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@@ -92,3 +92,6 @@ data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Subobject.C
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.RingTheory.TensorProduct.DirectLimitFG.sym.json filter=lfs diff=lfs merge=lfs -text
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Geometry.Manifold.VectorBundle.Riemannian.sym.json filter=lfs diff=lfs merge=lfs -text
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.MeasureTheory.Function.SimpleFuncDenseLp.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Bicategory.Functor.Oplax.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.CStarAlgebra.CStarMatrix.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor.sym.json filter=lfs diff=lfs merge=lfs -text

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.EuclideanDomain.Field.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Action.Pointwise.Set.Finite.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.MvPolynomial.SchwartzZippel.sym.json

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{a : Prop} {b : Prop} {c : Prop}, Eq.{1} Prop (Or (And a b) (And a c)) (And a (Or b c))","typeFull":"∀ {a b c : Prop}, (a ∧ b ∨ a ∧ c) = (a ∧ (b ∨ c))","typeReadable":"∀ {a b c : Prop}, (a ∧ b ∨ a ∧ c) = (a ∧ (b ∨ c))","typeReferences":[["Or"],["And"],["Eq"]],"valueReferences":[["Or"],["And"],["Eq","symm"],["and_or_left"],["propext"]]},{"isProp":true,"kind":"theorem","name":["MvPolynomial","schwartz_zippel_sup_sum"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.3 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.6 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.9 : DecidableEq.{succ u_1} R] {n : Nat} {p : MvPolynomial.{0, u_1} (Fin n) R (CommRing.toCommSemiring.{u_1} R 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x._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx.119.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.157 (OfNat.ofNat.{u_1} (MvPolynomial.{0, u_1} (Fin x._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx.118.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.154) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.3)) 0 (Zero.toOfNat0.{u_1} (MvPolynomial.{0, u_1} (Fin x._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx.118.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.154) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} (MvPolynomial.{0, u_1} (Fin x._@.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx.118.Mathlib.Algebra.MvPolynomial.SchwartzZippel.318105959._hygCtx._hyg.154) R (CommRing.toCommSemiring.{u_1} R 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Action.End.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Interval.Set.Monoid.sym.json

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[inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b c : M), (fun x => a + x) '' Set.Ioc b c = Set.Ioc (a + b) (a + c)","typeReferences":[["PartialOrder","toPreorder"],["instHAdd"],["Set"],["HAdd","hAdd"],["AddCommMonoid"],["Set","image"],["AddCommMonoid","toAddCommSemigroup"],["Set","Ioc"],["PartialOrder"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["ExistsAddOfLE"],["IsOrderedCancelAddMonoid"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["True"],["Set"],["instHAdd"],["Set","image_congr"],["congrArg"],["HAdd","hAdd"],["eq_self"],["Set","image_add_const_Ioc"],["Set","image"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["Set","Ioc"],["congr"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["add_comm"]]},{"isProp":true,"kind":"theorem","name":["Set","Ioo_add_bij"],"typeFallback":"forall {M : Type.{u_1}} 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[inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.9 : IsOrderedCancelAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.3 (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.3))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.4194817259._hygCtx._hyg.6))] (a : M) (b : M), Eq.{succ u_1} (Set.{u_1} M) (Set.image.{u_1, u_1} M M (fun (x : M) => HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M 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(a b : M), (fun x => a + x) '' Set.Ici b = Set.Ici (a + b)","typeReferences":[["PartialOrder","toPreorder"],["instHAdd"],["Set"],["HAdd","hAdd"],["Set","Ici"],["AddCommMonoid"],["Set","image"],["AddCommMonoid","toAddCommSemigroup"],["PartialOrder"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["ExistsAddOfLE"],["IsOrderedCancelAddMonoid"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["True"],["Set"],["instHAdd"],["Set","image_congr"],["congrArg"],["HAdd","hAdd"],["eq_self"],["Set","Ici"],["Set","image"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["congr"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Set","image_add_const_Ici"],["add_comm"]]},{"isProp":true,"kind":"theorem","name":["Set","image_const_add_Icc"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.9 : IsOrderedCancelAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.3 (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.3))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.2783916306._hygCtx._hyg.6))] (a : M) (b : M) (c : M), Eq.{succ u_1} (Set.{u_1} M) (Set.image.{u_1, u_1} M M (fun (x : M) => HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M 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M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b c : M), (fun x => a + x) '' Set.Icc b c = Set.Icc (a + b) (a + c)","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b c : M), (fun x => a + x) '' Set.Icc b c = Set.Icc (a + b) (a + 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b)","typeReferences":[["PartialOrder","toPreorder"],["instHAdd"],["Set"],["HAdd","hAdd"],["AddCommMonoid"],["Set","image"],["AddCommMonoid","toAddCommSemigroup"],["PartialOrder"],["Set","Ioi"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["ExistsAddOfLE"],["IsOrderedCancelAddMonoid"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["True"],["Set"],["instHAdd"],["Set","image_congr"],["congrArg"],["HAdd","hAdd"],["eq_self"],["Set","image"],["Set","image_add_const_Ioi"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["congr"],["Set","Ioi"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["add_comm"]]},{"isProp":true,"kind":"theorem","name":["Set","image_const_add_Ico"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.3658185842._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.3658185842._hygCtx._hyg.6 : 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[inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.9 : IsOrderedCancelAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.3 (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.3))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.6))] (a : M) (b : M) (d : M), Set.BijOn.{u_1, u_1} M M (fun (x._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1947212296._hygCtx._hyg.22 : M) => HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M 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AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b d : M), Set.BijOn (fun x => x + d) (Set.Ioc a b) (Set.Ioc (a + d) (b + d))","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b d : M), Set.BijOn (fun x => x + d) (Set.Ioc a b) (Set.Ioc (a + d) (b + 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{M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.9 : IsOrderedCancelAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.3 (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.3))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.490965580._hygCtx._hyg.6))] (a : M) (b : M) (c : M), Eq.{succ u_1} (Set.{u_1} M) 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[IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b c : M), (fun x => x + a) '' Set.Icc b c = Set.Icc (b + a) (c + a)","typeReferences":[["PartialOrder","toPreorder"],["Set","Icc"],["instHAdd"],["Set"],["HAdd","hAdd"],["AddCommMonoid"],["Set","image"],["AddCommMonoid","toAddCommSemigroup"],["PartialOrder"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["ExistsAddOfLE"],["IsOrderedCancelAddMonoid"]],"valueReferences":[["HAdd","hAdd"],["Set","BijOn","image_eq"],["PartialOrder","toPreorder"],["Set","Icc_add_bij"],["AddCommMonoid","toAddCommSemigroup"],["instHAdd"],["Set","Icc"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"]]},{"isProp":true,"kind":"theorem","name":["Set","Ico_add_bij"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.3200236335._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.3200236335._hygCtx._hyg.6 : PartialOrder.{u_1} M] 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[inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]\n  (a b c : M), (fun x => x + a) '' Set.Ioc b c = Set.Ioc (b + a) (c + a)","typeReferences":[["PartialOrder","toPreorder"],["instHAdd"],["Set"],["HAdd","hAdd"],["AddCommMonoid"],["Set","image"],["AddCommMonoid","toAddCommSemigroup"],["Set","Ioc"],["PartialOrder"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["ExistsAddOfLE"],["IsOrderedCancelAddMonoid"]],"valueReferences":[["HAdd","hAdd"],["Set","BijOn","image_eq"],["PartialOrder","toPreorder"],["AddCommMonoid","toAddCommSemigroup"],["instHAdd"],["Set","Ioc"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Set","Ioc_add_bij"]]},{"isProp":true,"kind":"theorem","name":["Set","Ioi_add_bij"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1783370553._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Interval.Set.Monoid.1783370553._hygCtx._hyg.6 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Monoid.ToMulBot.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.SuccPred.WithBot.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["WithBot","one_le_iff_pos"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.16 : PartialOrder.{u_2} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19 : AddMonoidWithOne.{u_2} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.22 : ZeroLEOneClass.{u_2} α (AddZero.toZero.{u_2} α (AddZeroClass.toAddZero.{u_2} α (AddMonoid.toAddZeroClass.{u_2} α (AddMonoidWithOne.toAddMonoid.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19)))) (AddMonoidWithOne.toOne.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19) (Preorder.toLE.{u_2} α (PartialOrder.toPreorder.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.16))] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.25 : NeZero.{u_2} α (AddZero.toZero.{u_2} α (AddZeroClass.toAddZero.{u_2} α (AddMonoid.toAddZeroClass.{u_2} α (AddMonoidWithOne.toAddMonoid.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19)))) (OfNat.ofNat.{u_2} α 1 (One.toOfNat1.{u_2} α (AddMonoidWithOne.toOne.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19)))] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.31 : SuccAddOrder.{u_2} α (PartialOrder.toPreorder.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.16) (AddSemigroup.toAdd.{u_2} α (AddMonoid.toAddSemigroup.{u_2} α (AddMonoidWithOne.toAddMonoid.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19))) (AddMonoidWithOne.toOne.{u_2} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.480605062._hygCtx._hyg.19)] (a : WithBot.{u_2} α), Iff (LE.le.{u_2} (WithBot.{u_2} α) (Preorder.toLE.{u_2} (WithBot.{u_2} α) (WithBot.instPreorder.{u_2} α (PartialOrder.toPreorder.{u_2} α 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{α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.3 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.6 : OrderBot.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.9 : AddMonoidWithOne.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.12 : SuccAddOrder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.3 (AddSemigroup.toAdd.{u_1} α (AddMonoid.toAddSemigroup.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.9))) (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.9)], Eq.{succ u_1} α (WithBot.succ.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.344280375._hygCtx._hyg.9)))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : OrderBot α] [inst_2 : AddMonoidWithOne α] [inst_3 : SuccAddOrder α],\n  WithBot.succ 0 = 1","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : OrderBot α] [inst_2 : AddMonoidWithOne α] [inst_3 : SuccAddOrder α],\n  WithBot.succ 0 = 1","typeReferences":[["WithBot","succ"],["WithBot","zero"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["OfNat","ofNat"],["Preorder"],["One","toOfNat1"],["WithBot"],["SuccAddOrder"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["Zero","toOfNat0"],["SuccAddOrder","toSuccOrder"],["Preorder","toLE"],["Eq"],["AddZero","toZero"],["AddMonoidWithOne"],["OrderBot"],["AddMonoid","toAddZeroClass"],["AddSemigroup","toAdd"]],"valueReferences":[["WithBot","natCast"],["Nat","cast"],["Eq","trans"],["Eq","mp"],["WithBot","succ"],["AddMonoidWithOne","toAddMonoid"],["congrArg"],["WithBot"],["instOfNatNat"],["congr"],["congrFun'"],["SuccAddOrder","toSuccOrder"],["Zero","toOfNat0"],["Eq"],["AddSemigroup","toAdd"],["WithBot","addMonoidWithOne"],["instHAdd"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HAdd","hAdd"],["zero_add"],["Nat"],["AddMonoidWithOne","toNatCast"],["One","toOfNat1"],["Nat","cast_zero"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["WithBot","succ_natCast"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","SuccPred","WithBot",0,"WithBot","one_le_iff_pos","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {x : α} [inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.5 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8 : AddMonoidWithOne.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.11 : ZeroLEOneClass.{u_1} α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8)))) (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.5))] [inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.14 : NeZero.{u_1} α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8)))) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8)))] [inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.20 : SuccAddOrder.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.5) (AddSemigroup.toAdd.{u_1} α (AddMonoid.toAddSemigroup.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8))) (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8)], Eq.{1} Prop (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.5)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8))) x) (LT.lt.{u_1} �� (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.5)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.480605062._hygCtx._hyg.8)))))) x)","typeFull":"∀ {α : Type u_1} {x : α} [inst : PartialOrder α] [inst_1 : AddMonoidWithOne α] [ZeroLEOneClass α] [NeZero 1]\n  [SuccAddOrder α], (1 ≤ x) = (0 < x)","typeReadable":"∀ {α : Type u_1} {x : α} [inst : PartialOrder α] [inst_1 : AddMonoidWithOne α] [ZeroLEOneClass α] [NeZero 1]\n  [SuccAddOrder α], (1 ≤ x) = (0 < x)","typeReferences":[["PartialOrder","toPreorder"],["Preorder","toLT"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["OfNat","ofNat"],["ZeroLEOneClass"],["LT","lt"],["NeZero"],["One","toOfNat1"],["SuccAddOrder"],["AddMonoid","toAddSemigroup"],["PartialOrder"],["LE","le"],["AddMonoidWithOne","toOne"],["Zero","toOfNat0"],["Eq"],["Preorder","toLE"],["AddMonoidWithOne"],["AddZero","toZero"],["AddSemigroup","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Preorder","toLT"],["AddMonoidWithOne","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["LT","lt"],["One","toOfNat1"],["LE","le"],["AddMonoidWithOne","toOne"],["Zero","toOfNat0"],["Preorder","toLE"],["AddZero","toZero"],["Order","one_le_iff_pos"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["WithBot","succ_one"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.3 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.6 : OrderBot.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.9 : AddMonoidWithOne.{u_1} α] [inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.12 : SuccAddOrder.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.3 (AddSemigroup.toAdd.{u_1} α (AddMonoid.toAddSemigroup.{u_1} α (AddMonoidWithOne.toAddMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.9))) (AddMonoidWithOne.toOne.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.9)], Eq.{succ u_1} α (WithBot.succ.{u_1} α inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.SuccPred.WithBot.649153498._hygCtx._hyg.6 (SuccAddOrder.toSuccOrder.{u_1} α 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Ring.PUnit.sym.json

@@ -0,0 +1 @@
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{E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.15 : ContinuousAdd.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.12 (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3))))] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.18 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.12] {f : E -> Real} {a : E}, (IsLocalExtr.{u_1, 0} E Real inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.12 Real.instPreorder f a) -> (Eq.{succ u_1} (E -> Real) (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966537._hygCtx._hyg.6 f a) (OfNat.ofNat.{u_1} (E -> Real) 0 (Zero.toOfNat0.{u_1} (E -> Real) (Pi.instZero.{u_1, 0} E (fun (v : E) => Real) (fun (i : E) => Real.instZero)))))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalExtr f a → lineDeriv ℝ f a = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalExtr f a → lineDeriv ℝ f a = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["ContinuousSMul"],["DenselyNormedField","toNontriviallyNormedField"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["IsLocalExtr"],["Real"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["ContinuousAdd"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Continuous","const_add"],["Real","instMonoidWithZero"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Continuous","tendsto'"],["DenselyNormedField","toNontriviallyNormedField"],["funext"],["AddGroup","toSubNegMonoid"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["Real"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["instSeparatelyContinuousAddOfContinuousAdd"],["add_zero"],["AddZeroClass","toAddZero"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["zero_smul"],["AddZero","toZero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"],["IsExtrFilter","lineDeriv_eq_zero"],["continuous_id'"],["UniformSpace","toTopologicalSpace"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["NontriviallyNormedField","toNormedField"],["congrArg"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["continuous_const"],["congrFun'"],["Eq"],["MulActionWithZero","toSMulWithZero"],["NormedField","toNormedSpace"],["True"],["instHAdd"],["Module","toMulActionWithZero"],["OfNat","ofNat"],["Real","semiring"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["Continuous","smul"],["AddCommGroup","toDivisionAddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["of_eq_true"],["Real","denselyNormedField"]]},{"isProp":true,"kind":"theorem","name":["IsMinOn","lineDerivWithin_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsMinOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDerivWithin.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096284._hygCtx._hyg.6 f s a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMinOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMinOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["IsMinOn"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["lineDerivWithin"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMinOn","isExtr"],["IsExtrOn","lineDerivWithin_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsExtrFilter","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3)] {f : E -> Real} {a : E} {b : E} {l : Filter.{u_1} E}, (IsExtrFilter.{u_1, 0} E Real Real.instPreorder f l a) -> (Filter.Tendsto.{0, u_1} Real E (fun (t : Real) => HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.6))))) t b)) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))) l) -> (Eq.{1} Real (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966533._hygCtx._hyg.6 f a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {a b : E} {l : Filter E},\n  IsExtrFilter f l a → Filter.Tendsto (fun t => a + t • b) (nhds 0) l → lineDeriv ℝ f a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {a b : E} {l : Filter E},\n  IsExtrFilter f l a → Filter.Tendsto (fun t => a + t • b) (nhds 0) l → lineDeriv ℝ f a b = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["IsExtrFilter"],["instHSMul"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["Filter","Tendsto"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Filter"],["Real","semiring"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["LineDifferentiableAt","hasLineDerivAt"],["Classical","propDecidable"],["lineDeriv"],["OfNat","ofNat"],["LineDifferentiableAt"],["IsExtrFilter","hasLineDerivAt_eq_zero"],["NontriviallyNormedField","toNormedField"],["Real","instZero"],["DenselyNormedField","toNontriviallyNormedField"],["lineDeriv_zero_of_not_lineDifferentiableAt"],["Real","denselyNormedField"],["Zero","toOfNat0"],["dite"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["IsLocalExtr","hasLineDerivAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.15 : ContinuousAdd.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.12 (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3))))] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.18 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.12] {f : E -> Real} {a : E} {b : E} {f' : Real}, (IsLocalExtr.{u_1, 0} E Real inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.12 Real.instPreorder f a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419197._hygCtx._hyg.6 f f' a b) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ}, IsLocalExtr f a → HasLineDerivAt ℝ f f' a b → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ}, IsLocalExtr f a → HasLineDerivAt ℝ f f' a b → f' = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["ContinuousSMul"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["IsLocalExtr"],["Real"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["ContinuousAdd"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Continuous","const_add"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Real","instMonoidWithZero"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Continuous","tendsto'"],["AddGroup","toSubNegMonoid"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real"],["DistribMulAction","toDistribSMul"],["instSeparatelyContinuousAddOfContinuousAdd"],["add_zero"],["AddZeroClass","toAddZero"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["continuous_id'"],["UniformSpace","toTopologicalSpace"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["continuous_const"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["MulActionWithZero","toSMulWithZero"],["True"],["instHAdd"],["Module","toMulActionWithZero"],["Real","semiring"],["OfNat","ofNat"],["IsExtrFilter","hasLineDerivAt_eq_zero"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["Continuous","smul"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","instZero"],["of_eq_true"],["AddCommGroup","toDivisionAddCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsMaxOn","lineDerivWithin_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsMaxOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDerivWithin.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096285._hygCtx._hyg.6 f s a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMaxOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMaxOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["IsMaxOn"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["lineDerivWithin"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMaxOn","isExtr"],["IsExtrOn","lineDerivWithin_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsLocalMax","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.15 : ContinuousAdd.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.12 (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3))))] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.18 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.12] {f : E -> Real} {a : E}, (IsLocalMax.{u_1, 0} E Real inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.12 Real.instPreorder f a) -> (Eq.{succ u_1} (E -> Real) (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966539._hygCtx._hyg.6 f a) (OfNat.ofNat.{u_1} (E -> Real) 0 (Zero.toOfNat0.{u_1} (E -> Real) (Pi.instZero.{u_1, 0} E (fun (v : E) => Real) (fun (i : E) => Real.instZero)))))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalMax f a → lineDeriv ℝ f a = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalMax f a → lineDeriv ℝ f a = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["IsLocalMax"],["NontriviallyNormedField","toNormedField"],["ContinuousSMul"],["DenselyNormedField","toNontriviallyNormedField"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["ContinuousAdd"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsMaxFilter"],["IsMinFilter"],["Real","instPreorder"],["Or","inr"],["Real"],["nhds"],["IsLocalExtr","lineDeriv_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsExtrOn","hasLineDerivWithinAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsExtrOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.6 f f' s a b) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213246._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsExtrOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsExtrOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["HasLineDerivWithinAt"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["Set"],["instHAdd"],["IsExtrOn"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["DenselyNormedField","toNontriviallyNormedField"],["Real","denselyNormedField"],["HasLineDerivWithinAt","hasLineDerivAt'"],["NontriviallyNormedField","toNormedField"],["IsExtrOn","hasLineDerivAt_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsExtrOn","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsExtrOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966534._hygCtx._hyg.6 f a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsExtrOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsExtrOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 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{E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsMaxOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.6 f f' a b) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419196._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMaxOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMaxOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["IsMaxOn"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMaxOn","isExtr"],["IsExtrOn","hasLineDerivAt_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsExtrFilter","hasLineDerivAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3)] {f : E -> Real} {a : E} {b : E} {f' : Real} {l : Filter.{u_1} E}, (IsExtrFilter.{u_1, 0} E Real Real.instPreorder f l a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.6 f f' a b) -> (Filter.Tendsto.{0, u_1} Real E (fun (t : Real) => HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419193._hygCtx._hyg.6))))) t b)) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))) l) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} {l : Filter E},\n  IsExtrFilter f l a → HasLineDerivAt ℝ f f' a b → Filter.Tendsto (fun t => a + t • b) (nhds 0) l → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} {l : Filter E},\n  IsExtrFilter f l a → HasLineDerivAt ℝ f f' a b → Filter.Tendsto (fun t => a + t • b) (nhds 0) l → f' = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["IsExtrFilter"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["Filter","Tendsto"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Filter"],["Real","semiring"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["Real","instMonoidWithZero"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["IsExtrFilter"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["MulActionWithZero","toSMulWithZero"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Module","toMulActionWithZero"],["Function","comp"],["add_zero"],["AddZeroClass","toAddZero"],["IsLocalExtr","hasDerivAt_eq_zero"],["Real","semiring"],["OfNat","ofNat"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["IsExtrFilter","comp_tendsto"],["Real","instMonoid"],["HSMul","hSMul"],["NegZeroClass","toZero"],["id"],["AddCommGroup","toAddCommMonoid"],["zero_smul"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsMaxOn","hasLineDerivWithinAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsMaxOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.6 f f' s a b) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213248._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMaxOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMaxOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["HasLineDerivWithinAt"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["IsMaxOn"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsExtrOn","hasLineDerivWithinAt_eq_zero"],["Real","instPreorder"],["Real"],["IsMaxOn","isExtr"]]},{"isProp":true,"kind":"theorem","name":["IsExtrOn","hasLineDerivAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsExtrOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.6 f f' a b) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419194._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsExtrOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsExtrOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["Set"],["instHAdd"],["IsExtrOn"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Filter","Eventually"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["instHSMul"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["nhds"],["Filter","Tendsto"],["DistribSMul","toSMulZeroClass"],["Filter","tendsto_principal"],["Real"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Filter","principal"],["Real","semiring"],["Set","instMembership"],["IsExtrFilter","hasLineDerivAt_eq_zero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Iff","mpr"],["Real","instMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalMin","hasLineDerivAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.15 : ContinuousAdd.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.12 (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3))))] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.18 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.12] {f : E -> Real} {a : E} {b : E} {f' : Real}, (IsLocalMin.{u_1, 0} E Real inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.12 Real.instPreorder f a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419198._hygCtx._hyg.6 f f' a b) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ}, IsLocalMin f a → HasLineDerivAt ℝ f f' a b → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ}, IsLocalMin f a → HasLineDerivAt ℝ f f' a b → f' = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["ContinuousSMul"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["Zero","toOfNat0"],["IsLocalMin"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["ContinuousAdd"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsMaxFilter"],["IsMinFilter"],["Real","instPreorder"],["Real"],["Or","inl"],["nhds"],["IsLocalExtr","hasLineDerivAt_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsMinOn","hasLineDerivAt_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsMinOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.6 f f' a b) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3468419195._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real f' (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMinOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMinOn f s a → HasLineDerivAt ℝ f f' a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["HasLineDerivAt"],["IsMinOn"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMinOn","isExtr"],["IsExtrOn","hasLineDerivAt_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsMinOn","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsMinOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966535._hygCtx._hyg.6 f a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMinOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMinOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["IsMinOn"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["Set"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMinOn","isExtr"],["IsExtrOn","lineDeriv_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsMaxOn","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsMaxOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966536._hygCtx._hyg.6 f a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMaxOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsMaxOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDeriv ℝ f a b = 0","typeReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["DenselyNormedField","toNontriviallyNormedField"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["instHAdd"],["Set"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Real","instZero"],["IsMaxOn"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["HSMul","hSMul"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Real"],["IsMaxOn","isExtr"],["IsExtrOn","lineDeriv_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsLocalMin","lineDeriv_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.15 : ContinuousAdd.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.12 (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3))))] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.18 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.12] {f : E -> Real} {a : E}, (IsLocalMin.{u_1, 0} E Real inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.12 Real.instPreorder f a) -> (Eq.{succ u_1} (E -> Real) (lineDeriv.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2458966538._hygCtx._hyg.6 f a) (OfNat.ofNat.{u_1} (E -> Real) 0 (Zero.toOfNat0.{u_1} (E -> Real) (Pi.instZero.{u_1, 0} E (fun (v : E) => Real) (fun (i : E) => Real.instZero)))))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalMin f a → lineDeriv ℝ f a = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]\n  [ContinuousSMul ℝ E] {f : E → ℝ} {a : E}, IsLocalMin f a → lineDeriv ℝ f a = 0","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NontriviallyNormedField","toNormedField"],["ContinuousSMul"],["DenselyNormedField","toNontriviallyNormedField"],["Zero","toOfNat0"],["IsLocalMin"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Real","normedAddCommGroup"],["NormedField","toNormedSpace"],["Real"],["lineDeriv"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["Real","instZero"],["AddCommMonoid","toAddCommSemigroup"],["ContinuousAdd"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Real","denselyNormedField"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsMaxFilter"],["IsMinFilter"],["Real","instPreorder"],["Real"],["Or","inl"],["nhds"],["IsLocalExtr","lineDeriv_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["IsExtrOn","lineDerivWithin_eq_zero"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E}, (IsExtrOn.{u_1, 0} E Real Real.instPreorder f s a) -> (Filter.Eventually.{0} Real (fun (t : Real) => Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) s (HAdd.hAdd.{u_1, u_1, u_1} E E E (instHAdd.{u_1} E (AddCommMagma.toAdd.{u_1} E (AddCommSemigroup.toAddCommMagma.{u_1} E (AddCommMonoid.toAddCommSemigroup.{u_1} E (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3))))) a (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.6))))) t b))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) -> (Eq.{1} Real (lineDerivWithin.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.3060096283._hygCtx._hyg.6 f s a b) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsExtrOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E},\n  IsExtrOn f s a → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → lineDerivWithin ℝ f s a b = 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{E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213247._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213247._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213247._hygCtx._hyg.3)] {f : E -> Real} {s : Set.{u_1} E} {a : E} {b : E} {f' : Real}, (IsMinOn.{u_1, 0} E Real Real.instPreorder f s a) -> (HasLineDerivWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.normedAddCommGroup (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) E inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213247._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.LocalExtr.LineDeriv.2164213247._hygCtx._hyg.6 f f' s a b) -> (Filter.Eventually.{0} Real 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IsMinOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 0","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ},\n  IsMinOn f s a → HasLineDerivWithinAt ℝ f f' s a b → (∀ᶠ (t : ℝ) in nhds 0, a + t • b ∈ s) → f' = 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Normed.Group.InfiniteSum.sym.json

@@ -0,0 +1 @@
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{ι : Type.{u_1}} {α : Type.{u_2}} {E : Type.{u_3}} [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.2453316630._hygCtx._hyg.7 : SeminormedAddCommGroup.{u_3} E] {f : ι -> E}, (Summable.{0, u_1} Real ι Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (a : ι) => Norm.norm.{u_3} E (SeminormedAddCommGroup.toNorm.{u_3} E inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.2453316630._hygCtx._hyg.7) (f a)) (SummationFilter.unconditional.{u_1} ι)) -> (forall {s : α -> (Finset.{u_1} ι)} {p : Filter.{u_2} α} [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.2453316630._hygCtx._hyg.37 : Filter.NeBot.{u_2} α p], (Filter.Tendsto.{u_2, u_1} α (Finset.{u_1} ι) s p (Filter.atTop.{u_1} (Finset.{u_1} ι) (PartialOrder.toPreorder.{u_1} (Finset.{u_1} ι) (Finset.partialOrder.{u_1} ι)))) -> (forall {a : E}, (Filter.Tendsto.{u_2, u_3} α E (fun (b : α) => Finset.sum.{u_1, u_3} ι E (AddCommGroup.toAddCommMonoid.{u_3} E 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{E : Type.{u_3}} [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.1616744540._hygCtx._hyg.7 : SeminormedAddCommGroup.{u_3} E] {f : Nat -> E} {g : Nat -> Real}, (CauchySeq.{0, 0} Real Nat (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace) Nat.instPreorder (fun (n : Nat) => Finset.sum.{0, 0} Nat Real Real.instAddCommMonoid (Finset.range n) (fun (i : Nat) => g i))) -> (forall (i : Nat), LE.le.{0} Real Real.instLE (Norm.norm.{u_3} E (SeminormedAddCommGroup.toNorm.{u_3} E inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.1616744540._hygCtx._hyg.7) (f i)) (g i)) -> (CauchySeq.{u_3, 0} E Nat (PseudoMetricSpace.toUniformSpace.{u_3} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_3} E inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.1616744540._hygCtx._hyg.7)) Nat.instPreorder (fun (n : Nat) => Finset.sum.{0, u_3} Nat E (AddCommGroup.toAddCommMonoid.{u_3} E (SeminormedAddCommGroup.toAddCommGroup.{u_3} E 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{r₁ : NNReal} {r₂ : NNReal}, Eq.{1} Prop (LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal NNReal.instPartialOrder)) r₁ r₂) (LE.le.{0} Real Real.instLE (NNReal.toReal r₁) (NNReal.toReal r₂))","typeFull":"∀ {r₁ r₂ : NNReal}, (r₁ ≤ r₂) = (↑r₁ ≤ ↑r₂)","typeReadable":"∀ {r₁ r₂ : NNReal}, (r₁ ≤ r₂) = (↑r₁ ≤ ↑r₂)","typeReferences":[["PartialOrder","toPreorder"],["Real"],["LE","le"],["NNReal"],["NNReal","toReal"],["NNReal","instPartialOrder"],["Preorder","toLE"],["Eq"],["Real","instLE"]],"valueReferences":[["NNReal","coe_le_coe"],["PartialOrder","toPreorder"],["Real"],["NNReal"],["LE","le"],["NNReal","toReal"],["Eq","symm"],["NNReal","instPartialOrder"],["Preorder","toLE"],["propext"],["Real","instLE"]]},{"isProp":true,"kind":"theorem","name":["Summable","of_norm"],"typeFallback":"forall {ι : Type.{u_1}} {E : Type.{u_3}} [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.196094043._hygCtx._hyg.7 : SeminormedAddCommGroup.{u_3} E] 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f","typeReferences":[["SeminormedAddCommGroup","toNorm"],["SeminormedAddCommGroup","toAddCommGroup"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Norm","norm"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["SummationFilter","unconditional"],["SeminormedAddCommGroup"],["Summable"],["CompleteSpace"],["Real","pseudoMetricSpace"],["AddCommGroup","toAddCommMonoid"],["Real","instAddCommMonoid"]],"valueReferences":[["le_rfl"],["SeminormedAddCommGroup","toNorm"],["Real","instPreorder"],["Real"],["Norm","norm"],["Summable","of_norm_bounded"]]},{"isProp":true,"kind":"theorem","name":["enorm_tsum_le_tsum_enorm"],"typeFallback":"forall {ι : Type.{u_1}} {ε : Type.{u_5}} [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.686788960._hygCtx._hyg.13 : TopologicalSpace.{u_5} ε] [inst._@.Mathlib.Analysis.Normed.Group.InfiniteSum.686788960._hygCtx._hyg.16 : ESeminormedAddCommMonoid.{u_5} ε 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.SpecialFunctions.MulExpNegMulSq.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","SpecialFunctions","MulExpNegMulSq",0,"Real","norm_deriv_mulExpNegMulSq_le_one","_simp_1_2"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.3 : AddCommGroup.{u} α] [inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.6 : LE.{u} α] [inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.9 : AddLeftMono.{u} α (AddZero.toAdd.{u} α (AddZeroClass.toAddZero.{u} α (AddMonoid.toAddZeroClass.{u} α (SubNegMonoid.toAddMonoid.{u} α (AddGroup.toSubNegMonoid.{u} α (AddCommGroup.toAddGroup.{u} α inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.3)))))) inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.6] {a : α} {b : α} {c : α}, Eq.{1} Prop (LE.le.{u} α inst._@.Mathlib.Algebra.Order.Group.Unbundled.Basic.1008422319._hygCtx._hyg.6 (Neg.neg.{u} α 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c : α}, (-b ≤ a - c) = (c ≤ a + b)","typeReadable":"∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LE α] [AddLeftMono α] {a b c : α}, (-b ≤ a - c) = (c ≤ a + b)","typeReferences":[["AddLeftMono"],["SubtractionMonoid","toSubNegZeroMonoid"],["instHAdd"],["Neg","neg"],["AddCommGroup","toAddGroup"],["LE"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["SubNegZeroMonoid","toNegZeroClass"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["LE","le"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["neg_le_sub_iff_le_add"],["instHAdd"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["SubNegZeroMonoid","toNegZeroClass"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toSub"],["LE","le"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Real","mulExpNegMulSq","eq_1"],"typeFallback":"forall (ε : Real) (x : Real), Eq.{1} Real (Real.mulExpNegMulSq ε x) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) x (Real.exp (Neg.neg.{0} Real Real.instNeg (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) ε x) x))))","typeFull":"∀ (ε x : ℝ), ε.mulExpNegMulSq x = x * Real.exp (-(ε * x * x))","typeReadable":"∀ (ε x : ℝ), ε.mulExpNegMulSq x = x * Real.exp (-(ε * x * x))","typeReferences":[["Real","instMul"],["Real","exp"],["Real"],["Neg","neg"],["Real","instNeg"],["instHMul"],["HMul","hMul"],["Real","mulExpNegMulSq"],["Eq"]],"valueReferences":[["Real"],["Eq","refl"],["Real","mulExpNegMulSq"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","SpecialFunctions","MulExpNegMulSq",0,"Real","norm_deriv_mulExpNegMulSq_le_one","_simp_1_3"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Sub.Defs.3276953686._hygCtx._hyg.3 : LE.{u_1} α] 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{ε : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) ε) -> (forall (x : Real), LE.le.{0} NNReal (Preorder.toLE.{0} NNReal (PartialOrder.toPreorder.{0} NNReal NNReal.instPartialOrder)) (NNNorm.nnnorm.{0} Real (SeminormedAddGroup.toNNNorm.{0} Real (SeminormedAddCommGroup.toSeminormedAddGroup.{0} Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) (deriv.{0, 0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real 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{ε : Real} (y : Real), HasDerivAt.{0, 0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ContinuousMul.to_continuousSMul.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real (SeminormedRing.toPseudoMetricSpace.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real (NormedField.toNormedCommRing.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) Real.instMul (IsTopologicalSemiring.toContinuousMul.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real (SeminormedRing.toPseudoMetricSpace.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real (NormedField.toNormedCommRing.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Real (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing))))) (IsTopologicalRing.toIsTopologicalSemiring.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Idempotents.SimplicialObject.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square.sym.json

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CategoryTheory.Category.{v, u} C] {sq₁ sq₂ : CategoryTheory.Square C} (e : sq₁ ≅ sq₂),\n  sq₁.IsPullback ↔ sq₂.IsPullback","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {sq₁ sq₂ : CategoryTheory.Square C} (e : sq₁ ≅ sq₂),\n  sq₁.IsPullback ↔ sq₂.IsPullback","typeReferences":[["CategoryTheory","Iso"],["CategoryTheory","Square"],["Iff"],["CategoryTheory","Category"],["CategoryTheory","Square","IsPullback"],["CategoryTheory","Square","category"]],"valueReferences":[["CategoryTheory","Square"],["CategoryTheory","Iso","symm"],["CategoryTheory","Square","IsPullback","of_iso"],["CategoryTheory","Square","category"],["CategoryTheory","Square","IsPullback"],["Iff","intro"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Square","IsPullback","mono_f₁₂"],"typeFallback":"forall {C : Type.{u}} [inst._@.Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square.2484416161._hygCtx._hyg.3 : CategoryTheory.Category.{v, u} C] {sq : CategoryTheory.Square.{v, u} C 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Int.Lemmas.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Multiset.Lattice.sym.json

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