂ : Type u_3} →\n {M : Type u_5} →\n {M₂ : Type u_7} →\n [inst : Semiring R] →\n [inst_1 : Semiring R₂] →\n [inst_2 : AddCommMonoid M] →\n [inst_3 : AddCommMonoid M₂] →\n [inst_4 : _root_.Module R M] →\n [inst_5 : _root_.Module R₂ M₂] → {σ₁₂ : R →+* R₂} → (M →ₛₗ[σ₁₂] M₂) → Submodule R₂ M₂ → Submodule R M", "constCategory": "Definition"}, {"references": ["instAddNat", "RelSeries.toFun", "instHAdd", "Fin.instOfNat", "RelSeries", "Fin", "SetRel", "RelSeries.head._proof_1", "OfNat.ofNat", "HAdd.hAdd", "Nat", "instOfNatNat", "RelSeries.length"], "name": "RelSeries.head", "constType": "{α : Type u_1} → {r : SetRel α α} → RelSeries r → α", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing"], "name": "HarderNarasimhan.IsCoprimary", "constType": "(R : Type u_1) → [inst : CommRing R] → (M : Type u_2) → [inst_1 : AddCommGroup M] → [_root_.Module R M] → Prop", "constCategory": "Other"}, {"references": ["Submodule.toAddSubmonoid", "Submodule.Quotient.module", "Membership.mem", "Inter.inter", "Algebra.id", "Semiring.toNonAssocSemiring", "RingHom.id", "DistribMulAction.toMulAction", "IsScalarTower.right", "Semiring.toModule", "Set.instInter", "LinearMap.ker", "LocalizedModule", "Submonoid.instSetLike", "Submodule.Quotient.addCommMonoid", "AddZeroClass.toAddZero", "CommSemiring.toCommMonoid", "Set.instMembership", "Submonoid", "SetLike.coe", "AddCommGroup.toAddCommMonoid", "AddSubmonoid.toAddSubsemigroup", "AddMonoid.toAddZeroClass", "Submodule.hasQuotient", "associatedPrimes", "HasQuotient.Quotient", "OreLocalization.oreSetComm", "Module", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "Set.instEmptyCollection", "EmptyCollection.emptyCollection", "Submodule", "LocalizedModule.mkLinearMap", "AddSubsemigroup.carrier", "Ideal", "instMulZeroOneClassOfSemiring", "Eq", "CommRing.toCommSemiring", "Semiring.toMonoid", "OreLocalization.instAddCommMonoidOreLocalization", "IsScalarTower.left", "Set", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "AddCommGroup", "CommRing", "AddZero.toAdd", "Semiring.toAddCommMonoid", "CommRing.toRing", "Module.toDistribMulAction"], "name": "HarderNarasimhan.inter_eq_empty_of_mem_associatedPrimes_quot_ker", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (S : Submonoid R) {p : Ideal R},\n p ∈ associatedPrimes R (M ⧸ (LocalizedModule.mkLinearMap S M).ker) → p.carrier ∩ ↑S = ∅", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "PartialOrder.toPreorder", "CompleteLattice.toLattice", "SemilatticeSup.toPartialOrder", "BoundedOrder", "Preorder.toLE", "CompleteLattice"], "name": "CompleteLattice.toBoundedOrder", "constType": "{α : Type u_8} → [self : CompleteLattice α] → BoundedOrder α", "constCategory": "Definition"}, {"references": ["Equiv.instEquivLike", "PartialOrder.toPreorder", "Finset", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Preorder.toLT", "DFunLike.coe", "Equiv", "Submodule", "instDistribLatticeOfLinearOrder", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Colex", "Eq", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "toColex", "Lattice.toSemilatticeInf", "LinearExtension", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Set.toFinset", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "DedekindCut.principal", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.payoff_apply", "constType": "∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_2) [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (HarderNarasimhan.Coprimary.payoff R M) I =\n DedekindCut.principal (toColex (HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I).toFinset)", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "PartialOrder.toPreorder", "Module", "CommSemiring.toSemiring", "Submodule.instPartialOrder", "Preorder.toLT", "AddCommGroup", "CommRing", "Submodule", "IsNoetherianRing", "AddCommGroup.toAddCommMonoid", "WellFoundedGT", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "constType": "∀ (R : Type u_2) [inst : CommRing R] [IsNoetherianRing R] (M : Type u_1) [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [Module.Finite R M], WellFoundedGT (Submodule R M)", "constCategory": "Theorem"}, {"references": ["FunLike", "DFunLike.mk", "Preorder", "OrderHom.toFun", "OrderHom.instFunLike._proof_1", "OrderHom", "Eq"], "name": "OrderHom.instFunLike", "constType": "{α : Type u_2} → {β : Type u_3} → [inst : Preorder α] → [inst_1 : Preorder β] → FunLike (α →o β) α β", "constCategory": "Definition"}, {"references": ["CompletelyDistribLattice.toCompleteLattice", "HarderNarasimhan.StrictIntvl", "Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Lattice", "Preorder.toLT", "CompleteLinearOrder.toCompletelyDistribLattice", "CompleteLinearOrder", "Eq", "HarderNarasimhan.PayoffFunction", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsBreakpoint.eq", "constType": "∀ {ℒ : Type u_1} [inst : Lattice ℒ] {S : Type u_3} [inst_1 : CompleteLinearOrder S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x y : ℒ},\n μ.IsBreakpoint I x → μ.IsBreakpoint I y → x = y", "constCategory": "Theorem"}, {"references": ["Quotient", "HasQuotient.mk", "Module", "Submodule.quotientRel", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "HasQuotient", "Submodule", "Ring.toSemiring", "Ring"], "name": "Submodule.hasQuotient", "constType": "{R : Type u_1} →\n {M : Type u_2} →\n [inst : Ring R] → [inst_1 : AddCommGroup M] → [inst_2 : _root_.Module R M] → HasQuotient M (Submodule R M)", "constCategory": "Definition"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "LT.lt", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.ofSub._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}\n (J : HarderNarasimhan.StrictIntvl { x // x ∈ I }), ↑J.left < ↑J.right", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.EventuallyTopDCC", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.PayoffFunction.StrongDCC", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.instStrongDCCOfEventuallyTopDCC", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [h : μ.EventuallyTopDCC], μ.StrongDCC", "constCategory": "Theorem"}, {"references": ["SubtractionMonoid", "SubtractionCommMonoid"], "name": "SubtractionCommMonoid.toSubtractionMonoid", "constType": "{G : Type u} → [self : SubtractionCommMonoid G] → SubtractionMonoid G", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "Eq", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsSemistable.congr_simp", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] (μ μ_1 : HarderNarasimhan.PayoffFunction ℒ S),\n μ = μ_1 → μ.IsSemistable = μ_1.IsSemistable", "constCategory": "Theorem"}, {"references": ["Equiv.refl", "OrderDual", "Equiv"], "name": "OrderDual.ofDual", "constType": "{α : Type u_1} → αᵒᵈ ≃ α", "constCategory": "Definition"}, {"references": ["CommRing.toCommSemiring", "IsNoetherianRing", "Module", "Nontrivial", "CommSemiring.toSemiring", "AddCommGroup.toAddCommMonoid", "AddCommGroup", "CommRing", "Module.Finite"], "name": "HarderNarasimhan.CoprimaryFiltration", "constType": "(R : Type u_1) →\n [inst : CommRing R] →\n [IsNoetherianRing R] →\n (M : Type u_2) →\n [Nontrivial M] → [inst_3 : AddCommGroup M] → [inst_4 : _root_.Module R M] → [Module.Finite R M] → Type u_2", "constCategory": "Other"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Submodule", "HarderNarasimhan.CoprimaryFiltration.toFun", "IsNoetherianRing", "Nat", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Module.Finite", "Function.Injective"], "name": "HarderNarasimhan.CoprimaryFiltration.instFunLikeNatSubmodule._proof_1", "constType": "∀ {R : Type u_2} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_1} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n Function.Injective HarderNarasimhan.CoprimaryFiltration.toFun", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "HarderNarasimhan.PayoffFunction.mk", "PartialOrder.toPreorder", "AddCommGroup.toAddGroup", "Preorder.toLT", "SMulZeroClass.toSMul", "DedekindCut.instCompleteLinearOrder", "NNReal.instLinearOrder", "NNReal.instZero", "PartialOrder", "AddGroup.toSubNegMonoid", "CoheytingAlgebra.toOrderTop", "SemilatticeInf.toPartialOrder", "DistribSMul.toSMulZeroClass", "Real", "CoheytingAlgebra.toGeneralizedCoheytingAlgebra", "NNReal", "LinearOrder", "DistribMulAction.toDistribSMul", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl", "LinearOrder.toDecidableLT", "Real.instMonoid", "HSMul.hSMul", "AddCommGroup.toAddCommMonoid", "NNReal.instInv", "Top.top", "CompleteDistribLattice.toCoframe", "AddZero.toZero", "AddMonoid.toAddZeroClass", "GeneralizedCoheytingAlgebra.toLattice", "Module", "Order.Coframe.toCoheytingAlgebra", "SemilatticeSup.toPartialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "NNReal.instPartialOrder", "instDistribLatticeOfLinearOrder", "CompletelyDistribLattice.toCompleteDistribLattice", "NNReal.instSMulOfReal", "instHSMul", "Zero.toOfNat0", "Preorder.toLE", "Not", "Inv.inv", "Lattice.toSemilatticeInf", "AddCommGroup", "DedekindCut", "OfNat.ofNat", "Real.semiring", "LT.lt", "Module.toDistribMulAction", "DistribLattice.toLattice", "SubNegMonoid.toAddMonoid", "DedekindCut.principal", "HarderNarasimhan.PayoffFunction", "dite", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.slope", "constType": "{ℒ : Type u_1} →\n [inst : PartialOrder ℒ] →\n {V : Type u_2} →\n [inst_1 : AddCommGroup V] →\n [_root_.Module ℝ V] →\n [inst_3 : LinearOrder V] →\n (HarderNarasimhan.StrictIntvl ℒ → NNReal) →\n (HarderNarasimhan.StrictIntvl ℒ → V) → HarderNarasimhan.PayoffFunction ℒ (DedekindCut V)", "constCategory": "Definition"}, {"references": ["LT.lt", "IsWellFounded", "LT"], "name": "WellFoundedGT", "constType": "(α : Type u_1) → [LT α] → Prop", "constCategory": "Definition"}, {"references": ["InfSet", "InfSet.sInf", "Set.range"], "name": "iInf", "constType": "{α : Type u} → {ι : Sort v} → [InfSet α] → (ι → α) → α", "constCategory": "Definition"}, {"references": ["Exists", "HarderNarasimhan.PayoffFunction.min._proof_1", "Set", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Set.Ico", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsAttained", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Preorder ℒ] →\n [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → HarderNarasimhan.StrictIntvl ℒ → Prop", "constCategory": "Definition"}, {"references": ["Set.Finite.fintype", "Submodule.hasQuotient", "associatedPrimes", "PartialOrder.toPreorder", "Subtype", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Preorder.toLT", "Set.Elem", "Submodule", "Submodule.Quotient.addCommGroup", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "CommRing.toCommSemiring", "Set.preimage", "SetLike.instMembership", "LinearExtension", "HarderNarasimhan.StrictIntvl.right", "Submodule.addCommGroup", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "AddCommGroup", "Fintype", "CommRing", "PrimeSpectrum", "CommRing.toRing", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "Submodule.setLike", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite._proof_1", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n {M : Type u_2} →\n [inst_1 : AddCommGroup M] →\n [inst_2 : _root_.Module R M] →\n [IsNoetherianRing R] →\n [Module.Finite R M] →\n (I : HarderNarasimhan.StrictIntvl (Submodule R M)) →\n Fintype ↑(HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I)", "constCategory": "Definition"}, {"references": ["Top", "LE.le", "Top.top", "LE", "OrderTop"], "name": "OrderTop.mk", "constType": "{α : Type u} → [inst : LE α] → [toTop : Top α] → (∀ (a : α), a ≤ ⊤) → OrderTop α", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.rec", "Set.Iic", "Bot.bot", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.mk", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.JordanHolderFiltration → Sort u} →\n (t : μ.JordanHolderFiltration) →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (antitone : Antitone toFun) →\n (head_eq_top : toFun 0 = ⊤) →\n (length_eq_bot : toFun length = ⊥) →\n (strictAntiOn : StrictAntiOn toFun (Set.Iic length)) →\n (step_payoff_eq :\n ∀ (i : ℕ) (hi : i < length),\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ } = μ ⊤) →\n (payoff_lt_of_between :\n ∀ (i : ℕ) (hi : i < length) (z : ℒ) (h' : toFun (i + 1) < z),\n z < toFun i →\n μ { left := toFun (i + 1), right := z, lt := h' } <\n μ { left := toFun (i + 1), right := toFun i, lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, antitone := antitone,\n head_eq_top := head_eq_top, length_eq_bot := length_eq_bot,\n strictAntiOn := strictAntiOn, step_payoff_eq := step_payoff_eq,\n payoff_lt_of_between := payoff_lt_of_between }) →\n motive t", "constCategory": "Definition"}, {"references": ["Add", "Distrib"], "name": "Distrib.toAdd", "constType": "{R : Type u_1} → [self : Distrib R] → Add R", "constCategory": "Definition"}, {"references": ["lt_trans", "HarderNarasimhan.StrictIntvl.lt", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "Or", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.mk", "BoundedOrder.toOrderTop", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop", "CompleteLattice.toConditionallyCompleteLattice", "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.rec", "HarderNarasimhan.StrictIntvl.right", "BoundedOrder", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.WeakSlopeLikeAtTop.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakSlopeLikeAtTop → Sort u} →\n (t : μ.WeakSlopeLikeAtTop) →\n ((le_or_le :\n ∀ (z : HarderNarasimhan.StrictIntvl ℒ) (hz : z.right < ⊤),\n μ z ≤ μ { left := z.left, right := ⊤, lt := ⋯ } ∨\n μ { left := z.right, right := ⊤, lt := hz } ≤ μ { left := z.left, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Subtype", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "HarderNarasimhan.StrictIntvl.instLatticeSubtypeMem", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Iff", "Lattice", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.isConvexOn_iff_isConvex_restrict", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ},\n μ.IsConvexOn I ↔ (μ.restrict I).IsConvex", "constCategory": "Theorem"}, {"references": ["Nat", "OfNat"], "name": "OfNat.ofNat", "constType": "{α : Type u} → (x : ℕ) → [self : OfNat α x] → α", "constCategory": "Definition"}, {"references": ["Submodule.toAddSubmonoid", "Submodule.Quotient.module", "Membership.mem", "Inter.inter", "Algebra.id", "Semiring.toNonAssocSemiring", "RingHom.id", "DistribMulAction.toMulAction", "IsScalarTower.right", "Semiring.toModule", "Set.instInter", "LinearMap.ker", "LocalizedModule", "Submonoid.instSetLike", "Submodule.Quotient.addCommMonoid", "AddZeroClass.toAddZero", "CommSemiring.toCommMonoid", "Set.instMembership", "Submonoid", "SetLike.coe", "AddCommGroup.toAddCommMonoid", "AddSubmonoid.toAddSubsemigroup", "AddMonoid.toAddZeroClass", "Submodule.hasQuotient", "associatedPrimes", "HasQuotient.Quotient", "OreLocalization.oreSetComm", "Module", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "Set.instEmptyCollection", "EmptyCollection.emptyCollection", "Submodule", "LocalizedModule.mkLinearMap", "AddSubsemigroup.carrier", "Ideal", "instMulZeroOneClassOfSemiring", "Eq", "CommRing.toCommSemiring", "Semiring.toMonoid", "OreLocalization.instAddCommMonoidOreLocalization", "IsScalarTower.left", "Set", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "AddCommGroup", "CommRing", "AddZero.toAdd", "Semiring.toAddCommMonoid", "CommRing.toRing", "IsNoetherianRing", "Module.toDistribMulAction"], "name": "HarderNarasimhan.mem_associatedPrimes_of_mem_associatedPrimes_quot_ker", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (S : Submonoid R) [IsNoetherianRing R] {p : Ideal R},\n p ∈ associatedPrimes R (M ⧸ (LocalizedModule.mkLinearMap S M).ker) → p.carrier ∩ ↑S = ∅ → p ∈ associatedPrimes R M", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "Module", "CommSemiring.toSemiring", "AddCommGroup", "CommRing", "Inhabited.default", "IsNoetherianRing", "HarderNarasimhan.CoprimaryFiltration", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Eq", "HarderNarasimhan.Coprimary.instInhabitedCoprimaryFiltration", "Module.Finite"], "name": "_private.HarderNarasimhan.Coprimary.Filtration.0.HarderNarasimhan.CoprimaryFiltration.instUnique._proof_1", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M]\n (a : HarderNarasimhan.CoprimaryFiltration R M), a = default", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "HarderNarasimhan.StrictIntvl.right", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, I.right ∈ I", "constCategory": "Theorem"}, {"references": ["HAdd", "outParam"], "name": "HAdd.hAdd", "constType": "{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ", "constCategory": "Definition"}, {"references": ["FunLike", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "DFunLike.mk", "Nat", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat._proof_1", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} → FunLike μ.JordanHolderFiltration ℕ ℒ", "constCategory": "Definition"}, {"references": ["Nat", "Nat.succ", "LE.le", "instLENat"], "name": "Nat.le_of_succ_le", "constType": "∀ {n m : ℕ}, n.succ ≤ m → n ≤ m", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Set", "HarderNarasimhan.StrictIntvl.right", "And.right", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "DFunLike.coe", "Set.instMembership", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Preorder", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Set.Ico", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {u : ℒ}\n (hu : u ∈ Set.Ico I.left I.right), μ.min I ≤ μ { left := u, right := I.right, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instLTNat", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "LT.lt", "Nat", "BoundedOrder.toOrderBot", "PartialOrder", "Nontrivial", "Ne", "OrderBot.toBot", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.ne_bot_of_lt", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} {F : μ.JordanHolderFiltration} {m : ℕ},\n m < F.length → F m ≠ ⊥", "constCategory": "Theorem"}, {"references": ["LT.lt", "Preorder", "Set", "Membership.mem", "Preorder.toLT", "Set.instMembership"], "name": "StrictMonoOn", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Set α → Prop", "constCategory": "Definition"}, {"references": ["Prod"], "name": "Prod.mk", "constType": "{α : Type u} → {β : Type v} → α → β → α × β", "constCategory": "Other"}, {"references": ["Equiv.instEquivLike", "PartialOrder.toPreorder", "Finset", "Module", "instLinearOrderLinearExtensionOfPartialOrder", "Singleton.singleton", "Preorder.toLT", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes_nonempty", "DFunLike.coe", "Equiv", "Submodule", "instDistribLatticeOfLinearOrder", "Finset.min'", "EquivLike.toFunLike", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "HarderNarasimhan.Coprimary.instFintypeElemLinearExtensionPrimeSpectrumSubquotientAssociatedPrimesOfIsNoetherianRingOfFinite", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "Eq", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "toColex", "Lattice.toSemilatticeInf", "LinearExtension", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "Set.toFinset", "AddCommGroup", "DedekindCut", "CommRing", "PrimeSpectrum", "HarderNarasimhan.StrictIntvl", "IsNoetherianRing", "Finset.instSingleton", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "DedekindCut.principal", "LE.le", "AddCommGroup.toAddCommMonoid", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.A_payoff", "constType": "∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M]\n [inst_3 : _root_.Module R M] [inst_4 : Module.Finite R M] (I : HarderNarasimhan.StrictIntvl (Submodule R M)),\n (HarderNarasimhan.Coprimary.payoff R M).A I =\n DedekindCut.principal (toColex {(HarderNarasimhan.Coprimary.subquotientAssociatedPrimes I).toFinset.min' ⋯})", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HarderNarasimhanFiltration → Sort u} →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable) →\n (not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot,\n length_eq_top := length_eq_top, strictMonoOn := strictMonoOn,\n piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ }) →\n (t : μ.HarderNarasimhanFiltration) → motive t", "constCategory": "Other"}, {"references": ["Nat", "HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.ctorIdx", "constType": "{ℒ : Type u_1} → {inst : LT ℒ} → HarderNarasimhan.StrictIntvl ℒ → ℕ", "constCategory": "Definition"}, {"references": ["SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "lt_of_le_of_lt", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Min.min", "Iff.mpr", "HarderNarasimhan.PayoffFunction.max", "LE.le", "Lattice", "HarderNarasimhan.PayoffFunction.A", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.A_le_max_inf", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) {x w u : ℒ} (hxw : ¬x ≤ w) (huxw : u ≤ x ⊓ w),\n μ.A { left := u, right := x, lt := ⋯ } ≤ μ.max { left := x ⊓ w, right := x, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "Membership.mem", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.StrictIntvl.left", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "HarderNarasimhan.PayoffFunction.B", "Set", "HarderNarasimhan.StrictIntvl.right", "HarderNarasimhan.PayoffFunction.min", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "Set.Ioc", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.B_le", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {s : S},\n (∀ (b : ℒ) (hb : b ∈ Set.Ioc I.left I.right), μ.min { left := I.left, right := b, lt := ⋯ } ≤ s) → μ.B I ≤ s", "constCategory": "Theorem"}, {"references": ["LT.lt", "OrderDual", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "OrderDual.instLT", "HarderNarasimhan.StrictIntvl.right", "LT"], "name": "HarderNarasimhan.PayoffFunction.dual._proof_1", "constType": "∀ {ℒ : Type u_1} [inst : LT ℒ] (p : HarderNarasimhan.StrictIntvl ℒᵒᵈ), p.left < p.right", "constCategory": "Theorem"}, {"references": ["GeneralizedCoheytingAlgebra", "CoheytingAlgebra"], "name": "CoheytingAlgebra.toGeneralizedCoheytingAlgebra", "constType": "{α : Type u_4} → [self : CoheytingAlgebra α] → GeneralizedCoheytingAlgebra α", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.FiniteTotalPayoff", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [Nontrivial ℒ] →\n [inst : PartialOrder ℒ] → [BoundedOrder ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["DistribSMul.mk", "SMulZeroClass.mk", "DistribMulAction.smul_zero", "SemigroupAction.toSMul", "AddMonoid", "AddZeroClass.toAddZero", "DistribSMul", "DistribMulAction.smul_add", "Monoid", "DistribMulAction.toMulAction", "Monoid.toSemigroup", "AddZero.toZero", "MulAction.toSemigroupAction", "DistribMulAction", "AddMonoid.toAddZeroClass"], "name": "DistribMulAction.toDistribSMul", "constType": "{M : Type u_1} → {A : Type u_7} → [inst : Monoid M] → [inst_1 : AddMonoid A] → [DistribMulAction M A] → DistribSMul M A", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.instFunLikeNat", "Nat", "PartialOrder", "Iff", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.ext_iff", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {F G : μ.HarderNarasimhanFiltration}, F = G ↔ ∀ (n : ℕ), F n = G n", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder.toPreorder", "PartialOrder", "HarderNarasimhan.StrictIntvl.instMembership", "Membership.mem", "Preorder.toLT", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem._proof_3", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {I : HarderNarasimhan.StrictIntvl ℒ}, I.left ∈ I", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsBreakpoint", "Set", "Iff", "PartialOrder", "Membership.mem", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.breakpoints", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "Set.instMembership"], "name": "HarderNarasimhan.PayoffFunction.mem_breakpoints", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ} {x : ℒ},\n x ∈ μ.breakpoints I ↔ μ.IsBreakpoint I x", "constCategory": "Theorem"}, {"references": [], "name": "id", "constType": "{α : Sort u} → α → α", "constCategory": "Definition"}, {"references": ["ConditionallyCompleteLattice", "Lattice"], "name": "ConditionallyCompleteLattice.toLattice", "constType": "{α : Type u_5} → [self : ConditionallyCompleteLattice α] → Lattice α", "constCategory": "Definition"}, {"references": ["Nat", "PartialOrder.toPreorder", "PartialOrder", "Nontrivial", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.length", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Nontrivial ℒ] →\n [inst_1 : PartialOrder ℒ] →\n [inst_2 : BoundedOrder ℒ] →\n [inst_3 : CompleteLattice S] → {μ : HarderNarasimhan.PayoffFunction ℒ S} → μ.JordanHolderFiltration → ℕ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "Nat", "Monotone", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "Nat.instPreorder"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.monotone", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration), Monotone self.toFun", "constCategory": "Theorem"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.IsConvex.mk", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "SemilatticeSup.toMax", "LE.le", "Lattice", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.rec", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : Lattice ℒ] →\n [inst_1 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.IsConvex → Sort u} →\n ((le :\n ∀ (x y : ℒ) (h : ¬x ≤ y),\n μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n motive ⋯) →\n (t : μ.IsConvex) → motive t", "constCategory": "Other"}, {"references": ["Subtype", "SetLike.instMembership", "Module", "Submodule.module", "Membership.mem", "Submodule.subtype", "Submodule", "AddCommMonoid", "Semiring.toNonAssocSemiring", "Submodule.comap", "Submodule.setLike", "RingHom.id", "Submodule.addCommMonoid", "Semiring"], "name": "Submodule.submoduleOf", "constType": "{R : Type u_1} →\n {M : Type u_5} →\n [inst : Semiring R] →\n [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → Submodule R M → (q : Submodule R M) → Submodule R ↥q", "constCategory": "Definition"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "And", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk.injEq", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (toFun : ℕ → ℒ) (length : ℕ) (monotone : Monotone toFun)\n (head_eq_bot : toFun 0 = ⊥) (length_eq_top : toFun length = ⊤) (strictMonoOn : StrictMonoOn toFun (Set.Iic length))\n (piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length), (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable)\n (not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ })\n (toFun_1 : ℕ → ℒ) (length_1 : ℕ) (monotone_1 : Monotone toFun_1) (head_eq_bot_1 : toFun_1 0 = ⊥)\n (length_eq_top_1 : toFun_1 length_1 = ⊤) (strictMonoOn_1 : StrictMonoOn toFun_1 (Set.Iic length_1))\n (piecewise_isSemistable_1 :\n ∀ (i : ℕ) (hi : i < length_1), (μ.restrict { left := toFun_1 i, right := toFun_1 (i + 1), lt := ⋯ }).IsSemistable)\n (not_A_le_succ_1 :\n ∀ (i : ℕ) (hi : i + 1 < length_1),\n ¬μ.A { left := toFun_1 i, right := toFun_1 (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun_1 (i + 1), right := toFun_1 (i + 2), lt := ⋯ }),\n ({ toFun := toFun, length := length, monotone := monotone, head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ } =\n { toFun := toFun_1, length := length_1, monotone := monotone_1, head_eq_bot := head_eq_bot_1,\n length_eq_top := length_eq_top_1, strictMonoOn := strictMonoOn_1,\n piecewise_isSemistable := piecewise_isSemistable_1, not_A_le_succ := not_A_le_succ_1 }) =\n (toFun = toFun_1 ∧ length = length_1)", "constCategory": "Theorem"}, {"references": ["CommRing.toCommSemiring", "associatedPrimes", "Module", "Set", "CommSemiring.toSemiring", "Membership.mem", "AddCommGroup", "CommRing", "Set.instMembership", "Ideal", "HarderNarasimhan.IsCoprimary", "AddCommGroup.toAddCommMonoid", "ExistsUnique"], "name": "HarderNarasimhan.IsCoprimary.mk", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M],\n (∃! p, p ∈ associatedPrimes R M) → HarderNarasimhan.IsCoprimary R M", "constCategory": "Other"}, {"references": ["MulOneClass.toMulOne", "Semiring.toMonoid", "Module", "IsScalarTower", "instSMulOfMul", "SemigroupAction.toSMul", "AddCommGroup", "AddCommMonoid.toAddMonoid", "Ring.toSemiring", "Module.toDistribMulAction", "MulOne.toMul", "Monoid.toMulOneClass", "DistribMulAction.toMulAction", "AddCommGroup.toAddCommMonoid", "Monoid.toSemigroup", "MulAction.toSemigroupAction", "Ring"], "name": "Submodule.Quotient.instSMul._proof_1", "constType": "∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M],\n IsScalarTower R R M", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.StrictIntvl.instBoundedOrderSubtypeMem", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.IsSemistable", "StrictMonoOn", "Membership.mem", "HarderNarasimhan.PayoffFunction.restrict", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "Monotone", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.mk", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.rec", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Set.Iic", "Bot.bot", "HarderNarasimhan.StrictIntvl.instNontrivialSubtypeMem", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "Subtype", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Subtype.partialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "HarderNarasimhan.StrictIntvl.instMembership", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "LE.le", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.HarderNarasimhanFiltration → Sort u} →\n (t : μ.HarderNarasimhanFiltration) →\n ((toFun : ℕ → ℒ) →\n (length : ℕ) →\n (monotone : Monotone toFun) →\n (head_eq_bot : toFun 0 = ⊥) →\n (length_eq_top : toFun length = ⊤) →\n (strictMonoOn : StrictMonoOn toFun (Set.Iic length)) →\n (piecewise_isSemistable :\n ∀ (i : ℕ) (hi : i < length),\n (μ.restrict { left := toFun i, right := toFun (i + 1), lt := ⋯ }).IsSemistable) →\n (not_A_le_succ :\n ∀ (i : ℕ) (hi : i + 1 < length),\n ¬μ.A { left := toFun i, right := toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := toFun (i + 1), right := toFun (i + 2), lt := ⋯ }) →\n motive\n { toFun := toFun, length := length, monotone := monotone,\n head_eq_bot := head_eq_bot, length_eq_top := length_eq_top,\n strictMonoOn := strictMonoOn, piecewise_isSemistable := piecewise_isSemistable,\n not_A_le_succ := not_A_le_succ }) →\n motive t", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeInf", "PartialOrder.toPreorder", "Lattice", "Preorder.toLT", "HarderNarasimhan.PayoffFunction", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvex", "constType": "{ℒ : Type u_1} → {S : Type u_2} → [inst : Lattice ℒ] → [CompleteLattice S] → HarderNarasimhan.PayoffFunction ℒ S → Prop", "constCategory": "Other"}, {"references": ["HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "LT"], "name": "HarderNarasimhan.StrictIntvl.mk._flat_ctor", "constType": "{ℒ : Type u_1} → [inst : LT ℒ] → (left right : ℒ) → left < right → HarderNarasimhan.StrictIntvl ℒ", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "Bot.bot", "OfNat.ofNat", "Nat", "BoundedOrder.toOrderBot", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "OrderBot.toBot", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.head_eq_bot", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration), self.toFun 0 = ⊥", "constCategory": "Theorem"}, {"references": ["ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Membership.mem", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "iSup", "HarderNarasimhan.StrictIntvl.left", "ConditionallyCompletePartialOrderSup.toSupSet", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Eq", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "And.left", "Set", "HarderNarasimhan.StrictIntvl.right", "Set.instMembership", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Preorder", "Set.Ioc", "HarderNarasimhan.PayoffFunction.max", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.max_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n (μ : HarderNarasimhan.PayoffFunction ℒ S) (I : HarderNarasimhan.StrictIntvl ℒ),\n μ.max I = ⨆ u, ⨆ (hu : u ∈ Set.Ioc I.left I.right), μ { left := I.left, right := u, lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "Nat.instIsOrderedAddMonoid", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl", "Nat", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration", "HarderNarasimhan.PayoffFunction.A", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "Nat.instOne", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Not", "Nat.instAddMonoid", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "lt_add_one", "LE.le", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.strictMonoOn", "Nat.le_of_succ_le", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.not_A_le_succ", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} (self : μ.HarderNarasimhanFiltration) (i : ℕ) (hi : i + 1 < self.length),\n ¬μ.A { left := self.toFun i, right := self.toFun (i + 1), lt := ⋯ } ≤\n μ.A { left := self.toFun (i + 1), right := self.toFun (i + 2), lt := ⋯ }", "constCategory": "Theorem"}, {"references": ["Semiring.toAddCommMonoid", "Semiring.toModule", "Submodule", "Semiring"], "name": "Ideal", "constType": "(R : Type u) → [Semiring R] → Type u", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "SemilatticeInf.toMin", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "HarderNarasimhan.PayoffFunction.IsConvex", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "inf_lt_left", "Not", "Lattice.toSemilatticeInf", "HarderNarasimhan.StrictIntvl.mk", "LT.lt", "HarderNarasimhan.StrictIntvl", "Max.max", "Min.min", "Iff.mpr", "LE.le", "Lattice", "SemilatticeSup.toMax", "right_lt_sup", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.IsConvex.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x y : ℒ) (h : ¬x ≤ y), μ { left := x ⊓ y, right := x, lt := ⋯ } ≤ μ { left := y, right := x ⊔ y, lt := ⋯ }) →\n μ.IsConvex", "constCategory": "Other"}, {"references": ["LE", "Bot", "OrderBot"], "name": "OrderBot.toBot", "constType": "{α : Type u} → {inst : LE α} → [self : OrderBot α] → Bot α", "constCategory": "Definition"}, {"references": ["LT.lt", "Preorder", "LE.le", "Preorder.toLT", "Preorder.toLE"], "name": "LT.lt.le", "constType": "∀ {α : Type u_1} [inst : Preorder α] {a b : α}, a < b → a ≤ b", "constCategory": "Theorem"}, {"references": ["LT.lt", "PartialOrder.toPreorder", "Iff", "PartialOrder", "Preorder.toLT", "Ne", "OrderBot.toBot", "Preorder.toLE", "Bot.bot", "OrderBot"], "name": "bot_lt_iff_ne_bot", "constType": "∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, ⊥ < a ↔ a ≠ ⊥", "constCategory": "Theorem"}, {"references": [], "name": "True", "constType": "Prop", "constCategory": "Other"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "Preorder.toLT", "Nat.instZeroLEOneClass", "Nat.instAddCommMonoid", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.noConfusionType", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "PartialOrder", "eq_of_heq", "Nat.instIsOrderedAddMonoid", "Eq.ndrec", "instLTNat", "Nat.instPartialOrder", "BoundedOrder", "IsOrderedAddMonoid.toAddLeftMono", "AddZeroClass.toAddZero", "Bot.bot", "Set.Iic", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.casesOn", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "Eq.refl", "Antitone", "Nontrivial", "instIsLeftCancelAddOfAddLeftReflectLE", "IsOrderedCancelAddMonoid.toAddLeftReflectLE", "Top.top", "HEq", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "StrictAntiOn", "AddMonoid.toAddZeroClass", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.StrictIntvl.instPartialOrder", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Nat.instNeZeroSucc", "Nat.instOne", "BoundedOrder.toOrderTop", "instOfNatNat", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Nat.instIsOrderedCancelAddMonoid", "OrderBot.toBot", "Preorder.toLE", "Eq", "CompleteLattice.toConditionallyCompleteLattice", "LT.lt.le", "Nat.instAddMonoid", "HEq.refl", "instHAdd", "AddZero.toAdd", "OfNat.ofNat", "LT.lt", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "lt_add_one", "HarderNarasimhan.StrictIntvl.instOrderTop", "HarderNarasimhan.PayoffFunction", "IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono", "OrderTop.toTop", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.noConfusion", "constType": "{P : Sort u} →\n {ℒ : Type u_1} →\n {S : Type u_2} →\n {inst : Nontrivial ℒ} →\n {inst_1 : PartialOrder ℒ} →\n {inst_2 : BoundedOrder ℒ} →\n {inst_3 : CompleteLattice S} →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {t : μ.JordanHolderFiltration} →\n {ℒ' : Type u_1} →\n {S' : Type u_2} →\n {inst' : Nontrivial ℒ'} →\n {inst'_1 : PartialOrder ℒ'} →\n {inst'_2 : BoundedOrder ℒ'} →\n {inst'_3 : CompleteLattice S'} →\n {μ' : HarderNarasimhan.PayoffFunction ℒ' S'} →\n {t' : μ'.JordanHolderFiltration} →\n ℒ = ℒ' →\n S = S' →\n inst ≍ inst' →\n inst_1 ≍ inst'_1 →\n inst_2 ≍ inst'_2 →\n inst_3 ≍ inst'_3 →\n μ ≍ μ' →\n t ≍ t' →\n HarderNarasimhan.PayoffFunction.JordanHolderFiltration.noConfusionType\n P t t'", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Preorder.toLT", "BoundedOrder", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.instFunLikeNat", "Nat", "PartialOrder", "Nontrivial", "HarderNarasimhan.PayoffFunction.JordanHolderFiltration", "Preorder.toLE", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.JordanHolderFiltration.toFun_eq_coe", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Nontrivial ℒ] [inst_1 : PartialOrder ℒ] [inst_2 : BoundedOrder ℒ]\n [inst_3 : CompleteLattice S] {μ : HarderNarasimhan.PayoffFunction ℒ S} (F : μ.JordanHolderFiltration), F.toFun = ⇑F", "constCategory": "Theorem"}, {"references": ["CommSemiring", "Semiring"], "name": "CommSemiring.toSemiring", "constType": "{R : Type u} → [self : CommSemiring R] → Semiring R", "constCategory": "Definition"}, {"references": ["AddCommMonoid", "Submodule.setLike", "Module", "PartialOrder", "PartialOrder.ofSetLike", "Submodule", "Semiring"], "name": "Submodule.instPartialOrder", "constType": "{R : Type u} →\n {M : Type v} →\n [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : _root_.Module R M] → PartialOrder (Submodule R M)", "constCategory": "Definition"}, {"references": ["Submodule.hasQuotient", "associatedPrimes", "PartialOrder.toPreorder", "Subtype", "Set.ofPred", "HasQuotient.Quotient", "Module", "Submodule.Quotient.module", "Submodule.module", "Membership.mem", "Preorder.toLT", "Submodule", "HarderNarasimhan.StrictIntvl.left", "Ideal", "PrimeSpectrum.asIdeal", "CommRing.toCommSemiring", "SetLike.instMembership", "LinearExtension", "Set", "HarderNarasimhan.StrictIntvl.right", "Submodule.addCommGroup", "Submodule.instPartialOrder", "CommSemiring.toSemiring", "Submodule.Quotient.addCommMonoid", "AddCommGroup", "CommRing", "Set.instMembership", "PrimeSpectrum", "CommRing.toRing", "HarderNarasimhan.StrictIntvl", "Submodule.setLike", "AddCommGroup.toAddCommMonoid", "Submodule.addCommMonoid", "Submodule.submoduleOf"], "name": "HarderNarasimhan.Coprimary.subquotientAssociatedPrimes", "constType": "{R : Type u_1} →\n [inst : CommRing R] →\n {M : Type u_2} →\n [inst_1 : AddCommGroup M] →\n [inst_2 : _root_.Module R M] →\n HarderNarasimhan.StrictIntvl (Submodule R M) → Set (LinearExtension (PrimeSpectrum R))", "constCategory": "Definition"}, {"references": ["Lattice.toSemilatticeSup", "HarderNarasimhan.PayoffFunction.mk", "PartialOrder.toPreorder", "AddCommGroup.toAddGroup", "Preorder.toLT", "SMulZeroClass.toSMul", "DedekindCut.instCompleteLinearOrder", "NNReal.instLinearOrder", "NNReal.instZero", "PartialOrder", "AddGroup.toSubNegMonoid", "CoheytingAlgebra.toOrderTop", "SemilatticeInf.toPartialOrder", "DistribSMul.toSMulZeroClass", "Real", "CoheytingAlgebra.toGeneralizedCoheytingAlgebra", "NNReal", "LinearOrder", "DistribMulAction.toDistribSMul", "AddZeroClass.toAddZero", "HarderNarasimhan.StrictIntvl", "LinearOrder.toDecidableLT", "Real.instMonoid", "HSMul.hSMul", "AddCommGroup.toAddCommMonoid", "NNReal.instInv", "Top.top", "CompleteDistribLattice.toCoframe", "AddZero.toZero", "AddMonoid.toAddZeroClass", "GeneralizedCoheytingAlgebra.toLattice", "Module", "Order.Coframe.toCoheytingAlgebra", "SemilatticeSup.toPartialOrder", "CompleteLinearOrder.toCompletelyDistribLattice", "NNReal.instPartialOrder", "instDistribLatticeOfLinearOrder", "CompletelyDistribLattice.toCompleteDistribLattice", "NNReal.instSMulOfReal", "HarderNarasimhan.PayoffFunction.slope", "instHSMul", "Zero.toOfNat0", "Eq", "Preorder.toLE", "Not", "Inv.inv", "Lattice.toSemilatticeInf", "DedekindCut", "AddCommGroup", "OfNat.ofNat", "Real.semiring", "LT.lt", "Module.toDistribMulAction", "DistribLattice.toLattice", "SubNegMonoid.toAddMonoid", "DedekindCut.principal", "dite", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop"], "name": "HarderNarasimhan.PayoffFunction.slope.eq_1", "constType": "∀ {ℒ : Type u_1} [inst : PartialOrder ℒ] {V : Type u_2} [inst_1 : AddCommGroup V] [inst_2 : _root_.Module ℝ V]\n [inst_3 : LinearOrder V] (r : HarderNarasimhan.StrictIntvl ℒ → NNReal) (d : HarderNarasimhan.StrictIntvl ℒ → V),\n HarderNarasimhan.PayoffFunction.slope r d =\n { toFun := fun I => if x : 0 < r I then DedekindCut.principal ((r I)⁻¹ • d I) else ⊤ }", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.toFun", "HarderNarasimhan.StrictIntvl", "Eq", "HarderNarasimhan.PayoffFunction", "LT"], "name": "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl._proof_1", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : LT ℒ] (μ ν : HarderNarasimhan.PayoffFunction ℒ S), μ.toFun = ν.toFun → μ = ν", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "PartialOrder", "Preorder.toLT", "WellFoundedGT", "BoundedOrder", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.instWeakACCOfWellFoundedGT", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} [WellFoundedGT ℒ], μ.WeakACC", "constCategory": "Theorem"}, {"references": ["PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "HarderNarasimhan.PayoffFunction.min", "Preorder.toLT", "DFunLike.coe", "ConditionallyCompletePartialOrderSup.toPartialOrder", "Preorder", "HarderNarasimhan.StrictIntvl", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "LE.le", "HarderNarasimhan.PayoffFunction", "Preorder.toLE", "CompleteLattice", "CompleteLattice.toConditionallyCompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.min_le_apply", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Preorder ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, μ.min I ≤ μ I", "constCategory": "Theorem"}, {"references": ["AddCommMonoid", "Semiring"], "name": "Semiring.toAddCommMonoid", "constType": "{α : Type u} → [self : Semiring α] → AddCommMonoid α", "constCategory": "Definition"}, {"references": ["HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.toFun", "PartialOrder.toPreorder", "Finset", "instLinearOrderLinearExtensionOfPartialOrder", "StrictMonoOn", "Module", "HarderNarasimhan.Coprimary.instADCCSubmoduleDedekindCutColexFinsetLinearExtensionPrimeSpectrumPayoff", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_5", "Nat.instPreorder", "Submodule", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_4", "instDistribLatticeOfLinearOrder", "HarderNarasimhan.PayoffFunction.HarderNarasimhanFiltration.length", "HarderNarasimhan.Coprimary.payoff", "PrimeSpectrum.instPartialOrder", "Concept.instCompleteLattice", "Colex", "Preorder.toLE", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_2", "CompleteLattice.toConditionallyCompleteLattice", "SemilatticeInf.toPartialOrder", "CommRing.toCommSemiring", "Lattice.toSemilatticeInf", "LinearExtension", "HarderNarasimhan.PayoffFunction.hnFiltration", "CommSemiring.toSemiring", "AddCommGroup", "DedekindCut", "CommRing", "Set.Iic", "PrimeSpectrum", "Nat", "IsNoetherianRing", "DistribLattice.toLattice", "Finset.Colex.instPartialOrder", "LE.le", "Nontrivial", "AddCommGroup.toAddCommMonoid", "Submodule.completeLattice", "ConditionallyCompleteLattice.toLattice", "CompleteLattice.toBoundedOrder", "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_3", "Module.Finite"], "name": "HarderNarasimhan.Coprimary.coprimaryFiltration._proof_9", "constType": "∀ (R : Type u_2) [inst : CommRing R] [inst_1 : IsNoetherianRing R] (M : Type u_1) [inst_2 : Nontrivial M]\n [inst_3 : AddCommGroup M] [inst_4 : _root_.Module R M] [inst_5 : Module.Finite R M],\n StrictMonoOn (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.toFun\n (Set.Iic (HarderNarasimhan.Coprimary.payoff R M).hnFiltration.length)", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.mk", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : PartialOrder ℒ] [inst_1 : BoundedOrder ℒ] [inst_2 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S},\n (∀ (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }) →\n μ.WeakACC", "constCategory": "Other"}, {"references": ["LT.lt", "Preorder", "Preorder.toLT"], "name": "StrictAnti", "constType": "{α : Type u} → {β : Type v} → [Preorder α] → [Preorder β] → (α → β) → Prop", "constCategory": "Definition"}, {"references": ["Nat", "SizeOf"], "name": "SizeOf.sizeOf", "constType": "{α : Sort u} → [self : SizeOf α] → α → ℕ", "constCategory": "Definition"}, {"references": ["PartialOrder.toPreorder", "Lattice.toSemilatticeInf", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.PayoffFunction.IsConvexOn", "Lattice", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "HarderNarasimhan.PayoffFunction.max", "HarderNarasimhan.PayoffFunction", "Eq", "CompleteLattice", "SemilatticeInf.toPartialOrder"], "name": "HarderNarasimhan.PayoffFunction.IsConvexOn.max_max", "constType": "∀ {ℒ : Type u_1} {S : Type u_2} [inst : Lattice ℒ] [inst_1 : CompleteLattice S]\n {μ : HarderNarasimhan.PayoffFunction ℒ S} {I : HarderNarasimhan.StrictIntvl ℒ}, μ.IsConvexOn I → μ.max.max I = μ.max I", "constCategory": "Theorem"}, {"references": ["Lattice", "DistribLattice"], "name": "DistribLattice.toLattice", "constType": "{α : Type u_1} → [self : DistribLattice α] → Lattice α", "constCategory": "Definition"}, {"references": ["instAddNat", "PartialOrder.toPreorder", "HarderNarasimhan.PayoffFunction.StrongDCC", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "bot_le", "DFunLike.coe", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "OrderBot.toBot", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "Exists", "instHAdd", "lt_of_le_of_lt", "BoundedOrder", "Bot.bot", "Nat.lt_add_one", "OfNat.ofNat", "HAdd.hAdd", "HarderNarasimhan.StrictIntvl.mk", "StrictAnti", "Nat", "HarderNarasimhan.StrictIntvl", "BoundedOrder.toOrderBot", "HarderNarasimhan.PayoffFunction.StrongDCC.rec", "HarderNarasimhan.PayoffFunction.StrongDCC.mk", "LE.le", "HarderNarasimhan.PayoffFunction", "CompleteLattice"], "name": "HarderNarasimhan.PayoffFunction.StrongDCC.casesOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.StrongDCC → Sort u} →\n (t : μ.StrongDCC) →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (saf : StrictAnti x),\n ∃ N, μ { left := ⊥, right := x N, lt := ⋯ } ≤ μ { left := x (N + 1), right := x N, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["SubNegMonoid", "AddMonoid"], "name": "SubNegMonoid.toAddMonoid", "constType": "{G : Type u} → [self : SubNegMonoid G] → AddMonoid G", "constCategory": "Definition"}, {"references": ["Submodule.toAddSubmonoid", "Membership.mem", "Inter.inter", "Algebra.id", "Semiring.toNonAssocSemiring", "RingHom.id", "DistribMulAction.toMulAction", "IsScalarTower.right", "Semiring.toModule", "Set.instInter", "LinearMap.ker", "SetLike.instMembership", "LocalizedModule", "Submonoid.instSetLike", "AddZeroClass.toAddZero", "CommSemiring.toCommMonoid", "Set.instMembership", "Submonoid", "SetLike.coe", "AddCommGroup.toAddCommMonoid", "AddSubmonoid.toAddSubsemigroup", "AddMonoid.toAddZeroClass", "associatedPrimes", "Subtype", "OreLocalization.oreSetComm", "Module", "Submodule.module", "MulZeroOneClass.toMulOneClass", "AddCommMonoid.toAddMonoid", "Submodule", "LocalizedModule.mkLinearMap", "AddSubsemigroup.carrier", "Ideal", "instMulZeroOneClassOfSemiring", "CommRing.toCommSemiring", "Set.Nonempty", "Semiring.toMonoid", "OreLocalization.instAddCommMonoidOreLocalization", "IsScalarTower.left", "Set", "CommSemiring.toSemiring", "OreLocalization.instModuleOfIsScalarTower", "AddCommGroup", "CommRing", "AddZero.toAdd", "Semiring.toAddCommMonoid", "Module.toDistribMulAction", "Submodule.setLike", "Submodule.addCommMonoid"], "name": "HarderNarasimhan.inter_nonempty_of_mem_associatedPrimes_ker", "constType": "∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : _root_.Module R M]\n (S : Submonoid R) {p : Ideal R},\n p ∈ associatedPrimes R ↥(LocalizedModule.mkLinearMap S M).ker → (p.carrier ∩ ↑S).Nonempty", "constCategory": "Theorem"}, {"references": ["instAddNat", "HarderNarasimhan.PayoffFunction.WeakACC", "PartialOrder.toPreorder", "ConditionallyCompleteLattice.toConditionallyCompletePartialOrder", "Preorder.toLT", "DFunLike.coe", "HarderNarasimhan.PayoffFunction.WeakACC.rec", "Nat.instPreorder", "ConditionallyCompletePartialOrderSup.toPartialOrder", "ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup", "BoundedOrder.toOrderTop", "instOfNatNat", "PartialOrder", "HarderNarasimhan.PayoffFunction.instFunLikeStrictIntvl", "Preorder.toLE", "CompleteLattice.toConditionallyCompleteLattice", "le_top", "Exists", "instHAdd", "BoundedOrder", "Nat.lt_add_one", "OfNat.ofNat", "lt_of_lt_of_le", "HarderNarasimhan.PayoffFunction.WeakACC.mk", "HarderNarasimhan.StrictIntvl.mk", "HAdd.hAdd", "Nat", "HarderNarasimhan.StrictIntvl", "LE.le", "Top.top", "HarderNarasimhan.PayoffFunction", "OrderTop.toTop", "CompleteLattice", "StrictMono"], "name": "HarderNarasimhan.PayoffFunction.WeakACC.recOn", "constType": "{ℒ : Type u_1} →\n {S : Type u_2} →\n [inst : PartialOrder ℒ] →\n [inst_1 : BoundedOrder ℒ] →\n [inst_2 : CompleteLattice S] →\n {μ : HarderNarasimhan.PayoffFunction ℒ S} →\n {motive : μ.WeakACC → Sort u} →\n (t : μ.WeakACC) →\n ((exists_le :\n ∀ (x : ℕ → ℒ) (smf : StrictMono x),\n ∃ N, μ { left := x N, right := x (N + 1), lt := ⋯ } ≤ μ { left := x N, right := ⊤, lt := ⋯ }) →\n motive ⋯) →\n motive t", "constCategory": "Definition"}, {"references": ["Subtype.instLT", "PartialOrder.toPreorder", "Subtype", "HarderNarasimhan.StrictIntvl.ofSub._proof_1", "HarderNarasimhan.StrictIntvl.right", "Membership.mem", "HarderNarasimhan.StrictIntvl.instMembership", "Preorder.toLT", "Subtype.val", "HarderNarasimhan.StrictIntvl.mk", "HarderNarasimhan.StrictIntvl", "HarderNarasimhan.StrictIntvl.left", "PartialOrder", "Preorder.toLE"], "name": "HarderNarasimhan.StrictIntvl.ofSub", "constType": "{ℒ : Type u_1} →\n [inst : PartialOrder ℒ] →\n {I : HarderNarasimhan.StrictIntvl ℒ} → HarderNarasimhan.StrictIntvl { x // x ∈ I } → HarderNarasimhan.StrictIntvl ℒ", "constCategory": "Definition"}, {"references": ["LT.lt", "Max.max", "PartialOrder.toPreorder", "SemilatticeSup.toMax", "SemilatticeSup.toPartialOrder", "Preorder.toLT", "SemilatticeSup"], "name": "lt_sup_of_lt_left", "constType": "∀ {α : Type u} [inst : SemilatticeSup α] {a b c : α}, c < a → c < a ⊔ b", "constCategory": "Theorem"}, {"references": ["HarderNarasimhan.StrictIntvl", "SizeOf", "LT", "HarderNarasimhan.StrictIntvl._sizeOf_1", "SizeOf.mk"], "name": "HarderNarasimhan.StrictIntvl._sizeOf_inst", "constType": "(ℒ : Type u_1) → {inst : LT ℒ} → [SizeOf ℒ] → SizeOf (HarderNarasimhan.StrictIntvl ℒ)", "constCategory": "Definition"}]