module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 148,
"column": 55
} | {
"line": 148,
"column": 66
} | {
"line": 148,
"column": 67
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nih :\n ∀ (f g : A × A ... | [
"R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nih :\n ∀ (f g : A × A → R), ∑ x ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 575,
"column": 2
} | {
"line": 578,
"column": 73
} | {
"line": 578,
"column": 74
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X.sum fun g f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ClassGroup.ExtendedHom | {
"line": 123,
"column": 68
} | {
"line": 123,
"column": 79
} | {
"line": 123,
"column": 80
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Module.IsTorsionFree A B\ninst✝¹ : IsDedekindDomain A\ninst✝ : IsDedekindDomain B\nh : ∀ (I : Ideal A), Submodule.IsPrincipal (Ideal.map (algebraMap A B) I)\nI : ↥(Ideal A)⁰\n⊢ Submodule.IsPrincipal ↑(e... | [
"A : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Module.IsTorsionFree A B\ninst✝¹ : IsDedekindDomain A\ninst✝ : IsDedekindDomain B\nh : ∀ (I : Ideal A), Submodule.IsPrincipal (Ideal.map (algebraMap A B) I)\nI : ↥(Ideal A)⁰\n⊢ Submodule.IsPrincipal (Ideal.map (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 15
} | {
"line": 73,
"column": 16
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : ¬p.coeff n = 0\n⊢ p.coeff n ∈ ↑p.coeffs",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset",
"Membership.mem",
"id",
"Polynomial.coeff",
... | [
"case neg\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : ¬p.coeff n = 0\n⊢ p.coeff n ∈ p.coeffs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 47
} | {
"line": 180,
"column": 48
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\np q : R[X]\nP : Ideal R\nhpq : (p * q).contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P)\nhPprime : P.IsPrime\n⊢ p.contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P) ∨ q.contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P)",
"ppTerm": "?m.71",
"assigned": true,
"u... | [
"R : Type u_3\ninst✝ : CommRing R\np q : R[X]\nP : Ideal R\nhpq : (p * q).contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P)\nhPprime : P.IsPrime\n⊢ map (Ideal.Quotient.mk P) p = 0 ∨ map (Ideal.Quotient.mk P) q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 187,
"column": 43
} | {
"line": 187,
"column": 59
} | {
"line": 188,
"column": 4
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\np q : R[X]\nhp : p.contentIdeal = ⊤\nhq : q.contentIdeal = ⊤\n⊢ ⊤ = p.contentIdeal * q.contentIdeal",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Polynomial.contentIdeal",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower... | [] | by simp [hp, hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 719,
"column": 4
} | {
"line": 720,
"column": 11
} | {
"line": 720,
"column": 12
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₃ f φ ≫ d₃₂ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₃₂ A ≫ chainsMap₂ f φ) ∘ₗ lsingle a✝) x✝",
"ppTe... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G × G\nx✝ : ↑A\n⊢ single (Prod.map (⇑f) (⇑f) a✝.2) ((B.ρ (f a✝.1⁻¹)) ((Hom.hom φ) x✝)) =\n single (Prod.map (⇑f) (⇑f) a✝.2) ((Hom.hom φ) ((A.ρ a✝.1⁻¹) x✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 719,
"column": 4
} | {
"line": 720,
"column": 66
} | {
"line": 721,
"column": 2
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₃ f φ ≫ d₃₂ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₃₂ A ≫ chainsMap₂ f φ) ∘ₗ lsingle a✝) x✝",
"ppTe... | [] | simpa [d₃₂, map_add, map_sub, ← map_inv]
using congr(Finsupp.single _ $((hom_comm_apply φ _ _).symm)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 724,
"column": 4
} | {
"line": 725,
"column": 11
} | {
"line": 725,
"column": 12
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₂ f φ ≫ d₂₁ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₂₁ A ≫ chainsMap₁ f φ) ∘ₗ lsingle a✝) x✝",
"ppTerm":... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ single (f a✝.2) ((B.ρ (f a✝.1⁻¹)) ((Hom.hom φ) x✝)) = single (f a✝.2) ((Hom.hom φ) ((A.ρ a✝.1⁻¹) x✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Category | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 62
} | {
"line": 46,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nr : R\n⊢ Epi (M.smulShortComplex r).g",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"Submodule",
"instHSMul",
"ModuleCat.smulShortComplex._proof_1",
... | [
"R : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nr : R\n⊢ Function.Surjective ⇑(r • ⊤).mkQ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 17
} | {
"line": 243,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_8\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R[M]\nhx : IsGroupLikeElem R x\nh : x ∉ Set.range fun x ↦ single x 1\nthis : LinearIndepOn R id (insert x (Set.range fun x ↦ single x 1))\n⊢ x.coeff.sum single ∉ span R (Set.range fun x ↦ single x 1)",
"ppTerm": "?m.107",
... | [
"R : Type u_1\nM : Type u_8\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R[M]\nhx : IsGroupLikeElem R x\nh : x ∉ Set.range fun x ↦ single x 1\nthis : LinearIndepOn R id (insert x (Set.range fun x ↦ single x 1))\n⊢ x ∉ span R (Set.range fun x ↦ single x 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 55
} | {
"line": 76,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np : Ideal R\ninst✝ : p.IsPrime\nass : p ∈ associatedPrimes R M\n⊢ Ideal.comap (algebraMap R (Localization.AtPrime p)) (maximalIdeal (Localization.AtPrime p)) ∈ associatedPrimes R M",
"ppTerm": "?m.89",
... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np : Ideal R\ninst✝ : p.IsPrime\nass : p ∈ associatedPrimes R M\n⊢ p ∈ associatedPrimes R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 41
} | {
"line": 88,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModu... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 84
} | {
"line": 222,
"column": 9
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn : ℕ\nhn : Fact (0 < n)\nval✝ inv✝ : R\nval_inv✝ : val✝ * inv✝ = 1\ninv_val✝ : inv✝ * val✝ = 1\nhx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val :... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn : ℕ\nhn : Fact (0 < n)\nval✝ inv✝ : R\nval_inv✝ : val✝ * inv✝ = 1\ninv_val✝ : inv✝ * val✝ = 1\nhx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | {
"line": 44,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nr : R\nreg : IsSMulRegular M r\nmem_ann : r ∈ annihilator R N\nf : N →ₗ[R] M\nx : N\nthis : r • f x = r • 0\n⊢ f x = 0 x",
"ppTerm": "?m.68",
... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nr : R\nreg : IsSMulRegular M r\nmem_ann : r ∈ annihilator R N\nf : N →ₗ[R] M\nx : N\nthis : r • f x = r • 0\n⊢ f x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | [] | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | [] | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | [] | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 72
} | {
"line": 78,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ annihilator R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Depth.Rees | {
"line": 53,
"column": 2
} | {
"line": 55,
"column": 48
} | {
"line": 55,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ I\nhI : I • ⊤ ≠ ⊤\neq : I • ⊤ = ⊤\n⊢ ¬Submodule.comap (r • ⊤).mkQ (I • ⊤) = Submodule.comap (r • ⊤).mkQ ⊤",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ I\nhI : I • ⊤ ≠ ⊤\neq : I • ⊤ = ⊤\n⊢ ¬I • ⊤ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Depth.Rees | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 25
} | {
"line": 95,
"column": 26
} | [
{
"pp": "case h\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : (Module.annihilator R ↑N).radical = I.radical\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i < n, Su... | [
"case h\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : (Module.annihilator R ↑N).radical = I.radical\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i < n, Subsingleton (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 33
} | {
"line": 233,
"column": 34
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\nh : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)\n⊢ (fun g ↦ g ∘ ⇑(RingCon.mkₐ R (ringCon R M))) ⇑f = ... | [
"R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\nh : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)\n⊢ f.comp (RingCon.mkₐ R (ringCon R M)) = g.comp (RingCon.mkₐ R (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 229,
"column": 50
} | {
"line": 233,
"column": 76
} | {
"line": 235,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\n⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)",
"ppTerm": "?m.35",
"assigned": true,
... | [] | by
refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right mkAlgHom_surjective
simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 148,
"column": 23
} | {
"line": 148,
"column": 27
} | {
"line": 148,
"column": 27
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | [
"A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+... | hy.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 41
} | {
"line": 254,
"column": 42
} | [
{
"pp": "case pos.refine_2\nA : Type u_3\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nS : Set A\nhS : I = span S\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, hI.dpow n a = hI'.dpow n a\nn : ℕ\na : A\nx✝ ha : a ∈ I\nm : ℕ\nb : A\nhb : b ∈ S\n⊢ (RingHom.id A) (hI.dpow m b) = hI'.dpow m ((RingHom.id A) b)",
"... | [
"case pos.refine_2\nA : Type u_3\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nS : Set A\nhS : I = span S\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, hI.dpow n a = hI'.dpow n a\nn : ℕ\na : A\nx✝ ha : a ∈ I\nm : ℕ\nb : A\nhb : b ∈ S\n⊢ hI.dpow m b = hI'.dpow m b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 11
} | {
"line": 51,
"column": 12
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nx✝ : A\nhx : x✝ ∈ I\n⊢ (if hx : x✝ ∈... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nx✝ : A\nhx : x✝ ∈ I\n⊢ hJ.dpow 1 (f x✝) = f x✝"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 58
} | {
"line": 52,
"column": 59
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx : A\nhn : n ≠ 0\nhx : x ∈ I... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx : A\nhn : n ≠ 0\nhx : x ∈ I\n⊢ ⋯.choose... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 55,
"column": 4
} | {
"line": 57,
"column": 11
} | {
"line": 57,
"column": 12
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx y : A\nhx : x ∈ I\nhy : y ∈... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx y : A\nhx : x ∈ I\nhy : y ∈ I\nhxy : x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 40
} | {
"line": 61,
"column": 41
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\na x : A\nhx : x ∈ I\nhax : a ... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\na x : A\nhx : x ∈ I\nhax : a * x ∈ I\n⊢ h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 62,
"column": 20
} | {
"line": 63,
"column": 31
} | {
"line": 63,
"column": 32
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nm✝ n✝ : ℕ\nx✝ : A\nhx : x✝ ∈ I\n⊢ ((... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nm✝ n✝ : ℕ\nx✝ : A\nhx : x✝ ∈ I\n⊢ hJ.dpow m✝ (f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | {
"line": 107,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | {
"line": 107,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | {
"line": 107,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 9
} | {
"line": 126,
"column": 10
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\ninst✝ : (x : A) → Decidable (x ∈ J)\nhJ : hI.IsSubDPIdeal J\n⊢ (dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semirin... | [
"A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\ninst✝ : (x : A) → Decidable (x ∈ J)\nhJ : hI.IsSubDPIdeal J\n⊢ J ≤ I ∧ ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ J, hI.dpow n a = (dividedPowers hI hJ).dpow n a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DualNumber | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 38
} | {
"line": 123,
"column": 39
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x\n⊢ I = Ideal.span {ε}",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"MulZeroClass.toMul",
"DistribMulA... | [
"case refine_1\nK : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x\nx : K[ε]\n⊢ x ∈ I → x ∈ Ideal.span {ε}",
"case refine_2\nK : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' ... | refine le_antisymm ?_ ?_ <;> intro x | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 351,
"column": 21
} | {
"line": 351,
"column": 76
} | {
"line": 351,
"column": 77
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ J' : hI.SubDPIdeal\nh : (fun J ↦ ⟨J.carrier, ⋯⟩) J = (fun J ↦ ⟨J.carrier, ⋯⟩) J'\n⊢ J = J'",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DividedPowers.SubDPIdeal",
"CommSemiri... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ J' : hI.SubDPIdeal\nh : (fun J ↦ ⟨J.carrier, ⋯⟩) J = (fun J ↦ ⟨J.carrier, ⋯⟩) J'\n⊢ J.carrier = J'.carrier"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DualNumber | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | {
"line": 134,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x\np y : K[ε]\nhyI : y ∈ I\nhy0 : y ≠ 0\nr : K[ε]\nhr : ε = r * y\n⊢ p * r * y ∈ I",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants"... | [] | exact Ideal.mul_mem_left _ _ hyI | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DualNumber | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 39
} | {
"line": 164,
"column": 40
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx y : K[ε]\nthis : ∃ c, fst x * c.fst = fst y ∨ fst y * c.fst = fst x\n⊢ ∃ c, ε * x * c = ε * y ∨ ε * y * c = ε * x",
"ppTerm": "?m.192",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"TrivSq... | [
"K : Type u_2\ninst✝ : DivisionRing K\nx y : K[ε]\nthis : ∃ c, fst x * c.fst = fst y ∨ fst y * c.fst = fst x\n⊢ ∃ c, fst x * fst c = fst y ∨ fst y * fst c = fst x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TotallySplit | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 49
} | {
"line": 97,
"column": 50
} | [
{
"pp": "k : Type u_1\nR : Type u_2\ninst✝⁴ : Field k\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsFiniteSplit k R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ\ne : R ≃ₐ[k] Fin n → k\np' : Ideal (Fin n → k) := comap e.symm p\ni : Fin n\nhq : PrimeSpectrum.comap (Pi.evalRingHom (fun i ↦ k) i) ⊥ = { asIdeal :... | [
"k : Type u_1\nR : Type u_2\ninst✝⁴ : Field k\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsFiniteSplit k R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ\ne : R ≃ₐ[k] Fin n → k\np' : Ideal (Fin n → k) := comap e.symm p\ni : Fin n\nhq : PrimeSpectrum.comap (Pi.evalRingHom (fun i ↦ k) i) ⊥ = { asIdeal := p', isPrim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TotallySplit | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 57
} | {
"line": 165,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [
"n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝⁴ : CommRing... | apply Module.nontrivial_of_rankAtStalk_pos (R := R) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Etale.Finite | {
"line": 177,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 185,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R Ω✝\nS : Type w\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Ω✝\ninst✝² : IsScalarTower R S Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nX : FintypeCat\n⊢ { obj... | [] | dsimp [FiniteEtale.finiteSpec]
apply Quiver.Hom.unop_inj
ext x i
dsimp
rw [FintypeCat.equivEquivIso_apply_hom, FintypeCat.homMk_apply]
dsimp
rw [← Pi.coe_evalAlgHom Ω]
simp [Algebra.FormallyEtale.equivPiOfIsSepClosed_comap,
Algebra.FormallyEtale.equivPiOfIsSepClosed_sel... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.Finite | {
"line": 177,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 185,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R Ω✝\nS : Type w\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Ω✝\ninst✝² : IsScalarTower R S Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nX : FintypeCat\n⊢ { obj... | [] | dsimp [FiniteEtale.finiteSpec]
apply Quiver.Hom.unop_inj
ext x i
dsimp
rw [FintypeCat.equivEquivIso_apply_hom, FintypeCat.homMk_apply]
dsimp
rw [← Pi.coe_evalAlgHom Ω]
simp [Algebra.FormallyEtale.equivPiOfIsSepClosed_comap,
Algebra.FormallyEtale.equivPiOfIsSepClosed_sel... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Frobenius | {
"line": 69,
"column": 27
} | {
"line": 69,
"column": 93
} | {
"line": 69,
"column": 94
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nh : Infinite (R ⧸ Ideal.under R Q)\n⊢ Q = ⊤",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nh : Infinite (R ⧸ Ideal.under R Q)\n⊢ Q = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 46
} | {
"line": 100,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : S ⧸ Q\nhx : H.restrict x = 0\n⊢ x = 0",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : S ⧸ Q\nhx : H.restrict x = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 54
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : S\nhx : x ∈ Ideal.comap φ Q\n⊢ x ∈ Q",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemi... | [] | rwa [← Ideal.Quotient.eq_zero_iff_mem, ← H.restrict_injective.eq_iff, map_zero, restrict_mk,
Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 642,
"column": 2
} | {
"line": 642,
"column": 28
} | {
"line": 642,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\n⊢ hI.IsSubDPIdeal (RingHom.ker (Ideal.Quotient.mk J) ⊓ I)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semir... | [
"A : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\n⊢ hI.IsSubDPIdeal (J ⊓ I)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 90,
"column": 25
} | {
"line": 90,
"column": 36
} | {
"line": 90,
"column": 37
} | [
{
"pp": "case add\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\ns : Set A\nh : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s\nx x✝ y✝ : A\nhx✝ : x✝ ∈ ... | [
"case add\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\ns : Set A\nh : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s\nx x✝ y✝ : A\nhx✝ : x✝ ∈ Subsemiring.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 24
} | {
"line": 203,
"column": 25
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ σ' : G\nH : IsArithFrobAt R σ Q\nH' : IsArithFrobAt R σ' Q\nx : S\n⊢ (σ * σ'⁻¹) • x - x ∈ Submodule.toA... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ σ' : G\nH : IsArithFrobAt R σ Q\nH' : IsArithFrobAt R σ' Q\nx : S\n⊢ σ • σ'⁻¹ • x - x ∈ Q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.TensorProduct | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | {
"line": 44,
"column": 16
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ni j : ι\nx : A\nhx : x ∈ ↑(𝒜 i)\ny ... | [
"ι : Type u_1\nR : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ni j : ι\nx : A\nhx : x ∈ ↑(𝒜 i)\ny : A\nhy : y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 40
} | {
"line": 212,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\nthis : Ideal.under R (Ideal.map ((MulSemiringAction.toRingE... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\nthis : Ideal.under R (Ideal.map ((MulSemiringAction.toRingEquiv G S) τ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.TensorProduct | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 33
} | {
"line": 128,
"column": 34
} | [
{
"pp": "case tmul\nι : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝¹² : DecidableEq ι\ninst✝¹¹ : AddMonoid ι\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S B\n𝒜 : ι → Submodule R A\nℬ : ι → Submo... | [
"case tmul\nι : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝¹² : DecidableEq ι\ninst✝¹¹ : AddMonoid ι\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule S B\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Grassmannian | {
"line": 133,
"column": 10
} | {
"line": 133,
"column": 21
} | {
"line": 133,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | [
"R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R A B := IsSc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Grassmannian | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 21
} | {
"line": 136,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | [
"R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R A B := IsSc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 164,
"column": 68
} | {
"line": 165,
"column": 52
} | {
"line": 167,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝ : CommRing S\nd e : ℕ\n⊢ invOneSubPow S (d + e) = invOneSubPow S d * invOneSubPow S e",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"MulOne.toOne",
"NonUnitalCommRi... | [] | by
simp_rw [invOneSubPow_eq_inv_one_sub_pow, pow_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 25
} | {
"line": 198,
"column": 0
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring A'\ninst✝¹ : Algebra ℚ A\ninst✝ : Algebra ℚ A'\nf : A →+* A'\n⊢ (map f) (sin A) = sin A'",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"RingHom.instRingHomClass",
"instHDiv",
"Se... | [] | ext
simp [sin, apply_ite f] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 25
} | {
"line": 198,
"column": 0
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring A'\ninst✝¹ : Algebra ℚ A\ninst✝ : Algebra ℚ A'\nf : A →+* A'\n⊢ (map f) (sin A) = sin A'",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"RingHom.instRingHomClass",
"instHDiv",
"Se... | [] | ext
simp [sin, apply_ite f] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Binomial | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 13
} | {
"line": 81,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : BinomialRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\n⊢ binomialSeries A 0 = 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : BinomialRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\n⊢ binomialSeries A 0 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 64
} | {
"line": 97,
"column": 65
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : 0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff g... | [
"Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : 0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff g ≠ 0\n⊢ ¬∑ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 27
} | {
"line": 111,
"column": 28
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff ... | [
"Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff g ≠ 0\nhz : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 110,
"column": 6
} | {
"line": 114,
"column": 55
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b))... | [] | have : g = 0 ∧ (a.constantCoeff * b.constantCoeff) • (1 : V) ≠ 0 := by
simpa [hz, h] using hn
simp only [coe_image, Set.mem_image]
use (0, 0)
simp [this.2, this.1, h, hz, smul_smul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 110,
"column": 6
} | {
"line": 114,
"column": 55
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b))... | [] | have : g = 0 ∧ (a.constantCoeff * b.constantCoeff) • (1 : V) ≠ 0 := by
simpa [hz, h] using hn
simp only [coe_image, Set.mem_image]
use (0, 0)
simp [this.2, this.1, h, hz, smul_smul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 125,
"column": 25
} | {
"line": 125,
"column": 73
} | {
"line": 125,
"column": 74
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\nh : 0 < x.orderTop\ng : Γ\ni : ℕ\nhi : i ∈ image (fun i ↦ i.1 + i.2) (((powerSer... | [
"Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\nh : 0 < x.orderTop\ng : Γ\ni : ℕ\nhi : i ∈ image (fun i ↦ i.1 + i.2) (((powerSeriesFamily x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 12
} | {
"line": 404,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : (⋃ a, (s a).... | intro ab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 12
} | {
"line": 404,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : (⋃ a, (s a).... | intro ab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 400,
"column": 2
} | {
"line": 412,
"column": 47
} | {
"line": 414,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [] | apply (hs.vadd ht).mono
have hsupp : ∀ ab : α × β, support ((fun ab ↦ (of R).symm (s ab.1 • (of R) (t ab.2))) ab) ⊆
(s ab.1).support +ᵥ (t ab.2).support := by
intro ab
refine Set.Subset.trans (fun x hx => ?_) (support_vaddAntidiagonal_subset_vadd fun a ↦
Set.VAddAntidiagonal.finite_of_isPWO (s ab.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 400,
"column": 2
} | {
"line": 412,
"column": 47
} | {
"line": 414,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [] | apply (hs.vadd ht).mono
have hsupp : ∀ ab : α × β, support ((fun ab ↦ (of R).symm (s ab.1 • (of R) (t ab.2))) ab) ⊆
(s ab.1).support +ᵥ (t ab.2).support := by
intro ab
refine Set.Subset.trans (fun x hx => ?_) (support_vaddAntidiagonal_subset_vadd fun a ↦
Set.VAddAntidiagonal.finite_of_isPWO (s ab.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | {
"line": 43,
"column": 16
} | [
{
"pp": "case h₁\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), F... | [
"case h₁\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), Function.Inje... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | {
"line": 44,
"column": 16
} | [
{
"pp": "case h₂\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), F... | [
"case h₂\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), Function.Inje... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 481,
"column": 6
} | {
"line": 481,
"column": 17
} | {
"line": 481,
"column": 18
} | [
{
"pp": "case refine_1.left\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α\... | [
"case refine_1.left\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α\nt : Summabl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 98,
"column": 31
} | {
"line": 98,
"column": 40
} | {
"line": 98,
"column": 41
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nf₁ : M →+o DivisibleHull M := DivisibleHull.coeOrderAddMonoidHom M\nhf₁ : Function.Injective ⇑f₁\nhf₁class :\n ∀ (a : M), ArchimedeanClass.mk a = (DivisibleHull.archimedeanClassOrderIso M).symm (ArchimedeanCla... | [
"M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nf₁ : M →+o DivisibleHull M := DivisibleHull.coeOrderAddMonoidHom M\nhf₁ : Function.Injective ⇑f₁\nhf₁class :\n ∀ (a : M), ArchimedeanClass.mk a = (DivisibleHull.archimedeanClassOrderIso M).symm (ArchimedeanClass.mk (f₁ a)... | hf₃class, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Henselian | {
"line": 83,
"column": 38
} | {
"line": 83,
"column": 49
} | {
"line": 83,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : IsUnit ((a * y - 1) * 1 + 1)\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n⊢ IsUnit a ∧ IsUnit y",
"ppTerm": "?m.323",
"assigned": false,
"usedConstants": [],
... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : IsUnit ((a * y - 1) * 1 + 1)\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n⊢ IsUnit a ∧ IsUnit y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 53
} | {
"line": 28,
"column": 54
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 53
} | {
"line": 32,
"column": 54
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ a * (antipode R) a = 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ a * (antipode R) a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | {
"line": 100,
"column": 14
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ ((single a) r * x).cardSupp ≤ x.cardSupp",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedF... | [
"Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ ((single a) r * x).cardSupp ≤ x.cardSupp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | {
"line": 105,
"column": 14
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ (x * (single a) r).cardSupp ≤ x.cardSupp",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedF... | [
"Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ (x * (single a) r).cardSupp ≤ x.cardSupp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 26
} | {
"line": 112,
"column": 27
} | [
{
"pp": "case succ\nΓ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\nIH : (x ^ n).cardSupp ≤ x.cardSupp ^ n\n⊢ (x ^ (n + 1)).cardSupp ≤ x.cardSupp ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
... | [
"case succ\nΓ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\nIH : (x ^ n).cardSupp ≤ x.cardSupp ^ n\n⊢ (x ^ n * x).cardSupp ≤ x.cardSupp ^ n * x.cardSupp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 706,
"column": 2
} | {
"line": 708,
"column": 61
} | {
"line": 709,
"column": 2
} | [
{
"pp": "case pos.refine_1\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n,... | [
"case pos.refine_2\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n, (x ^ n).sup... | · obtain ⟨hi, _, rfl⟩ := mem_antidiagonal.1 (mem_coe.1 hij)
exact lt_add_of_pos_left ij.2 <| lt_of_lt_of_le ((zero_lt_orderTop_iff h0).mp hx) <|
order_le_of_coeff_ne_zero <| Function.mem_support.mp hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 159,
"column": 18
} | {
"line": 159,
"column": 29
} | {
"line": 159,
"column": 30
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nhκ : Fact (ℵ₀ ≤ κ)\n⊢ 0 ∈ {x | x.cardSupp < κ}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"... | [
"Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nhκ : Fact (ℵ₀ ≤ κ)\n⊢ 0 < κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Henselian | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 87
} | {
"line": 181,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\n⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ I",
"ppTerm": "?m.30",
"ass... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Pol... | let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * (f'.eval b)⁻¹ʳ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 906,
"column": 2
} | {
"line": 906,
"column": 13
} | {
"line": 906,
"column": 14
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nm : ℕ\ne : Bool\ns : ℕ\n⊢ (single 0) (OfScientific.ofScientific m e s) = OfScientific.ofScientific m e s",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
... | [
"Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nm : ℕ\ne : Bool\ns : ℕ\n⊢ (single 0) (OfScientific.ofScientific m e s) = OfScientific.ofScientific m e s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 30
} | {
"line": 54,
"column": 31
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ((LinearMap.mul' R A[G] ∘ₗ LinearMap.rTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a) b =\n ((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a) b",
... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ∑ x ∈ (ℛ R b).index, single 1 ((antipode R) ((ℛ R b).left x) * (ℛ R b).right x) =\n single 1 ((algebraMap R A) (counit b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 31
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ((LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a) b =\n ((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a) b",
... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ∑ x ∈ (ℛ R b).index, single 1 ((ℛ R b).left x * (antipode R) ((ℛ R b).right x)) =\n single 1 ((algebraMap R A) (counit b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 39
} | {
"line": 77,
"column": 40
} | [
{
"pp": "case singleton\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nq : PrimeSpectrum R\nP : Ideal S\ninst✝ : P.IsPrime\nlo : P.LiesOver (RelSeries.singleton {(a, b) | a < b} q).head.asIdeal\n⊢ List.map (PrimeSpectrum.comap (algeb... | [
"case singleton\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nq : PrimeSpectrum R\nP : Ideal S\ninst✝ : P.IsPrime\nlo : P.LiesOver (RelSeries.singleton {(a, b) | a < b} q).head.asIdeal\n⊢ comap (algebraMap R S) P = q.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 85,
"column": 22
} | {
"line": 85,
"column": 33
} | {
"line": 85,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\n ∃ L,\n... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\n ∃ L,\n L.leng... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 45
} | {
"line": 86,
"column": 46
} | [
{
"pp": "case cons\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\... | [
"case cons\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\n ∃ L,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 39
} | {
"line": 113,
"column": 40
} | [
{
"pp": "case refine_2\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : SpecializingMap (PrimeSpectrum.comap (algebraMap R S))\nq : Ideal R\nhq : q.IsPrime\nP : Ideal S\nhP : P.IsPrime\nhlt : Ideal.under R P < q\nthis : PrimeSpectrum.comap (algebraMap R S) { asIdea... | [
"case refine_2\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : SpecializingMap (PrimeSpectrum.comap (algebraMap R S))\nq : Ideal R\nhq : q.IsPrime\nP : Ideal S\nhP : P.IsPrime\nhlt : Ideal.under R P < q\nthis : PrimeSpectrum.comap (algebraMap R S) { asIdeal := P, isPr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MonicSpan | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 46
} | {
"line": 50,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np : Ideal R[X]\nism : (comap C p).IsMaximal\nne : p ≠ Ideal.map C (comap C p)\nq : Ideal R := comap C p\nthis✝ : Field (R ⧸ q) := Quotient.field q\nne' : Ideal.map (mapRingHom (Ideal.Quotient.mk q)) p ≠ ⊥\ny : (R ⧸ q)[X]\nmony : y.Monic\nhy : Ideal.map (mapRingHom (Ide... | [
"R : Type u_1\ninst✝ : CommRing R\np : Ideal R[X]\nism : (comap C p).IsMaximal\nne : p ≠ Ideal.map C (comap C p)\nq : Ideal R := comap C p\nthis✝ : Field (R ⧸ q) := Quotient.field q\nne' : Ideal.map (mapRingHom (Ideal.Quotient.mk q)) p ≠ ⊥\ny : (R ⧸ q)[X]\nmony : y.Monic\nhy : Ideal.map (mapRingHom (Ideal.Quotient.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 40
} | {
"line": 110,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot (L ⊓ K) K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon (L ⊓ K) {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon L {k}",
"ppTerm": "?m.76",
"assig... | [
"A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot (L ⊓ K) K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon (L ⊓ K) {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon L {k}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 40
} | {
"line": 110,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot L K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon L {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon (L ⊓ K) {k}",
"ppTerm": "?m.147",
"assigned":... | [
"A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot L K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon L {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon L {k}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 81,
"column": 47
} | {
"line": 81,
"column": 58
} | {
"line": 81,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nx✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}\nB : RingFilterBasis A\nhB : B.sets = {x | ∃ I ∈ F, ↑I = x}\nI : Ideal A\nhI : I ∈ F\na : A\nJ : Ideal A\nhJ : J ∈ F\nhbasis : ↑J ∈ B\nhsub : ↑J ⊆ (fun x ↦ x * a) ⁻¹' ↑I\nx : A\nhx : x ∈ J\n⊢ x ∈ Submodule.colon I... | [
"A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nx✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}\nB : RingFilterBasis A\nhB : B.sets = {x | ∃ I ∈ F, ↑I = x}\nI : Ideal A\nhI : I ∈ F\na : A\nJ : Ideal A\nhJ : J ∈ F\nhbasis : ↑J ∈ B\nhsub : ↑J ⊆ (fun x ↦ x * a) ⁻¹' ↑I\nx : A\nhx : x ∈ J\n⊢ x * a ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 149,
"column": 2
} | {
"line": 157,
"column": 36
} | {
"line": 159,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF G : IdealFilter A\n⊢ Order.IsPFilter {L | ∃ K ∈ G, F.IsTorsionQuot L K}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Order.IsPFilter.of_def",
"Semiring.toModule",
"IdealFilter.IsTorsionQuot",
"Set.ofPred",
"Order.PF... | [] | refine Order.IsPFilter.of_def ?nonempty ?directed ?mem_of_le
· obtain ⟨J, hJ⟩ := G.nonempty
exact ⟨J, J, hJ, isTorsionQuot_self F J⟩
· rintro I ⟨K, hK, hIK⟩ J ⟨L, hL, hJL⟩
refine ⟨I ⊓ J, ?_, inf_le_left, inf_le_right⟩
exact ⟨K ⊓ L, G.inf_mem hK hL,
(hIK.anti_right inf_le_left).inf (hJL.anti_right ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 149,
"column": 2
} | {
"line": 157,
"column": 36
} | {
"line": 159,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF G : IdealFilter A\n⊢ Order.IsPFilter {L | ∃ K ∈ G, F.IsTorsionQuot L K}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Order.IsPFilter.of_def",
"Semiring.toModule",
"IdealFilter.IsTorsionQuot",
"Set.ofPred",
"Order.PF... | [] | refine Order.IsPFilter.of_def ?nonempty ?directed ?mem_of_le
· obtain ⟨J, hJ⟩ := G.nonempty
exact ⟨J, J, hJ, isTorsionQuot_self F J⟩
· rintro I ⟨K, hK, hIK⟩ J ⟨L, hL, hJL⟩
refine ⟨I ⊓ J, ?_, inf_le_left, inf_le_right⟩
exact ⟨K ⊓ L, G.inf_mem hK hL,
(hIK.anti_right inf_le_left).inf (hJL.anti_right ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 25
} | {
"line": 125,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\ns : Set (WithIdealFilter F)\n⊢ s ∈ 𝓝 0 ↔ ∃ I ∈ F, idealSet I ⊆ s",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\ns : Set (WithIdealFilter F)\n⊢ s ∈ 𝓝 0 ↔ ∃ I ∈ F, idealSet I ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 160,
"column": 6
} | {
"line": 161,
"column": 57
} | {
"line": 161,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surje... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surjective ⇑f\nhI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 204,
"column": 78
} | {
"line": 204,
"column": 89
} | {
"line": 204,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nthis✝ : p.IsPrime\nhp : maximalIdeal (Localization p.primeCompl) ∈ (span (⇑(algebraMap R (Localization p.primeCompl)) '' ↑s)).minimalPrimes\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [Is... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nthis✝ : p.IsPrime\nhp : maximalIdeal (Localization p.primeCompl) ∈ (span (⇑(algebraMap R (Localization p.primeCompl)) '' ↑s)).minimalPrimes\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 228,
"column": 10
} | {
"line": 228,
"column": 21
} | {
"line": 228,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),\n p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m\nw✝ : IsLocalRing R\nthis✝ : (maximalIdeal R).IsPrime\nq... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),\n p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m\nw✝ : IsLocalRing R\nthis✝ : (maximalIdeal R).IsPrime\nq : Ideal R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 28,
"column": 19
} | {
"line": 28,
"column": 30
} | {
"line": 28,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis : IsPrincipalIdealRing (R ⧸ P)\n⊢ RingHom.ker (Ideal.Quotient.mk P) ≤ I",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis : IsPrincipalIdealRing (R ⧸ P)\n⊢ P ≤ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 17
} | {
"line": 35,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝¹ : IsPrincipalIdealRing (R ⧸ P)\nthis✝ : (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal\nthis : (Ideal.comap (Ideal.Quotient.mk P) (Ideal.map (Ideal.Quotient.mk... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝¹ : IsPrincipalIdealRing (R ⧸ P)\nthis✝ : (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal\nthis : (Ideal.comap (Ideal.Quotient.mk P) (Ideal.map (Ideal.Quotient.mk P) I)).IsMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 8
} | {
"line": 56,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ m.height = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",... | [] | refine le_antisymm ?_ ?_
· suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
· apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 8
} | {
"line": 56,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ m.height = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",... | [] | refine le_antisymm ?_ ?_
· suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
· apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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