module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.W.Basic | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 46
} | [
{
"pp": "case zero.zero\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nh :\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n 0 =\n (fun n ↦\n have this := Nat.recOn n (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.W.Basic | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 46
} | [
{
"pp": "case zero.succ\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nm : ℕ\nh :\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n 0 =\n (fun n ↦\n have this := Nat.re... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.W.Basic | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 16
} | [
{
"pp": "case succ.zero\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.W.Basic | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 16
} | [
{
"pp": "case succ.zero\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.W.Basic | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 16
} | [
{
"pp": "case succ.zero\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 124,
"column": 41
} | {
"line": 125,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : ExactPairing X Y\n⊢ η_ X Y ▷ X ⊗≫ X ◁ ε_ X Y = ⊗𝟙.hom",
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",
"Eq.mpr",
"CategoryTheory.ExactPairing.evaluation_coevaluation",
"Cate... | by
convert! evaluation_coevaluation X Y <;> simp [monoidalComp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 219,
"column": 98
} | {
"line": 220,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\ng : ᘁX ⟶ Z\n⊢ (ᘁf) ≫ g = 𝟙 ᘁY ⊗≫ η_ (ᘁX) X ▷ ᘁY ⊗≫ ᘁX ◁ f ▷ ᘁY ⊗≫ (ᘁX ◁ ε_ (ᘁY) Y ≫ g ▷ 𝟙_ C) ⊗≫ 𝟙 Z",
"usedConstants": [
"CategoryTheory.MonoidalCoh... | by
dsimp only [leftAdjointMate]; monoidal | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 30
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns t : Submodule K V\nhdisjoint : Disjoint s t\n⊢ finrank K ↥(s ⊔ t) ≤ finrank K V",
"usedConstants": [
"IsNoetherianRing.strongRankCondition",
"Submodule",
... | exact Submodule.finrank_le _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 36
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional K V\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nne_top : ¬span K (Set.range b) = ⊤\n⊢ False",
"usedCons... | ← finrank_span_eq_card lin_ind | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 435,
"column": 50
} | {
"line": 435,
"column": 78
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : End K V\nm : ℕ\nhm : finrank K V ≤ m\nk : ℕ\nh_k_le : k ≤ finrank K V\nhk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)\n⊢ LinearMap.ker (f ^ (k + (finrank K V - k)))... | add_tsub_cancel_of_le h_k_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber | {
"line": 31,
"column": 4
} | {
"line": 37,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : IsStablyFiniteRing R\ninst✝ : Nontrivial R\nn m : ℕ\nf : (Fin n → R) →ₗ[R] Fin m → R\nhf : Surjective ⇑f\n⊢ m ≤ n",
"usedConstants": [
"Pi.Function.module",
"False",
"Preorder.toLT",
"Semiring.toModule",
"Pi.addCommMonoid",
... | by_contra! lt
let p : (Fin m → R) →ₗ[R] Fin n → R := funLeft R R (Fin.castLE lt.le)
have hp : Surjective p := funLeft_surjective_of_injective _ _ _ (Fin.castLE_injective lt.le)
have : Injective p := .of_comp_right
(Module.End.injective_of_surjective_fin (f := p ∘ₗ f) (hp.comp hf)) hf
have ⟨⟨i, lt⟩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber | {
"line": 31,
"column": 4
} | {
"line": 37,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : IsStablyFiniteRing R\ninst✝ : Nontrivial R\nn m : ℕ\nf : (Fin n → R) →ₗ[R] Fin m → R\nhf : Surjective ⇑f\n⊢ m ≤ n",
"usedConstants": [
"Pi.Function.module",
"False",
"Preorder.toLT",
"Semiring.toModule",
"Pi.addCommMonoid",
... | by_contra! lt
let p : (Fin m → R) →ₗ[R] Fin n → R := funLeft R R (Fin.castLE lt.le)
have hp : Surjective p := funLeft_surjective_of_injective _ _ _ (Fin.castLE_injective lt.le)
have : Injective p := .of_comp_right
(Module.End.injective_of_surjective_fin (f := p ∘ₗ f) (hp.comp hf)) hf
have ⟨⟨i, lt⟩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 47
} | [
{
"pp": "case a.h.h.a\nR : Type u_1\ninst✝ : Semiring R\nn m : ℕ\nf : Matrix (Fin m) (Fin n) R\ng : Matrix (Fin n) (Fin m) R\n⊢ opEquiv.mapMatrix.symm (opEquiv.mapMatrix f * opEquiv.mapMatrix g) = opEquiv.mapMatrix.symm 1 → m ≤ n ↔\n (transposeAddEquiv (Fin n) (Fin m) R).toEquiv g * (transposeAddEquiv (Fin m... | ← (transposeAddEquiv ..).injective.eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 564,
"column": 16
} | {
"line": 567,
"column": 7
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nX₁ X₂ Y : C\np₁ : ExactPairing X₁ Y\np₂ : ExactPairing X₂ Y\n⊢ ((ᘁ𝟙 Y) ≫ ᘁ𝟙 Y) = 𝟙 X₁",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryT... | by
-- Make all arguments explicit, because we want to find them by unification not synthesis.
rw [← @comp_leftAdjointMate C, Category.comp_id, @leftAdjointMate_id]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Contraction | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 99
} | [
{
"pp": "case neg\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nm : Type u_7\nn : Type u_8\ninst✝³ : Fintype m\ninst✝² : Finite n\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nbM : Basis m R M\n... | · rw [and_iff_not_or_not, Classical.not_not] at hij
rcases hij with hij | hij <;> simp [LinearMap.toMatrix_apply, Finsupp.single_eq_pi_single, hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Contraction | {
"line": 242,
"column": 84
} | {
"line": 243,
"column": 75
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nP : Type u_5\nQ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : AddCommMonoid Q\ninst✝⁴ : Module R M\ninst✝³ : Module R P\ninst✝² : Module R Q\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : (M →ₗ[R] P) ⊗[R] Q\n⊢ (rTensorH... | by
rw [← LinearEquiv.coe_toLinearMap, rTensorHomEquivHomRTensor_toLinearMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 90,
"column": 18
} | {
"line": 90,
"column": 54
} | [
{
"pp": "case mp\nR : Type u_1\nR₁ : Type u_2\nM : Type u_5\nM₁ : Type u_6\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI₁ I₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M\nv : n → M₁\n⊢ (∀ (i j : n... | exact fun h i j hij ↦ h j i hij.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 90,
"column": 18
} | {
"line": 90,
"column": 54
} | [
{
"pp": "case mpr\nR : Type u_1\nR₁ : Type u_2\nM : Type u_5\nM₁ : Type u_6\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI₁ I₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M\nv : n → M₁\n⊢ (∀ (i j : ... | exact fun h i j hij ↦ h j i hij.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 47
} | [
{
"pp": "case h.mpr\nK : Type u_13\nV : Type u_16\nV₂ : Type u_18\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nJ : K →+* K\nB : V →ₗ[K] V →ₛₗ[J] V₂\nx y : V\nh : (B x) y = 0\nz : K\n⊢ B.IsOrtho (z • x) y",
"usedConstants": [
"Eq.mpr",
... | rw [isOrtho_def, map_smulₛₗ₂, smul_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1022,
"column": 27
} | {
"line": 1022,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1022,
"column": 27
} | {
"line": 1022,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1022,
"column": 27
} | {
"line": 1022,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1023,
"column": 27
} | {
"line": 1023,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1023,
"column": 27
} | {
"line": 1023,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 1023,
"column": 27
} | {
"line": 1023,
"column": 35
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 123,
"column": 69
} | {
"line": 123,
"column": 72
} | [
{
"pp": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : ⟨j₁, x⟩.fst ⟶ l\ng : ⟨j₃, x'⟩.fst ⟶ l\nhfg : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'\ns : J\nα : Is... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 141,
"column": 52
} | {
"line": 141,
"column": 55
} | [
{
"pp": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : ⟨j₁, y⟩.fst ⟶ l\ng : ⟨j₃, y'⟩.fst ⟶ l\nhfg : (ConcreteCategory.hom (F.map f)) y = (ConcreteCategory.hom (F.map g)) y'\ns : J\nα : Is... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ne : S₁ ≅ S₂\nh : S₁.ShortExact\nthis : Epi S₁.g\n⊢ Epi (e.hom.τ₂ ≫ S₂.g)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.H... | rw [e.hom.comm₂₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 56
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\nhS : S.Exact\nh : S.HomologyData\nthis✝ : Epi h.left.f'\nthis : Mono h.right.g'\nS' : ShortComplex C := { X₁ := h.left.K, X₂ := S.X₂, X₃ := S.X₃, f := h.left.i, g := S.g, zero := ⋯ }\nS'' : ShortComplex C := { X₁ :... | ← ShortComplex.exact_iff_of_epi_of_isIso_of_mono a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 411,
"column": 72
} | {
"line": 412,
"column": 89
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nv : V\n⊢ (∀ φ ∈ dualAnnihilator W, φ v = 0) ↔ v ∈ W",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Submodule.dualCoannihilator",
"congrArg",
... | by
rw [← SetLike.ext_iff.mp dualAnnihilator_dualCoannihilator_eq v, mem_dualCoannihilator] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Filtered | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ IsCofiltered C ↔\n ∀ {J : Type v} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), ∃ X, Nonempty (limit (F ⋙ coyoneda.obj (op X)))",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits.Cone",
"Opposit... | rw [IsCofiltered.iff_cone_nonempty.{v}] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 62
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nα : Type u_1\nZ : α → C\nf f' : Fan Z\nc : Cofan fun x ↦ op (Z x)\nhf : IsLimit f\nhf' : IsLimit f'\nhc : IsColimit c\nj : α\n⊢ ((f'.proj j).op ≫ (opProductIsoCoproduct' hf' hc).hom) ≫ (opProductIsoCoproduct' hf' hc).inv =\n (f'.π.app { as := j }).... | simp only [Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nT : H\nf g : (StructuredArrow.right E).obj Y ⟶ T\nhfg :\n ∀ ⦃X : C⦄ (φ : L... | simpa using hfg j.hom | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nT : H\nf g : (StructuredArrow.right E).obj Y ⟶ T\nhfg :\n ∀ ⦃X : C⦄ (φ : L... | simpa using hfg j.hom | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nT : H\nf g : (StructuredArrow.right E).obj Y ⟶ T\nhfg :\n ∀ ⦃X : C⦄ (φ : L... | simpa using hfg j.hom | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 554,
"column": 4
} | {
"line": 555,
"column": 78
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nX₁ X₂ : C\nf : X₁ ⟶ X₂\n⊢ F.map f ≫ coli... | simp only [comp_obj, pointwiseLeftKanExtension_obj, comp_map,
pointwiseLeftKanExtension_map, colimit.ι_desc, CostructuredArrow.map_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseRightKanExtension F\nX₁ X₂ : C\nf : X₁ ⟶ X₂\n⊢ limit.π (Stru... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 68
} | [
{
"pp": "case mpr\nC : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_3, u_3} H\ninst✝ : Category.{v_4, u_4} D\nL : C ⥤ D\nF₁ F₂ : C ⥤ H\nF₁' F₂' : D ⥤ H\nα₁ : F₁ ⟶ L ⋙ F₁'\nα₂ : F₂ ⟶ L ⋙ F₂'\ne : F₁ ≅ F₂\ne' : F₁' ≅ F₂'\nh : α₁ ≫ L.whiskerLeft e'.hom = e.hom ≫ α₂\neq... | · exact fun _ => ⟨⟨eq.2 (isUniversalOfIsLeftKanExtension F₂' α₂)⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 786,
"column": 30
} | {
"line": 786,
"column": 66
} | [
{
"pp": "C : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_3, u_3} H\ninst✝³ : Category.{v_4, u_4} D\nF' : D ⥤ H\nL : C ⥤ D\nF : C ⥤ H\nα : F ⟶ L ⋙ F'\ninst✝² : F'.IsLeftKanExtension α\ninst✝¹ : HasColimit F\ninst✝ : HasColimit F'\ni : C\n⊢ colimit.ι F i = α.app i ≫ ... | ι_colimitIsoOfIsLeftKanExtension_hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Constructions.WeaklyInitial | {
"line": 51,
"column": 2
} | {
"line": 63,
"column": 29
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasWideEqualizers C\nT : C\nhT : ∀ (X : C), Nonempty (T ⟶ X)\nendos : Type v := T ⟶ T\ni : wideEqualizer id ⟶ T := wideEqualizer.ι id\nthis : Nonempty endos\n⊢ HasInitial C",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
... | have : ∀ X : C, Unique (wideEqualizer (id : endos → endos) ⟶ X) := by
intro X
refine ⟨⟨i ≫ Classical.choice (hT X)⟩, fun a => ?_⟩
let E := equalizer a (i ≫ Classical.choice (hT _))
let e : E ⟶ wideEqualizer id := equalizer.ι _ _
let h : T ⟶ E := Classical.choice (hT E)
have : ((i ≫ h) ≫ e) ≫ i =... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 121,
"column": 17
} | {
"line": 121,
"column": 53
} | [
{
"pp": "J : Type w\n⊢ Function.LeftInverse\n (fun x ↦\n match x with\n | none => Arrow.mk (𝟙 zero)\n | some none => Arrow.mk (𝟙 one)\n | some (some t) => Arrow.mk (line t))\n fun f ↦\n match f.left, f.right, f.hom with\n | zero, .(zero), Hom.id .(zero) => none\n | one, .(on... | rintro ⟨(_ | _), _, (_ | _)⟩ <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 121,
"column": 17
} | {
"line": 121,
"column": 53
} | [
{
"pp": "J : Type w\n⊢ Function.LeftInverse\n (fun x ↦\n match x with\n | none => Arrow.mk (𝟙 zero)\n | some none => Arrow.mk (𝟙 one)\n | some (some t) => Arrow.mk (line t))\n fun f ↦\n match f.left, f.right, f.hom with\n | zero, .(zero), Hom.id .(zero) => none\n | one, .(on... | rintro ⟨(_ | _), _, (_ | _)⟩ <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 121,
"column": 17
} | {
"line": 121,
"column": 53
} | [
{
"pp": "J : Type w\n⊢ Function.LeftInverse\n (fun x ↦\n match x with\n | none => Arrow.mk (𝟙 zero)\n | some none => Arrow.mk (𝟙 one)\n | some (some t) => Arrow.mk (line t))\n fun f ↦\n match f.left, f.right, f.hom with\n | zero, .(zero), Hom.id .(zero) => none\n | one, .(on... | rintro ⟨(_ | _), _, (_ | _)⟩ <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 476,
"column": 4
} | {
"line": 477,
"column": 35
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\n𝒢 : ObjectProperty C\nh𝒢 : 𝒢.IsDetecting\nX : C\nP Q : Subobject X\nh₁ : P ≤ Q\nh₂ : ∀ (G : C), 𝒢 G → ∀ {f : G ⟶ X}, Q.Factors f → P.Factors f\nG : C\nhG : 𝒢 G\nf : G ⟶ underlying.obj Q\nthis : P.Factors (f ≫ Q.arrow)\ng : G ⟶ underlying.obj... | simp only [← cancel_mono (Subobject.ofLE _ _ h₁), ← cancel_mono Q.arrow, hg, Category.assoc,
ofLE_arrow, factorThru_arrow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 476,
"column": 4
} | {
"line": 477,
"column": 35
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\n𝒢 : ObjectProperty C\nh𝒢 : 𝒢.IsDetecting\nX : C\nP Q : Subobject X\nh₁ : P ≤ Q\nh₂ : ∀ (G : C), 𝒢 G → ∀ {f : G ⟶ X}, Q.Factors f → P.Factors f\nG : C\nhG : 𝒢 G\nf : G ⟶ underlying.obj Q\nthis : P.Factors (f ≫ Q.arrow)\ng : G ⟶ underlying.obj... | simp only [← cancel_mono (Subobject.ofLE _ _ h₁), ← cancel_mono Q.arrow, hg, Category.assoc,
ofLE_arrow, factorThru_arrow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 476,
"column": 4
} | {
"line": 477,
"column": 35
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\n𝒢 : ObjectProperty C\nh𝒢 : 𝒢.IsDetecting\nX : C\nP Q : Subobject X\nh₁ : P ≤ Q\nh₂ : ∀ (G : C), 𝒢 G → ∀ {f : G ⟶ X}, Q.Factors f → P.Factors f\nG : C\nhG : 𝒢 G\nf : G ⟶ underlying.obj Q\nthis : P.Factors (f ≫ Q.arrow)\ng : G ⟶ underlying.obj... | simp only [← cancel_mono (Subobject.ofLE _ _ h₁), ← cancel_mono Q.arrow, hg, Category.assoc,
ofLE_arrow, factorThru_arrow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\nhf : Mono f\nhg : Mono g\ni : P ≅ Q\nhi : i.ho... | have := congr_arg Quiver.Hom.unop hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 705,
"column": 2
} | {
"line": 711,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasZeroMorphisms C\nG H : C\ninst✝ : HasBinaryCoproduct G H\n⊢ IsSeparator (G ⨿ H) ↔ (ObjectProperty.pair G H).IsSeparating",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.ObjectProperty.ofObj",
"_private.Mathlib.... | refine (isSeparator_iff_of_isColimit_cofan (coprodIsCoprod G H)).trans ?_
convert! Iff.rfl
ext X
simp only [ObjectProperty.pair_iff, ObjectProperty.ofObj_iff]
constructor
· rintro (rfl | rfl); exacts [⟨.left, rfl⟩, ⟨.right, rfl⟩]
· rintro ⟨⟨_ | _⟩, rfl⟩ <;> tauto | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 705,
"column": 2
} | {
"line": 711,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasZeroMorphisms C\nG H : C\ninst✝ : HasBinaryCoproduct G H\n⊢ IsSeparator (G ⨿ H) ↔ (ObjectProperty.pair G H).IsSeparating",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.ObjectProperty.ofObj",
"_private.Mathlib.... | refine (isSeparator_iff_of_isColimit_cofan (coprodIsCoprod G H)).trans ?_
convert! Iff.rfl
ext X
simp only [ObjectProperty.pair_iff, ObjectProperty.ofObj_iff]
constructor
· rintro (rfl | rfl); exacts [⟨.left, rfl⟩, ⟨.right, rfl⟩]
· rintro ⟨⟨_ | _⟩, rfl⟩ <;> tauto | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nh : HasSeparator Cᵒᵖ\n⊢ HasCoseparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasSeparat... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nh : HasSeparator Cᵒᵖ\n⊢ HasCoseparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasSeparat... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nh : HasSeparator Cᵒᵖ\n⊢ HasCoseparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasSeparat... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 972,
"column": 25
} | {
"line": 972,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCoseparator Cᵒᵖ\n⊢ HasSeparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 972,
"column": 25
} | {
"line": 972,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCoseparator Cᵒᵖ\n⊢ HasSeparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 972,
"column": 25
} | {
"line": 972,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCoseparator Cᵒᵖ\n⊢ HasSeparator C",
"usedConstants": [
"CategoryTheory.HasSeparator",
"Opposite",
"CategoryTheory.HasCoseparator",
"Eq.mp",
"id",
"CategoryTheory.Category.opposite",
"CategoryTheory.hasCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 976,
"column": 26
} | {
"line": 976,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasDetector Cᵒᵖ\n⊢ HasCodetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.Category.opposite",
"CategoryTheory.HasDetector",
"CategoryTheory.hasDetect... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 976,
"column": 26
} | {
"line": 976,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasDetector Cᵒᵖ\n⊢ HasCodetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.Category.opposite",
"CategoryTheory.HasDetector",
"CategoryTheory.hasDetect... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 976,
"column": 26
} | {
"line": 976,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasDetector Cᵒᵖ\n⊢ HasCodetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.Category.opposite",
"CategoryTheory.HasDetector",
"CategoryTheory.hasDetect... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 980,
"column": 24
} | {
"line": 980,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCodetector Cᵒᵖ\n⊢ HasDetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.hasCodetector_op_iff._simp_1",
"CategoryTheory.Category.opposite",
"Categor... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 980,
"column": 24
} | {
"line": 980,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCodetector Cᵒᵖ\n⊢ HasDetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.hasCodetector_op_iff._simp_1",
"CategoryTheory.Category.opposite",
"Categor... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 980,
"column": 24
} | {
"line": 980,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasCodetector Cᵒᵖ\n⊢ HasDetector C",
"usedConstants": [
"Opposite",
"Eq.mp",
"id",
"CategoryTheory.HasCodetector",
"CategoryTheory.hasCodetector_op_iff._simp_1",
"CategoryTheory.Category.opposite",
"Categor... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 115,
"column": 16
} | {
"line": 123,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁ v₂)\nE : ℰ\nf :\n CostructuredArrow.proj uliftYoneda.{max w v₂, v₁, u₁} P ⋙ A ⟶\n (Functor.const (CostructuredArrow uliftYoneda.{max w v₂, v₁, u₁} P)).obj E\n⊢ (fun g ↦ { app := ... | by
ext X
let e : CostructuredArrow.mk
(uliftYonedaEquiv.{max w v₂}.symm (X.hom.app (op X.left) ⟨𝟙 X.left⟩)) ≅ X :=
CostructuredArrow.isoMk (Iso.refl _) (by
ext Y x
dsimp
simp [← NatTrans.naturality_apply])
simpa [e] using f.naturality e.inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 98
} | [
{
"pp": "case hγ.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)\nφ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanEx... | exact _root_.congr_arg _ (compULiftYonedaIsoULiftYonedaCompLan_inv_app_app_apply_eq_id F X).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 45
} | [
{
"pp": "X Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective ⇑(ConcreteCategory.hom c.inl)\nh₂ : Injective ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (Set.range ⇑(ConcreteCategory.hom c.inl)) (Set.range ⇑(ConcreteCategory.hom c.inr))\n⊢ ∀ (x : (fun X ↦ X) (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { ... | rw [eq_compl_iff_isCompl.mpr h₃.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝ : Add α\nh : Periodic f c\ng : β → γ\n⊢ Periodic (g ∘ f) c",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"of_eq_true",
"congr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝ : Add α\nh : Periodic f c\ng : β → γ\n⊢ Periodic (g ∘ f) c",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"of_eq_true",
"congr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝ : Add α\nh : Periodic f c\ng : β → γ\n⊢ Periodic (g ∘ f) c",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"of_eq_true",
"congr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 51,
"column": 95
} | {
"line": 52,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝ : Add α\nh : Periodic f c\ng : β → γ\n⊢ Periodic (g ∘ f) c",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"of_eq_true",
"congr... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Periodic | {
"line": 60,
"column": 29
} | {
"line": 60,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Mul β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"congr",
"True",
"eq_self",
"o... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 60,
"column": 29
} | {
"line": 60,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Mul β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"congr",
"True",
"eq_self",
"o... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 60,
"column": 29
} | {
"line": 60,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Mul β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"congr",
"True",
"eq_self",
"o... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Div β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f / g) c",
"usedConstants": [
"instHDiv",
"congrArg",
"Pi.instDiv",
"id",
"HDiv.hDiv",
"instHAdd",
"HAdd.hAdd",
"congr",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Div β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f / g) c",
"usedConstants": [
"instHDiv",
"congrArg",
"Pi.instDiv",
"id",
"HDiv.hDiv",
"instHAdd",
"HAdd.hAdd",
"congr",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : Div β\nhf : Periodic f c\nhg : Periodic g c\n⊢ Periodic (f / g) c",
"usedConstants": [
"instHDiv",
"congrArg",
"Pi.instDiv",
"id",
"HDiv.hDiv",
"instHAdd",
"HAdd.hAdd",
"congr",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : SMul γ β\nh : Periodic f c\na : γ\n⊢ Periodic (a • f) c",
"usedConstants": [
"instHSMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"Pi.instSMul",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : SMul γ β\nh : Periodic f c\na : γ\n⊢ Periodic (a • f) c",
"usedConstants": [
"instHSMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"Pi.instSMul",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝¹ : Add α\ninst✝ : SMul γ β\nh : Periodic f c\na : γ\n⊢ Periodic (a • f) c",
"usedConstants": [
"instHSMul",
"congrArg",
"id",
"instHAdd",
"HAdd.hAdd",
"True",
"eq_self",
"Pi.instSMul",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 380,
"column": 66
} | {
"line": 380,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : Add α\ninst✝² : Monoid γ\ninst✝¹ : AddGroup β\ninst✝ : DistribMulAction γ β\nh : Antiperiodic f c\na : γ\n⊢ Antiperiodic (a • f) c",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"instH... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 380,
"column": 66
} | {
"line": 380,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : Add α\ninst✝² : Monoid γ\ninst✝¹ : AddGroup β\ninst✝ : DistribMulAction γ β\nh : Antiperiodic f c\na : γ\n⊢ Antiperiodic (a • f) c",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"instH... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 380,
"column": 66
} | {
"line": 380,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : Add α\ninst✝² : Monoid γ\ninst✝¹ : AddGroup β\ninst✝ : DistribMulAction γ β\nh : Antiperiodic f c\na : γ\n⊢ Antiperiodic (a • f) c",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"instH... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 413,
"column": 55
} | {
"line": 413,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝² : Add α\ninst✝¹ : Mul β\ninst✝ : HasDistribNeg β\nhf : Antiperiodic f c\nhg : Antiperiodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"neg_neg",
"id",
"instHAdd",
"HAdd.hAdd",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Ring.Periodic | {
"line": 413,
"column": 55
} | {
"line": 413,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝² : Add α\ninst✝¹ : Mul β\ninst✝ : HasDistribNeg β\nhf : Antiperiodic f c\nhg : Antiperiodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"neg_neg",
"id",
"instHAdd",
"HAdd.hAdd",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Periodic | {
"line": 413,
"column": 55
} | {
"line": 413,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\nc : α\ninst✝² : Add α\ninst✝¹ : Mul β\ninst✝ : HasDistribNeg β\nhf : Antiperiodic f c\nhg : Antiperiodic g c\n⊢ Periodic (f * g) c",
"usedConstants": [
"HMul.hMul",
"congrArg",
"neg_neg",
"id",
"instHAdd",
"HAdd.hAdd",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Count | {
"line": 41,
"column": 41
} | {
"line": 41,
"column": 53
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\n⊢ count p 0 = 0",
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_self",
"of_eq_true",
"OfNat.ofNat",
"Eq"
]
}
] | simp [count] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Count | {
"line": 41,
"column": 41
} | {
"line": 41,
"column": 53
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\n⊢ count p 0 = 0",
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_self",
"of_eq_true",
"OfNat.ofNat",
"Eq"
]
}
] | simp [count] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Count | {
"line": 41,
"column": 41
} | {
"line": 41,
"column": 53
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\n⊢ count p 0 = 0",
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_self",
"of_eq_true",
"OfNat.ofNat",
"Eq"
]
}
] | simp [count] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Totient | {
"line": 336,
"column": 8
} | {
"line": 337,
"column": 36
} | [
{
"pp": "a b a1 a2 b1 b2 c1 c2 : ℕ\nh1 : b1 ∣ a1\nh2 : b2 ∣ a2\n⊢ a1 / b1 * (a2 / b2) * (c1 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)",
"usedConstants": [
"instHDiv",
"HMul.hMul",
"HMul",
"Eq.rec",
"HDiv.hDiv",
"instMulNat",
"Nat",
"Nat.instDiv",
"Eq.refl"... | congr 1
exact div_mul_div_comm h1 h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Totient | {
"line": 336,
"column": 8
} | {
"line": 337,
"column": 36
} | [
{
"pp": "a b a1 a2 b1 b2 c1 c2 : ℕ\nh1 : b1 ∣ a1\nh2 : b2 ∣ a2\n⊢ a1 / b1 * (a2 / b2) * (c1 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)",
"usedConstants": [
"instHDiv",
"HMul.hMul",
"HMul",
"Eq.rec",
"HDiv.hDiv",
"instMulNat",
"Nat",
"Nat.instDiv",
"Eq.refl"... | congr 1
exact div_mul_div_comm h1 h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 74
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\ninst✝ : Finite ι\nF : ι → β → α\n⊢ UniformEquicontinuous F ↔ ∀ (i : ι), UniformContinuous (F i)",
"usedConstants": [
"Filter.instMembership",
"UniformContinuous",
"Eq.mpr",
"congrArg",
... | simp only [UniformEquicontinuous, eventually_all, @forall_comm _ ι]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 74
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\ninst✝ : Finite ι\nF : ι → β → α\n⊢ UniformEquicontinuous F ↔ ∀ (i : ι), UniformContinuous (F i)",
"usedConstants": [
"Filter.instMembership",
"UniformContinuous",
"Eq.mpr",
"congrArg",
... | simp only [UniformEquicontinuous, eventually_all, @forall_comm _ ι]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 30
} | [
{
"pp": "case mp\nι : Type u_1\nX : Type u_3\nα : Type u_6\ntX : TopologicalSpace X\nuα : UniformSpace α\nF : ι → X → α\nS : Set X\nx₀ : X\nhx₀ : x₀ ∈ S\nH : EquicontinuousWithinAt F S x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set (α × α)\nhV : V ∈ 𝓤 α\nhVsymm : SetRel.IsSymm V\nhVU : SetRel.comp V V ⊆ U\nx : X\... | exact SetRel.symm V (hx i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.AtTopBot.Group | {
"line": 155,
"column": 51
} | {
"line": 155,
"column": 66
} | [
{
"pp": "G : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : LinearOrder G\ninst✝ : IsOrderedMonoid G\na b x : G\nhx : x ≤ (max a⁻¹ b)⁻¹ ∨ a⁻¹ ≤ x ∧ b ≤ x\n⊢ x ∈ Iic (a, b).1 ∪ Ici (a, b).2",
"usedConstants": [
"Lattice.toSemilatticeSup",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.to... | ← min_inv_inv', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 684,
"column": 2
} | {
"line": 685,
"column": 71
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ UniformEquicontinuousOn F S ↔\n ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i... | rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 684,
"column": 2
} | {
"line": 686,
"column": 5
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ UniformEquicontinuousOn F S ↔\n ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i... | rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 684,
"column": 2
} | {
"line": 686,
"column": 5
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ UniformEquicontinuousOn F S ↔\n ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i... | rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LocalExtr | {
"line": 137,
"column": 72
} | {
"line": 138,
"column": 55
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder β\nf : α → β\na : α\n⊢ IsLocalMaxOn f univ a ↔ IsLocalMax f a",
"usedConstants": [
"nhdsWithin_univ",
"congrArg",
"Set.univ",
"nhdsWithin",
"nhds",
"IsLocalMaxOn",
"iff_self",
"Iff"... | by
simp only [IsLocalMaxOn, IsLocalMax, nhdsWithin_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Field | {
"line": 63,
"column": 28
} | {
"line": 67,
"column": 56
} | [
{
"pp": "K✝ : Type u_1\ninst✝⁴ : DivisionRing K✝\ninst✝³ : TopologicalSpace K✝\nα : Type u_2\ninst✝² : Field α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalDivisionRing α\nK : Subfield α\nx : α\nhx : x ∈ closure ↑K\n⊢ x⁻¹ ∈ closure ↑K",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidW... | by
rcases eq_or_ne x 0 with (rfl | h)
· rwa [inv_zero]
· rw [← inv_coe_set, ← Set.image_inv_eq_inv]
exact mem_closure_image (continuousAt_inv₀ h) hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 581,
"column": 4
} | {
"line": 581,
"column": 12
} | [
{
"pp": "case left.hf.hm\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹⁴ : TopologicalSpace α\ninst✝¹³ : AddCommGroup α\ninst✝¹² : IsTopologicalAddGroup α\ninst✝¹¹ : TopologicalSpace β\ninst✝¹⁰ : AddCommGroup β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : AddCommGroup γ\ninst✝⁷ : IsTopo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.Interval.Set.Pi | {
"line": 296,
"column": 34
} | {
"line": 296,
"column": 42
} | [
{
"pp": "case inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → LinearOrder (α i)\nx y x' y' a : (i : ι) → α i\nhxa : x ≤ a\nhay : a ≤ y\nw : ι\nhw : ¬(x' w < a w ∧ a w < y' w)\nh✝ : x' w < a w\n⊢ (True ∧ a w ≤ x' w ∧ ∀ (j : ι), j ≠ w → True) ∨ (y' w ≤ a w ∧ ∀ (j : ι), j ≠ w → True)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.Interval.Set.Pi | {
"line": 296,
"column": 34
} | {
"line": 296,
"column": 42
} | [
{
"pp": "case inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → LinearOrder (α i)\nx y x' y' a : (i : ι) → α i\nhxa : x ≤ a\nhay : a ≤ y\nw : ι\nhw : ¬(x' w < a w ∧ a w < y' w)\nh✝ : a w ≤ x' w\n⊢ (True ∧ a w ≤ x' w ∧ ∀ (j : ι), j ≠ w → True) ∨ (y' w ≤ a w ∧ ∀ (j : ι), j ≠ w → True)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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