module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Order.Basic | {
"line": 559,
"column": 2
} | {
"line": 561,
"column": 39
} | {
"line": 563,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : SeparableSpace α\n⊢ SecondCountableTopology α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"SecondCountableTopology",
"Parti... | [] | rcases exists_countable_dense α with ⟨s, hc, hd⟩
refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩
exact (hc.image _).union (hc.image _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 306,
"column": 6
} | {
"line": 309,
"column": 24
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case mpr.refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\nh : S.δlast.Exact\nh' : (mk₂ (S.map' n (n + 1) ⋯ ⋯) (S.map' (n + 1) (n + 2) ⋯ ⋯)).Exact\ni : ℕ\nhi : i + 2 ≤ n + 2\n⊢ (S.sc ⋯ i hi).Exact",
"ppTerm": "?mpr.refine_2",
... | [] | simp only [Nat.add_le_add_iff_right] at hi
obtain hi | rfl := hi.lt_or_eq
· exact h.exact i
· exact h'.exact 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 306,
"column": 6
} | {
"line": 309,
"column": 24
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case mpr.refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\nh : S.δlast.Exact\nh' : (mk₂ (S.map' n (n + 1) ⋯ ⋯) (S.map' (n + 1) (n + 2) ⋯ ⋯)).Exact\ni : ℕ\nhi : i + 2 ≤ n + 2\n⊢ (S.sc ⋯ i hi).Exact",
"ppTerm": "?mpr.refine_2",
... | [] | simp only [Nat.add_le_add_iff_right] at hi
obtain hi | rfl := hi.lt_or_eq
· exact h.exact i
· exact h'.exact 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 23
} | {
"line": 273,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\nA : C\nx₂ : A ⟶ S.L₀'.X₂\nhx₂ : x₂ ≫ S.L₀'.g = 0\n⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀'.f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\nA : C\nx₂ : A ⟶ S.P\nhx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0\n⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.L₀'.f"
] | dsimp [L₀'] at x₂ hx₂ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 357,
"column": 26
} | {
"line": 357,
"column": 69
} | {
"line": 358,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\n⊢ S.op.δ.unop = S.δ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.ShortComplex.SnakeInput.P'._proof_1",
"C... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\n⊢ S.op.δ.unop ≫ pushout.inr S.L₂.f S.v₂₃.τ₁ = S.δ ≫ pushout.inr S.L₂.f S.v₂₃.τ₁"
] | ← cancel_mono (pushout.inr _ _ : _ ⟶ S.P'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 96
} | {
"line": 530,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : SnakeInput C\nf : S₁ ⟶ S₂\n⊢ S₁.φ₁ ≫ f.f₂.τ₁ = functorP.map f ≫ S₂.φ₁",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
... | [] | simp only [← cancel_mono S₂.L₂.f, assoc, φ₁_L₂_f, ← naturality_φ₂, f.f₂.comm₁₂, φ₁_L₂_f_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 96
} | {
"line": 530,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : SnakeInput C\nf : S₁ ⟶ S₂\n⊢ S₁.φ₁ ≫ f.f₂.τ₁ = functorP.map f ≫ S₂.φ₁",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
... | [] | simp only [← cancel_mono S₂.L₂.f, assoc, φ₁_L₂_f, ← naturality_φ₂, f.f₂.comm₁₂, φ₁_L₂_f_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 96
} | {
"line": 530,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : SnakeInput C\nf : S₁ ⟶ S₂\n⊢ S₁.φ₁ ≫ f.f₂.τ₁ = functorP.map f ≫ S₂.φ₁",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
... | [] | simp only [← cancel_mono S₂.L₂.f, assoc, φ₁_L₂_f, ← naturality_φ₂, f.f₂.comm₁₂, φ₁_L₂_f_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | {
"line": 353,
"column": 2
} | {
"line": 355,
"column": 7
} | {
"line": 357,
"column": 0
} | [
{
"pp": "case some\nC : Type u\ninst✝⁴ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory C FC\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (... | [] | · right
use j, y
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 237,
"column": 14
} | {
"line": 237,
"column": 33
} | {
"line": 237,
"column": 33
} | [
{
"pp": "⊢ ∀ {W X Y Z : WalkingReflexivePair} (f : W.Hom X) (g : X.Hom Y) (h : Y.Hom Z), (f.comp g).comp h = f.comp (g.comp h)",
"ppTerm": "?m.444",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingReflexivePair.Hom",
"CategoryTheory.Limits.WalkingReflexivePair"
],
... | [
"W✝ X✝ Y✝ Z✝ : WalkingReflexivePair\nf : W✝.Hom X✝\ng : X✝.Hom Y✝\nh : Y✝.Hom Z✝\n⊢ (f.comp g).comp h = f.comp (g.comp h)"
] | intro _ _ _ _ f g h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 206,
"column": 8
} | {
"line": 209,
"column": 17
} | {
"line": 209,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nA B : J\nf : A ⟶ B\n⊢... | [] | ext1
dsimp
rw [comp_id]
apply c.w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 206,
"column": 8
} | {
"line": 209,
"column": 17
} | {
"line": 209,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nA B : J\nf : A ⟶ B\n⊢... | [] | ext1
dsimp
rw [comp_id]
apply c.w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 487,
"column": 11
} | {
"line": 487,
"column": 84
} | {
"line": 487,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF G : WalkingReflexivePair ⥤ C\ne₀ : F.obj zero ⟶ G.obj zero\ne₁ : F.obj one ⟶ G.obj one\nh₁ : F.map left ≫ e₀ = e₁ ≫ G.map left\nh₂ : F.map right ≫ e₀ = e₁ ≫ G.map right\nh₃ : F.map reflexion ≫ e₁ = e₀ ≫ G.map reflexion\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\... | [] | by simp only [Functor.comp_obj, Functor.comp_map, ← Functor.map_comp, h₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Over | {
"line": 363,
"column": 69
} | {
"line": 365,
"column": 71
} | {
"line": 367,
"column": 0
} | [
{
"pp": "R✝ : Type u_1\ninst✝¹⁶ : CommRing R✝\nR : Type u_2\ninst✝¹⁵ : CommSemiring R\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹⁴ : CommRing A\ninst✝¹³ : CommRing B\ninst✝¹² : CommRing C\ninst✝¹¹ : Algebra A B\ninst✝¹⁰ : Algebra A C\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra R B\ninst✝⁷ : IsScalarTower R A ... | [] | by
apply Ideal.isPrime_map_quotientMk_of_isPrime
rw [Ideal.map_le_iff_le_comap, Ideal.LiesOver.over (p := p) (P := P)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 148,
"column": 2
} | {
"line": 150,
"column": 85
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R N\np : Submonoid R\nf : M →ₗ[R] N\ninst✝ : IsLocalizedModule p f\nι : Type u_5\ng : ι → Submodule R M\n⊢ localized₀ p f (⨆ i, g i) = ⨆ i, localiz... | [] | let : Module (Localization p) N := IsLocalizedModule.module p f
have : IsScalarTower R (Localization p) N := IsLocalizedModule.isScalarTower_module p f
simpa using! congr_arg (restrictScalars R) (localized'_iSup (Localization p) p f g) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 148,
"column": 2
} | {
"line": 150,
"column": 85
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R N\np : Submonoid R\nf : M →ₗ[R] N\ninst✝ : IsLocalizedModule p f\nι : Type u_5\ng : ι → Submodule R M\n⊢ localized₀ p f (⨆ i, g i) = ⨆ i, localiz... | [] | let : Module (Localization p) N := IsLocalizedModule.module p f
have : IsScalarTower R (Localization p) N := IsLocalizedModule.isScalarTower_module p f
simpa using! congr_arg (restrictScalars R) (localized'_iSup (Localization p) p f g) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.Lemmas | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 7
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ≤ Submodule.comap f p\nk : ℕ\nhk : f ^ k = 0\n⊢ IsNilpotent (p.mapQ p f hp)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Submodule",
"Subm... | [
"case h\nR : Type u_1\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ≤ Submodule.comap f p\nk : ℕ\nhk : f ^ k = 0\n⊢ p.mapQ p f hp ^ k = 0"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Localization.Ideal | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 31
} | {
"line": 334,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : I.IsPrime\nJ : Ideal R\nH : J ≤ Ideal.under R I\nhI : IsField (R ⧸ Ideal.under R I)\nr m : R\nhm : m ∈ M\nhM : ¬(Ideal.Quotient.mk (Id... | [
"case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : I.IsPrime\nJ : Ideal R\nH : J ≤ Ideal.under R I\nhI : IsField (R ⧸ Ideal.under R I)\nr m : R\nhm : m ∈ M\nhM : ¬(Ideal.Quotient.mk (Ideal.comap (a... | rw [map_one, map_mul] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PolynomialAlgebra | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 55
} | {
"line": 91,
"column": 55
} | [
{
"pp": "R : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\np₁ p₂ : R[X]\nk : ℕ\n⊢ (if ¬(p₁ * p₂).coeff k = 0 then a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) else 0) =\n ∑ x ∈ Finset.antidiagonal k,\n if ¬p₂.coeff x.2 = 0 then\n (if ¬p₁.c... | [
"R : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\np₁ p₂ : R[X]\nk : ℕ\n⊢ (if ¬(p₁ * p₂).coeff k = 0 then a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) else 0) =\n ∑ x ∈ Finset.antidiagonal k,\n (if ¬p₁.coeff x.1 = 0 then a₁ * (algebraMap R A) (p₁.coe... | ← mul_ite_zero (¬coeff p₂ _ = 0) _ (_ * _) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 78
} | {
"line": 398,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | [] | simpa only [map_mul, ← RingHom.comp_apply, H] using DFunLike.congr_arg g e | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.LocalProperties.Exactness | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 88
} | {
"line": 96,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Idea... | [] | by rw [map_linearCombination]; exact H P | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalProperties.Exactness | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 59
} | {
"line": 266,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nRₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3\ninst✝⁹ : (p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)\ninst✝⁸ : (p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Rₚ p)\nSₚ : (p : Ideal R)... | [] | apply IsLocalizedModule.map_linearMap_of_isLocalization | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.LocalProperties.Exactness | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 59
} | {
"line": 277,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nRₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3\ninst✝⁹ : (p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)\ninst✝⁸ : (p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Rₚ p)\nSₚ : (p : Ideal R)... | [] | apply IsLocalizedModule.map_linearMap_of_isLocalization | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 37
} | {
"line": 339,
"column": 4
} | [
{
"pp": "case left\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\n⊢ ∀ a ∈ s,\n ∀ b ∈ s,\n ... | [
"case left\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\na✝⁵ : a ∈ s\nb : S\na✝⁴ : b ∈ s\n... | intro a _ b _ _ c _ d _ _ x hxP e | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 53
} | {
"line": 560,
"column": 4
} | [
{
"pp": "case refine_2\nR✝ : Type u_1\ninst✝¹⁸ : CommSemiring R✝\nS✝ : Type u_2\ninst✝¹⁷ : CommSemiring S✝\ninst✝¹⁶ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝¹⁵ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝¹⁴ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝¹³ : CommSemiring Rₚ✝\ninst✝¹² : Algebra R✝ Rₚ✝\ninst✝¹¹ : IsLocalization.AtPri... | [
"case refine_2\nR✝ : Type u_1\ninst✝¹⁸ : CommSemiring R✝\nS✝ : Type u_2\ninst✝¹⁷ : CommSemiring S✝\ninst✝¹⁶ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝¹⁵ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝¹⁴ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝¹³ : CommSemiring Rₚ✝\ninst✝¹² : Algebra R✝ Rₚ✝\ninst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝\ni... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.TensorProduct.Quotient | {
"line": 318,
"column": 23
} | {
"line": 318,
"column": 35
} | {
"line": 318,
"column": 36
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nM : Type u_4\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module A M\ninst✝⁴ : IsScalarTower R A M\nN : Type u_5\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module A N\ninst✝ : IsScalarTo... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.Submodule | {
"line": 148,
"column": 51
} | {
"line": 148,
"column": 60
} | {
"line": 148,
"column": 61
} | [
{
"pp": "case mp.refine_3.e_a\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ... | [
"case mp.refine_3.e_a\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Submodule.... | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Submodule | {
"line": 149,
"column": 56
} | {
"line": 149,
"column": 65
} | {
"line": 149,
"column": 66
} | [
{
"pp": "case mp.refine_3.e_a\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ... | [
"case mp.refine_3.e_a\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Submodule.... | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 589,
"column": 9
} | {
"line": 589,
"column": 34
} | {
"line": 589,
"column": 34
} | [
{
"pp": "R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\nh₁ : (Ideal.Quotient.mk p) ↑s ≠ 0\n⊢ (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk... | [
"R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\nh₁ : (Ideal.Quotient.mk p) ↑s ≠ 0\n⊢ (Ideal.Quotient.mk p) ↑s ≠ 0"
] | RingEquiv.map_ne_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 13
} | {
"line": 592,
"column": 14
} | [
{
"pp": "R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\nh₁ : (Ideal.Quotient.mk p) ↑s ≠ 0\nh₂ : (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient... | [
"R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\nh₁ : (Ideal.Quotient.mk p) ↑s ≠ 0\nh₂ : (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) ↑s) ≠... | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 30
} | {
"line": 306,
"column": 0
} | [
{
"pp": "case right\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P\nR✝ S T : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : CommRing S\ninst... | [] | exact hPc _ _ hf (hPl T s) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 33
} | {
"line": 509,
"column": 0
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : IsStableUnderBaseChange P\nR S Rᵣ Sᵣ : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing Rᵣ\ninst✝⁴ : CommRing Sᵣ\ninst✝³ : Algebra R Rᵣ\ninst✝² : Algebra S Sᵣ\nM : Submonoid R\ninst✝¹ : IsLocali... | [] | apply hP.of_isLocalization M hf | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 679,
"column": 4
} | {
"line": 679,
"column": 53
} | {
"line": 680,
"column": 4
} | [
{
"pp": "case refine_2\nR✝ : Type u_1\ninst✝²⁷ : CommSemiring R✝\nS✝ : Type u_2\ninst✝²⁶ : CommSemiring S✝\ninst✝²⁵ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝²⁴ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝²³ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝²² : CommSemiring Rₚ✝\ninst✝²¹ : Algebra R✝ Rₚ✝\ninst✝²⁰ : IsLocalization.AtPri... | [
"case refine_2\nR✝ : Type u_1\ninst✝²⁷ : CommSemiring R✝\nS✝ : Type u_2\ninst✝²⁶ : CommSemiring S✝\ninst✝²⁵ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝²⁴ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝²³ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝²² : CommSemiring Rₚ✝\ninst✝²¹ : Algebra R✝ Rₚ✝\ninst✝²⁰ : IsLocalization.AtPrime Rₚ✝ p✝\ni... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 69,
"column": 25
} | {
"line": 69,
"column": 73
} | {
"line": 69,
"column": 73
} | [
{
"pp": "case inr.a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function... | [
"case inr.a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Injective ⇑... | rw [Module.rank]; apply ciSup_le'; intro ⟨s, hs⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 69,
"column": 25
} | {
"line": 69,
"column": 73
} | {
"line": 69,
"column": 73
} | [
{
"pp": "case inr.a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function... | [
"case inr.a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Injective ⇑... | rw [Module.rank]; apply ciSup_le'; intro ⟨s, hs⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 69,
"column": 25
} | {
"line": 69,
"column": 73
} | {
"line": 69,
"column": 73
} | [
{
"pp": "case a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Inj... | [
"case a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Injective ⇑(alg... | rw [Module.rank]; apply ciSup_le'; intro ⟨s, hs⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 69,
"column": 25
} | {
"line": 69,
"column": 73
} | {
"line": 69,
"column": 73
} | [
{
"pp": "case a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Inj... | [
"case a\nR : Type uR\nS : Type uS\nN : Type uN\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nhp : p ≤ R⁰\nh✝ : Nontrivial R\ninj : Function.Injective ⇑(alg... | rw [Module.rank]; apply ciSup_le'; intro ⟨s, hs⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 20
} | {
"line": 502,
"column": 21
} | [
{
"pp": "R : Type u_1\nS✝ : Type u_2\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf✝ : R →+* S\nx : Sˣ\nf : R[X]\n⊢ ∃ c, ↑c * f = ↑c * f",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 401,
"column": 6
} | {
"line": 401,
"column": 32
} | {
"line": 402,
"column": 4
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCol M).lengt... | [] | simp [n', listTransvecCol] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 233,
"column": 2
} | {
"line": 237,
"column": 36
} | {
"line": 239,
"column": 0
} | [
{
"pp": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\ne : n ≃ m\nA : Matrix m m α\n⊢ (A.submatrix ⇑e ⇑e).adjugate = A.adjugate.submatrix ⇑e ⇑e",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | ext i j
have : (fun j ↦ Pi.single i 1 <| e.symm j) = Pi.single (e i) 1 :=
Function.update_comp_equiv (0 : n → α) e.symm i 1
rw [adjugate_apply, submatrix_apply, adjugate_apply, ← det_submatrix_equiv_self e,
updateRow_submatrix_equiv, this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 233,
"column": 2
} | {
"line": 237,
"column": 36
} | {
"line": 239,
"column": 0
} | [
{
"pp": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\ne : n ≃ m\nA : Matrix m m α\n⊢ (A.submatrix ⇑e ⇑e).adjugate = A.adjugate.submatrix ⇑e ⇑e",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | ext i j
have : (fun j ↦ Pi.single i 1 <| e.symm j) = Pi.single (e i) 1 :=
Function.update_comp_equiv (0 : n → α) e.symm i 1
rw [adjugate_apply, submatrix_apply, adjugate_apply, ← det_submatrix_equiv_self e,
updateRow_submatrix_equiv, this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Kronecker | {
"line": 140,
"column": 89
} | {
"line": 142,
"column": 82
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_5\nγ : Type u_7\nm : Type u_10\nn : Type u_11\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : Zero γ\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nf : α → β → γ\nhf₁ : ∀ (b : β), f 0 b = 0\nhf₂ : ∀ (a : α), f a 0 = 0\na : m → α\nb : n → β\n⊢ kroneckerMap f (diagonal a) (diagonal ... | [] | by
ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
simp [diagonal, apply_ite f, ite_and, ite_apply, apply_ite (f (a i₁)), hf₁, hf₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 271,
"column": 24
} | {
"line": 271,
"column": 57
} | {
"line": 271,
"column": 57
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\nB C : Matrix n m α\nh : A⁻¹ * B = C\n⊢ B = A * (A⁻¹ * B)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",... | [
"m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\nB C : Matrix n m α\nh : A⁻¹ * B = C\n⊢ B = B"
] | mul_inv_cancel_left_of_invertible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 33
} | {
"line": 429,
"column": 0
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nthis : Aᵀ.adjugate.map star = (Aᵀ.map star).adjugate\n⊢ A.adjugateᵀ.map star = (Aᵀ.map star).adjugate",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
... | [] | rw [A.adjugate_transpose, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 435,
"column": 8
} | {
"line": 435,
"column": 27
} | {
"line": 435,
"column": 28
} | [
{
"pp": "case left\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nhA : IsLeftRegular A.det\nB C : Matrix n n α\nh : A * B = A * C\n⊢ A.det • B = A.det • C",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matr... | [
"case left\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nhA : IsLeftRegular A.det\nB C : Matrix n n α\nh : A * B = A * C\n⊢ A.det • (1 * B) = A.det • C"
] | ← Matrix.one_mul B, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 728,
"column": 2
} | {
"line": 730,
"column": 36
} | {
"line": 732,
"column": 0
} | [
{
"pp": "case neg\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\ne₁ e₂ : n ≃ m\nh : ¬IsUnit A\n⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁",
"ppTerm": "?neg✝",
"assigned": true,
... | [] | · have := (isUnit_submatrix_equiv e₁ e₂).not.mpr h
simp_rw [nonsing_inv_eq_ringInverse, Ring.inverse_non_unit _ h, Ring.inverse_non_unit _ this,
submatrix_zero, Pi.zero_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 747,
"column": 6
} | {
"line": 747,
"column": 31
} | {
"line": 748,
"column": 6
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nB : Matrix n n α\nhA : ¬IsUnit A.det\nh✝ : Nonempty n\nhAB : IsUnit (kroneckerMap (fun x1 x2 ↦ x1 * x2) A B).det\n⊢ IsUnit A.det",
"ppTe... | [
"m : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nB : Matrix n n α\nhA : ¬IsUnit A.det\nh✝ : Nonempty n\nhAB : IsUnit (A.det ^ Fintype.card n * B.det ^ Fintype.card m)\n⊢ IsUnit A.det"
] | rw [det_kronecker] at hAB | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 755,
"column": 6
} | {
"line": 755,
"column": 31
} | {
"line": 756,
"column": 6
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nB : Matrix n n α\nhA : IsUnit A.det\nhB : ¬IsUnit B.det\nh✝ : Nonempty m\nhAB : IsUnit (kroneckerMap (fun x1 x2 ↦ x1 * x2) A B).det\n⊢ IsUni... | [
"m : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nB : Matrix n n α\nhA : IsUnit A.det\nhB : ¬IsUnit B.det\nh✝ : Nonempty m\nhAB : IsUnit (A.det ^ Fintype.card n * B.det ^ Fintype.card m)\n⊢ IsUnit B.det... | rw [det_kronecker] at hAB | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.Basic | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 11
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\n⊢ matPolyEquiv M.charmatrix = X - C M",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.ext",
"CommSemiring.toSemiring",
... | [
"R : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nk : ℕ\ni j : n\n⊢ (matPolyEquiv M.charmatrix).coeff k i j = (X - C M).coeff k i j"
] | ext k i j | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 7
} | {
"line": 56,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : MonoidWithZero R\ninst✝³ : MonoidWithZero S\ninst✝² : MulActionWithZero R S\ninst✝¹ : SMulCommClass R S S\ninst✝ : IsScalarTower R S S\na : S\nt : R\nk : ℕ\nha : a ^ k = 0\n⊢ IsNilpotent (t • a)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
... | [
"case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : MonoidWithZero R\ninst✝³ : MonoidWithZero S\ninst✝² : MulActionWithZero R S\ninst✝¹ : SMulCommClass R S S\ninst✝ : IsScalarTower R S S\na : S\nt : R\nk : ℕ\nha : a ^ k = 0\n⊢ (t • a) ^ k = 0"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 441,
"column": 7
} | {
"line": 441,
"column": 54
} | {
"line": 441,
"column": 55
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m m α\nU : Matrix m n α\nV : Matrix n m α\nhA : IsUnit A.det\n⊢ (A * 1 + U * V).det = A.det * (1 + V * A⁻¹ * U).det",
"ppTerm": "?m.63",
... | [
"m : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m m α\nU : Matrix m n α\nV : Matrix n m α\nhA : IsUnit A.det\n⊢ (A * 1 + A * (A⁻¹ * (U * V))).det = A.det * (1 + V * A⁻¹ * U).det",
"m : Type u_2\nn : T... | ← Matrix.mul_nonsing_inv_cancel_left A (U * V), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 84,
"column": 54
} | {
"line": 85,
"column": 69
} | {
"line": 87,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nr : R\n⊢ (p.scaleRoots r).leadingCoeff = p.leadingCoeff",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"congrArg",
"id",
"Polynomial.natDegree_scaleRoots",
"Po... | [] | by
rw [leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 55
} | {
"line": 213,
"column": 0
} | [
{
"pp": "case inr.inr\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nr : R\nn a b : ℕ\ne : a + b = n\nha : a ≤ p.natDegree\nhb : b ≤ q.natDegree\n⊢ p.coeff a * (q.coeff b * r ^ (p.natDegree + q.natDegree - (a + b))) =\n p.coeff a * (q.coeff b * r ^ (q.natDegree - b + (p.natDegree - a)))",
"ppTerm": "... | [] | rw [add_comm (_ - _), tsub_add_tsub_comm ha hb] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 46
} | {
"line": 158,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nA : Type u_1\ninst✝ : Semiring A\np : R[X]\nf : R →+* A\nz : A\nhz : eval₂ f z p = 0\nh₁ : Commute (f p.leadingCoeff) z\nh₂ : ∀ {r r' : R}, Commute (f r) (f r')\ninj : ∀ (x : R), f x = 0 → x = 0\n⊢ eval₂ f (f p.leadingCoeff * z) p.integralNormalization = 0",
"ppTerm... | [
"case inl\nR : Type u\ninst✝¹ : Semiring R\nA : Type u_1\ninst✝ : Semiring A\np : R[X]\nf : R →+* A\nz : A\nhz : eval₂ f z p = 0\nh₁ : Commute (f p.leadingCoeff) z\nh₂ : ∀ {r r' : R}, Commute (f r) (f r')\ninj : ∀ (x : R), f x = 0 → x = 0\nh : p.natDegree = 0\n⊢ eval₂ f (f p.leadingCoeff * z) p.integralNormalizatio... | obtain (h | h) := p.natDegree.eq_zero_or_pos | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 86,
"column": 26
} | {
"line": 86,
"column": 36
} | {
"line": 86,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nT : Subring R\ni : ℕ\n⊢ coeff 1 i = ↑(coeff 1 i)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.coeff_one",
"Polynomial.instOne",
"Subring.instSetLike",
... | [
"R : Type u_1\ninst✝ : Ring R\nT : Subring R\ni : ℕ\n⊢ (if i = 0 then 1 else 0) = ↑(coeff 1 i)"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 47
} | {
"line": 86,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nT : Subring R\ni : ℕ\n⊢ (if i = 0 then 1 else 0) = ↑(coeff 1 i)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.coeff_one",
"Polynomial.instOne",
"Subring.i... | [
"R : Type u_1\ninst✝ : Ring R\nT : Subring R\ni : ℕ\n⊢ (if i = 0 then 1 else 0) = ↑(if i = 0 then 1 else 0)"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 362,
"column": 2
} | {
"line": 363,
"column": 6
} | {
"line": 365,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nx : R\nhr : f.roots = {x}\n⊢ f = C f.leadingCoeff * (X - C x)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"IsDomain.to_noZeroDivisors",
"Polynomial... | [] | rw [hf.eq_prod_roots, hr]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 362,
"column": 2
} | {
"line": 363,
"column": 6
} | {
"line": 365,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nx : R\nhr : f.roots = {x}\n⊢ f = C f.leadingCoeff * (X - C x)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"IsDomain.to_noZeroDivisors",
"Polynomial... | [] | rw [hf.eq_prod_roots, hr]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 596,
"column": 24
} | {
"line": 596,
"column": 31
} | {
"line": 596,
"column": 32
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDomain B\ninst✝³ : Algebra R B\ninst✝² : Algebra A B\ninst✝¹ : FaithfulSMul A B\ninst✝ : IsScalarTower R A B\nf : R[X]\nhf : (map (algebraMap R ... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDomain B\ninst✝³ : Algebra R B\ninst✝² : Algebra A B\ninst✝¹ : FaithfulSMul A B\ninst✝ : IsScalarTower R A B\nf : R[X]\nhf : (map (algebraMap R A) f).Splits... | aroots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 596,
"column": 32
} | {
"line": 596,
"column": 39
} | {
"line": 597,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDomain B\ninst✝³ : Algebra R B\ninst✝² : Algebra A B\ninst✝¹ : FaithfulSMul A B\ninst✝ : IsScalarTower R A B\nf : R[X]\nhf : (map (algebraMap R ... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDomain B\ninst✝³ : Algebra R B\ninst✝² : Algebra A B\ninst✝¹ : FaithfulSMul A B\ninst✝ : IsScalarTower R A B\nf : R[X]\nhf : (map (algebraMap R A) f).Splits... | aroots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 615,
"column": 67
} | {
"line": 615,
"column": 78
} | {
"line": 615,
"column": 79
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\ninj : Function.Injective ⇑f\na : R\nha : IsUnit (f a)\np : R[X]\np_monic : p.Monic\nhp : eval₂ f (↑ha.unit⁻¹) p = 0\n⊢ eval a (X * p.reverse.divX) + eval ?m.109 (C (p.reverse.coeff 0)) = 0",
"ppTerm... | [
"R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\ninj : Function.Injective ⇑f\na : R\nha : IsUnit (f a)\np : R[X]\np_monic : p.Monic\nhp : eval₂ f (↑ha.unit⁻¹) p = 0\n⊢ eval a (X * p.reverse.divX + C (p.reverse.coeff 0)) = 0"
] | ← eval_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 250,
"column": 2
} | {
"line": 252,
"column": 42
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nRₘ : Type u_3\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\np : Rₘ[X]\nhp : p.leadingCoeff ∈ (algebraMap R Rₘ).range\nn : ℕ\nh₁ : n ∉ p.support\n⊢ p.coeff n * (algebraMap R Rₘ) ↑(commonDenom M p.support p.coeff) ... | [] | · rw [Polynomial.notMem_support_iff] at h₁
rw [h₁, zero_mul]
exact zero_mem (algebraMap R Rₘ).range | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 289,
"column": 61
} | {
"line": 289,
"column": 99
} | {
"line": 289,
"column": 99
} | [
{
"pp": "R : Type u_1\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nha : IsAlgebraic R a\np : R[X]\nh : p ≠ 0\neval0 : (aeval a) p = 0\n⊢ (aeval (-a)) (algEquivAevalNegX p) = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZero... | [] | by simpa [← comp_eq_aeval, aeval_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 26
} | {
"line": 474,
"column": 4
} | [
{
"pp": "case pos\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nM : Ideal A\nhgt : ∀ J > M, J ≠ ⊥ → ∃ Z, (Multiset.map asIdeal Z).prod ≤ J ∧ (Multiset.map asIdeal Z).prod ≠ ⊥\nh_nzI : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : M = ⊤\n⊢ ∃ Z, (Multiset.map asId... | [
"case pos\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nhA_nont : Nontrivial A\nhgt : ∀ J > ⊤, J ≠ ⊥ → ∃ Z, (Multiset.map asIdeal Z).prod ≤ J ∧ (Multiset.map asIdeal Z).prod ≠ ⊥\nh_nzI : ⊤ ≠ ⊥\n⊢ ∃ Z, (Multiset.map asIdeal Z).prod ≤ ⊤ ∧ (Multiset.map asIdeal Z... | rcases h_topM with rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 680,
"column": 2
} | {
"line": 683,
"column": 40
} | {
"line": 685,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Ring A\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nz : A\nz' : S\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : Module.Finit... | [] | obtain ⟨_, s, hs⟩ := Module.Finite.exists_fin (R := R) (M := S)
exact Module.finite_def.mpr <|
(span_eq_top_localization_localization (FractionRing R) R⁰ (FractionRing S) hs) ▸
Submodule.fg_span (Set.toFinite _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 680,
"column": 2
} | {
"line": 683,
"column": 40
} | {
"line": 685,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Ring A\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nz : A\nz' : S\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : Module.Finit... | [] | obtain ⟨_, s, hs⟩ := Module.Finite.exists_fin (R := R) (M := S)
exact Module.finite_def.mpr <|
(span_eq_top_localization_localization (FractionRing R) R⁰ (FractionRing S) hs) ▸
Submodule.fg_span (Set.toFinite _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 159,
"column": 30
} | {
"line": 159,
"column": 44
} | {
"line": 159,
"column": 44
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\nx₁ x₂ : S ⊗[R] T\nt₁ : T\nr₁ : R\na₁ : S\nhr₁ : r₁ • t₁ ∉ J\ne₁ : 1 ⊗ₜ[R]... | [
"case add\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\nx₁ x₂ : S ⊗[R] T\nt₁ : T\nr₁ : R\na₁ : S\nhr₁ : r₁ • t₁ ∉ J\ne₁ : 1 ⊗ₜ[R] (r₁ • t₁) *... | mul_comm t₁ t₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 103,
"column": 2
} | {
"line": 106,
"column": 5
} | {
"line": 108,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ KrullDimLE 0 R ↔ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Semiring.toModule",
"Equiv... | [] | simp_rw [Ring.KrullDimLE, Order.krullDimLE_iff, Nat.cast_zero,
Order.krullDim_nonpos_iff_forall_isMax,
(PrimeSpectrum.equivSubtype R).forall_congr_left, Subtype.forall, PrimeSpectrum.isMax_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 103,
"column": 2
} | {
"line": 106,
"column": 5
} | {
"line": 108,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ KrullDimLE 0 R ↔ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Semiring.toModule",
"Equiv... | [] | simp_rw [Ring.KrullDimLE, Order.krullDimLE_iff, Nat.cast_zero,
Order.krullDim_nonpos_iff_forall_isMax,
(PrimeSpectrum.equivSubtype R).forall_congr_left, Subtype.forall, PrimeSpectrum.isMax_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 65,
"column": 4
} | {
"line": 69,
"column": 48
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx : R\n⊢ (∃ y ∉ I.radical, x * y ∈ I.radical) → ∃ i, I.IsMinimalPrime i ∧ x ∈ i",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"Idea... | [] | rintro ⟨y, hy, hx⟩
obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
simpa [← Ideal.sInf_minimalPrimes] using hy
refine ⟨p, hp, (hp.isPrime.mem_or_mem ?_).resolve_right hyp⟩
exact hp.isPrime.radical_le_iff.mpr hp.le hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 65,
"column": 4
} | {
"line": 69,
"column": 48
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx : R\n⊢ (∃ y ∉ I.radical, x * y ∈ I.radical) → ∃ i, I.IsMinimalPrime i ∧ x ∈ i",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"Idea... | [] | rintro ⟨y, hy, hx⟩
obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
simpa [← Ideal.sInf_minimalPrimes] using hy
refine ⟨p, hp, (hp.isPrime.mem_or_mem ?_).resolve_right hyp⟩
exact hp.isPrime.radical_le_iff.mpr hp.le hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 118,
"column": 90
} | {
"line": 124,
"column": 72
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ (comap f I).minimalPrimes\n⊢ ∃ p' ∈ I.minimalPrimes, comap f p' = p",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"... | [] | by
obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H
obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂
refine ⟨q, hq, Eq.symm ?_⟩
have := hq.isPrime
have := (Ideal.comap_mono hq').trans_eq h₃
exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.le⟩ this).antisymm this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.ResidueField.Ideal | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 53
} | {
"line": 121,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\nI : Ideal R\ninst✝ : I.IsPrime\nx y : R ⧸ I\ne : (algebraMap (R ⧸ I) I.ResidueField) x = (algebraMap (R ⧸ I) I.ResidueF... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\nI : Ideal R\ninst✝ : I.IsPrime\ny : R ⧸ I\nx : R\ne : (algebraMap (R ⧸ I) I.ResidueField) ((Ideal.Quotient.mk I) x) = (algebraMap (... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 198,
"column": 16
} | {
"line": 198,
"column": 25
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case pos\nι : Type u_3\nR : ι → Type u_4\ninst✝¹ : (i : ι) → CommRing (R i)\ninst✝ : _root_.Finite ι\np : PrimeSpectrum ((i : ι) → R i)\nval✝ : Fintype ι\ne : ι → (i : ι) → R i := fun i ↦ Function.update 1 i 0\nH : ∏ i, e i = 0\ni : ι\nhi : e i ∈ p.asIdeal\nh₁ : Function.Surjective ⇑(Pi.evalRingHom R i... | [] | simpa [e] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 172,
"column": 21
} | {
"line": 172,
"column": 95
} | {
"line": 173,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ I.minimalPrimes = comap (Quotient.mk I) '' {p | IsMinimalPrime p}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ I.minimalPrimes = (comap (Quotient.mk I) ⊥).minimalPrimes"
] | ← Ideal.comap_minimalPrimes_eq_of_surjective Ideal.Quotient.mk_surjective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sets.Opens | {
"line": 275,
"column": 2
} | {
"line": 276,
"column": 66
} | {
"line": 278,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nU V : Opens α\nx : α\n⊢ (∃ W, ↑W ⊆ (↑U)ᶜ ∪ ↑V ∧ x ∈ W) ↔ ∃ t ⊆ ↑V ∪ (↑U)ᶜ, IsOpen[inst✝] t ∧ x ∈ t",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Compl.compl",
"TopologicalSpace.Opens",
"Membership.mem",
"Exists",... | [] | exact ⟨fun ⟨⟨W, hW⟩, hsub, hx⟩ => ⟨W, union_comm _ _ ▸ hsub, hW, hx⟩,
fun ⟨W, hsub, hW, hx⟩ => ⟨⟨W, hW⟩, union_comm _ _ ▸ hsub, hx⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.LocalAtTarget | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 53
} | {
"line": 136,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nH : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)\ns : Set α\nhs : IsOpen[inst✝¹] s\ni : ι\n⊢ IsOpen[instTopologicalSpaceSubtype] (Subtype.val ⁻¹'... | [
"case e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nH : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)\ns : Set α\nhs : IsOpen[inst✝¹] s\ni : ι\ne_1✝ : ↥(U i) = ↑(U i).carrier\n⊢ Subtype.val ⁻¹' f '' ... | convert! H i _ (hs.preimage continuous_subtype_val) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.LocalAtTarget | {
"line": 151,
"column": 67
} | {
"line": 160,
"column": 75
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\n⊢ IsInducing f ↔ ∀ (i : ι), IsInducing ((U i).carrier.restrictPreimage f)",
"ppTerm": "?m.23",
"assigned": true,
... | [] | by
simp_rw [← IsInducing.subtypeVal.of_comp_iff, isInducing_iff_nhds, restrictPreimage,
MapsTo.coe_restrict, restrict_eq, ← Filter.comap_comap]
constructor
· intro H i x
rw [Function.comp_apply, ← H, ← IsInducing.subtypeVal.nhds_eq_comap]
· intro H x
obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (show f x ∈ i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.Opens | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 94
} | {
"line": 482,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α ≃ₜ β\n⊢ ∀ {a b : Opens α},\n { toFun := ⇑(comap ↑f.symm), invFun := ⇑(comap ↑f), left_inv := ⋯, right_inv := ⋯ } a ≤\n { toFun := ⇑(comap ↑f.symm... | [] | simp only [← SetLike.coe_subset_coe]; exact f.symm.surjective.preimage_subset_preimage_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Opens | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 94
} | {
"line": 482,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α ≃ₜ β\n⊢ ∀ {a b : Opens α},\n { toFun := ⇑(comap ↑f.symm), invFun := ⇑(comap ↑f), left_inv := ⋯, right_inv := ⋯ } a ≤\n { toFun := ⇑(comap ↑f.symm... | [] | simp only [← SetLike.coe_subset_coe]; exact f.symm.surjective.preimage_subset_preimage_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Ideal | {
"line": 335,
"column": 46
} | {
"line": 335,
"column": 56
} | {
"line": 335,
"column": 56
} | [
{
"pp": "P : Type u_1\ninst✝¹ : PartialOrder P\ninst✝ : OrderTop P\na : P\n⊢ ¬⊤ ≤ a ↔ a ≠ ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Ne",
"LE.le",
"Iff",
"O... | [
"P : Type u_1\ninst✝¹ : PartialOrder P\ninst✝ : OrderTop P\na : P\n⊢ ¬a = ⊤ ↔ a ≠ ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Spectral.Prespectral | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 33
} | {
"line": 139,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PrespectralSpace X\nU V : Opens X\nH : ∀ (W : CompactOpens X), ↑W ⊆ ↑U → ↑W ⊆ ↑V\nx : X\nhxU : x ∈ U\nW : Set X\nh₁ : IsOpen[inst✝²] W\nh₂ : IsCompact W\nhxW : x ∈ W\nhWU : W ⊆ ↑U\n⊢ x ∈ V",
"ppTerm": "?m.... | [] | exact H ⟨⟨W, h₂⟩, h₁⟩ hWU hxW | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructible | {
"line": 283,
"column": 54
} | {
"line": 285,
"column": 99
} | {
"line": 286,
"column": 4
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\nhfU : IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)\nh : IsRetrocompact (f '' U... | [] | by
simpa [union_inter_distrib_right, inter_eq_left.2 (image_subset_range ..)]
using (h.isConstructible hfU).sdiff (hfcomp.isConstructible hf.isClosed_range.isOpen_compl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Descent | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 92
} | {
"line": 48,
"column": 0
} | [
{
"pp": "A B : Type u\ninst✝¹ : CommRing A\ninst✝ : CommRing B\nf : A →+* B\nhf : f.Flat\nthis : PreservesFiniteLimits (extendScalars f ⋙ restrictScalars f)\n⊢ PreservesFiniteLimits (extendScalars f)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"ModuleCat",
"CategoryTheory.L... | [] | exact preservesFiniteLimits_of_reflects_of_preserves (extendScalars f) (restrictScalars f) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RingHom.Flat | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 71
} | {
"line": 268,
"column": 4
} | [
{
"pp": "case refine_1\nR S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.... | [] | exact (CommRingCat.isPushout_tensorProduct R S S).isoPushout.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RingHom.Flat | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 71
} | {
"line": 268,
"column": 4
} | [
{
"pp": "case refine_1\nR S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.... | [] | exact (CommRingCat.isPushout_tensorProduct R S S).isoPushout.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Flat | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 71
} | {
"line": 268,
"column": 4
} | [
{
"pp": "case refine_1\nR S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.... | [] | exact (CommRingCat.isPushout_tensorProduct R S S).isoPushout.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 36
} | {
"line": 177,
"column": 37
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ IsClosed {x} ↔ x.asIdeal.IsMaximal",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"closure_subset_iff_isClosed",
"Set.instSingletonSet",
... | [
"R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ closure {x} ⊆ {x} ↔ x.asIdeal.IsMaximal"
] | ← closure_subset_iff_isClosed, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 83,
"column": 36
} | {
"line": 83,
"column": 53
} | {
"line": 83,
"column": 53
} | [
{
"pp": "case pos\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nle : b.factorization p ≤ a.factorization p\n⊢ p ^ (a.factorization ⊔ b.factorization) p ∣ p ^ a.factorization p",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Latti... | [
"case pos\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nle : b.factorization p ≤ a.factorization p\n⊢ (a.factorization ⊔ b.factorization) p ≤ a.factorization p"
] | apply pow_dvd_pow | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 98,
"column": 36
} | {
"line": 98,
"column": 53
} | {
"line": 98,
"column": 53
} | [
{
"pp": "case neg\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nle : ¬b.factorization p ≤ a.factorization p\n⊢ p ^ (a.factorization ⊔ b.factorization) p ∣ p ^ b.factorization p",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Latt... | [
"case neg\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nle : ¬b.factorization p ≤ a.factorization p\n⊢ (a.factorization ⊔ b.factorization) p ≤ b.factorization p"
] | apply pow_dvd_pow | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Derivation.Basic | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 53
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring A\ninst✝³ : AddCommMonoid M\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : Module R M\nD : Derivation R A M\na : A\n⊢ D (a ^ 0) = 0 • a ^ (0 - 1) • D a",
"ppTerm": "?zero",
"assigned": true,... | [] | rw [pow_zero, map_one_eq_zero, zero_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Derivation.Basic | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 53
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring A\ninst✝³ : AddCommMonoid M\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : Module R M\nD : Derivation R A M\na : A\n⊢ D (a ^ 0) = 0 • a ^ (0 - 1) • D a",
"ppTerm": "?zero",
"assigned": true,... | [] | rw [pow_zero, map_one_eq_zero, zero_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Derivation.Basic | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 53
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring A\ninst✝³ : AddCommMonoid M\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : Module R M\nD : Derivation R A M\na : A\n⊢ D (a ^ 0) = 0 • a ^ (0 - 1) • D a",
"ppTerm": "?zero",
"assigned": true,... | [] | rw [pow_zero, map_one_eq_zero, zero_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 92
} | {
"line": 507,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\n⊢ (IsClosedMap fun a ↦ comap (RingHom.fst R S) a) ∧ IsClosedMap fun b ↦ comap (RingHom.snd R S) b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.isClosedEmbedding_coma... | [] | exact ⟨isClosedEmbedding_comap_fst.isClosedMap, isClosedEmbedding_comap_snd.isClosedMap⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 645,
"column": 4
} | {
"line": 646,
"column": 81
} | {
"line": 647,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : R\np : PrimeSpectrum R\nh : (basicOpen f).carrier = {p}\n⊢ IsLocalization.AtPrime (Localization.Away f) p.asIdeal",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [
"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : R\np : PrimeSpectrum R\nh : (basicOpen f).carrier = {p}\nr : R\nhr : r ∈ p.asIdeal.primeCompl\n⊢ ∃ m ∈ Submonoid.powers f, r ∣ m"
] | refine .of_le_of_exists_dvd (.powers f) _
(Submonoid.powers_le.mpr <| by apply h ▸ Set.mem_singleton p) fun r hr ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 37,
"column": 2
} | {
"line": 38,
"column": 34
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\n⊢ IsCyclic α ↔ ∃ g, zpowers g = ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Membership.mem",
"Exists",
"id",
"Subgroup",
"Int",
"Group.toDivInvMo... | [] | simp only [eq_top_iff', mem_zpowers_iff]
exact ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 37,
"column": 2
} | {
"line": 38,
"column": 34
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\n⊢ IsCyclic α ↔ ∃ g, zpowers g = ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Membership.mem",
"Exists",
"id",
"Subgroup",
"Int",
"Group.toDivInvMo... | [] | simp only [eq_top_iff', mem_zpowers_iff]
exact ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 25
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\nh✝ : IsCyclic G\nσ : G →* G\nh : G\nhG : ∀ (x : G), x ∈ zpowers h\nm : ℤ\nhm : (fun x ↦ h ^ x) m = σ h\n⊢ ∃ m, ∀ (g : G), σ g = g ^ m",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMu... | [
"G : Type u_2\ninst✝ : Group G\nh✝ : IsCyclic G\nσ : G →* G\nh : G\nhG : ∀ (x : G), x ∈ zpowers h\nm : ℤ\nhm : (fun x ↦ h ^ x) m = σ h\ng : G\n⊢ σ g = g ^ m"
] | refine ⟨m, fun g => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 22
} | {
"line": 227,
"column": 0
} | [
{
"pp": "case e'_3\nG : Type u_2\ninst✝ : Group G\nk : ℕ\nk_pos : k ≠ 0\nthis : Finite G\na : G\nha : Function.Surjective fun x ↦ a ^ x\nk_lt_card_G : a ^ k = 1\n⊢ ⊤ ≤ zpowers a",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"M... | [] | exact fun x _ ↦ ha x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.PGroup | {
"line": 186,
"column": 76
} | {
"line": 187,
"column": 64
} | {
"line": 187,
"column": 64
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nthis✝ : Fintype α\nthis : Fintype ↑(fixedPoints G α)\nkey : ∀ (x : α), card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)\na : ↑(fixedPoints G α)\n... | [] | by
rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2] | [anonymous] | Lean.Parser.Term.byTactic |
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