module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Extension.Generators | {
"line": 668,
"column": 6
} | {
"line": 669,
"column": 87
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis✝ : DecidableEq (ι' →₀ ℕ) := Cl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 91,
"column": 40
} | {
"line": 91,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nι : Type u_4\ninst✝¹ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u_5\ninst✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nhay : ∀ (i : ι), n i = ∑ j, a i j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 96
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 729,
"column": 2
} | {
"line": 741,
"column": 12
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra R S\nP✝ : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁵ : CommRing R'\ninst✝¹⁴ : CommRing S'\ninst✝¹³ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type u_5\nι''... | refine ⟨x.1.sum fun n r ↦ ?_, ?_⟩
· -- The use of `refine` is intentional to control the elaboration order
-- so that the term has type `(Q.comp P).Ring` and not `MvPolynomial (Q.ι ⊕ P.ι) R`
refine rename ?_ (P.σ r) * monomial ?_ 1
exacts [Sum.inr, n.mapDomain Sum.inl]
· simp only [ker_eq_ker_aeval_val,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Generators | {
"line": 729,
"column": 2
} | {
"line": 741,
"column": 12
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra R S\nP✝ : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁵ : CommRing R'\ninst✝¹⁴ : CommRing S'\ninst✝¹³ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type u_5\nι''... | refine ⟨x.1.sum fun n r ↦ ?_, ?_⟩
· -- The use of `refine` is intentional to control the elaboration order
-- so that the term has type `(Q.comp P).Ring` and not `MvPolynomial (Q.ι ⊕ P.ι) R`
refine rename ?_ (P.σ r) * monomial ?_ 1
exacts [Sum.inr, n.mapDomain Sum.inl]
· simp only [ker_eq_ker_aeval_val,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 30,
"column": 7
} | {
"line": 30,
"column": 52
} | [
{
"pp": "case a.h.h\nR : Type u_1\ninst✝⁴ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : ι → Ideal R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx✝ : M\n⊢ ((AlgebraTensorModule.curry (rTensor M (LinearMap.pi fun i ↦ Submodule.mkQ (I i)))) 1) x✝ =\n ((AlgebraTensor... | simp [LinearMap.pi, LinearEquiv.piCongrRight] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\ninst✝ : Finite ι\nthis :\n Surjective\n ⇑(↑(piLeft R M fun i ↦ R ⧸ I i).symm ∘ₗ\n (LinearMap.pi fun i ↦ (TensorProduct.mk R (R ⧸ I i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 48
} | [
{
"pp": "case h.e'_3.refine_2.refine_1\nR : Type u_1\ninst✝³ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\ninst✝ : Finite ι\nthis :\n (rTensor M (LinearMap.pi fun i ↦ Submodule.mkQ (I i)) ∘ₗ ↑(TensorProduct.lid R M).symm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | [
{
"pp": "case h.e'_3.refine_2.refine_2\nR : Type u_1\ninst✝³ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\ninst✝ : Finite ι\nthis :\n (rTensor M (LinearMap.pi fun i ↦ Submodule.mkQ (I i)) ∘ₗ ↑(TensorProduct.lid R M).symm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 30
} | [
{
"pp": "case h.e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) ... | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | [
{
"pp": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) ... | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | [
{
"pp": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) ... | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | [
{
"pp": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) ... | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 105,
"column": 20
} | {
"line": 105,
"column": 31
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nR' : Type u_3\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : Algebra R R'\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module R' N\ninst✝¹ : IsScalarTower R R' N\ninst✝ : IsLocalization M R'\nx : N\nhs' : x ∈ Submodule.span ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → R\nx : ι → M\nh : IsTrivialRelation f x\n⊢ ∑ i, f i • x i = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | [
{
"pp": "case h\nR : Type u_3\nS : Type u_4\nRₚ : Type u_1\nSₚ : Type u_2\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : CommSemiring Sₚ\ninst✝⁸ : Algebra R S\ninst✝⁷ : Algebra R Rₚ\ninst✝⁶ : Algebra R Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : Algebra Rₚ Sₚ\ninst✝³ : IsScalar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | [
{
"pp": "case h\nR : Type u\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nRₚ : Type v\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : Algebra R Rₚ\ninst✝⁸ : IsLocalization S Rₚ\nM : Type w\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nMₚ : Type t\ninst✝⁵ : AddCommMonoid Mₚ\ninst✝⁴ : Module R Mₚ\ninst✝³ : Module Rₚ Mₚ\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuotSMulTop | {
"line": 114,
"column": 63
} | {
"line": 115,
"column": 64
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\nr : R\nM : Type u_1\nM' : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf : Surjective ⇑f\nH₁ : Surjective (⇑(r • ⊤).mkQ ∘ ⇑f)\n⊢ Surjective (⇑((map r) f) ∘ ⇑(r • ⊤).mkQ)",
"usedConstants": ... | by
rwa [← LinearMap.coe_comp, map_comp_mkQ, LinearMap.coe_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuotSMulTop | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\nm : QuotSMulTop x M\nm' : M\nhm' : Submodule.Quotient.mk m' = m\n⊢ x • m = 0",
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"Submodule",
"Submodule.Quo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 112,
"column": 30
} | {
"line": 112,
"column": 41
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : m = 0\n⊢ 1 • m = 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"CommSemiring.toSemiring",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 17
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : ¬(R ∙ m).annihilator = ⊤\np : Ideal R\nhp : (R ∙ m).annihilator ≤ p\nhp' : p.IsPrime\n⊢ f ∈ p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 56
} | [
{
"pp": "case h.refine_1\nR : Type u_1\ninst✝² : CommRing R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\np : PrimeSpectrum R\nx✝ : ∃ m, ∀ r ∉ p.asIdeal, ¬r • m = 0\nx : R\nhx : x ∈ RingHom.ker (algebraMap R A)\nm : A\nhm : ∀ r ∉ p.asIdeal, ¬r • m = 0\nhx' : x ∉ p.asIdeal\n⊢ False",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 160,
"column": 28
} | {
"line": 160,
"column": 39
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\n⊢ r • f m ≠ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 78
} | [
{
"pp": "case h.left\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 168,
"column": 37
} | {
"line": 168,
"column": 48
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • f m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\ne : r • m = 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 230,
"column": 15
} | {
"line": 230,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\np : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : Subsingleton (LocalizedModule p.primeCompl M)\nf : R\nhf : f ∈ ↑(Module.annihilator R M)\nhf' : { asIdeal := p, isPrime := ⋯ } ∈ (zeroLocus... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 156,
"column": 27
} | {
"line": 156,
"column": 85
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 248,
"column": 4
} | {
"line": 251,
"column": 87
} | [
{
"pp": "case a.refine_2\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nI : Ideal R\n⊢ support R (M ⧸ I • ⊤) ⊆ zeroLocus ↑I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Submodule.annihilator_quotie... | · rw [support_eq_zeroLocus]
apply PrimeSpectrum.zeroLocus_anti_mono_ideal
rw [Submodule.annihilator_quotient]
exact fun x hx ↦ Submodule.mem_colon.mpr fun p hp ↦ Submodule.smul_mem_smul hx hp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 58
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 293,
"column": 2
} | {
"line": 308,
"column": 86
} | [
{
"pp": "case h\nR : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing S'\ninst✝⁵ : Algebra R' S'\nP' : Extension R' S'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\n... | induction x using TensorProduct.induction_on with
| zero =>
simp only [map_zero]
| add =>
simp only [map_add, LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply, *]
| tmul x y =>
obtain ⟨y, rfl⟩ := KaehlerDifferential.tensorProductTo_surjective _ _ y
induction y with
| zero... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 70
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Algebra R S\nP✝ : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝²⁰ : CommRing R'\ninst✝¹⁹ : CommRing S'\ninst✝¹⁸ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁷ : Algebra R R'\ninst✝¹⁶ : Algebra S S'\ninst✝¹⁵ : Algebra R S'\... | simp only [LinearMap.mem_ker, Submodule.restrictScalars_mem] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 17
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 56
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\n⊢ Module.FinitePresentation S Ω[S⁄R]",
"usedConstants": [
"Algebra.Presentation.ofFinitePresentationVars",
"Algebra.Presentation.ofFinitePresentation",
"Algebra.... | let P := Algebra.Presentation.ofFinitePresentation R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 539,
"column": 82
} | {
"line": 544,
"column": 82
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\n⊢ Module.FinitePresentation S Ω[S⁄R]",
"usedConstants": [
"Function.Exact",
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"Module.Finite.fg_top",
... | by
let P := Algebra.Presentation.ofFinitePresentation R S
have : Algebra.FiniteType R P.toExtension.Ring := by simp [P]; infer_instance
refine Module.finitePresentation_of_surjective _ P.toExtension.toKaehler_surjective ?_
rw [LinearMap.exact_iff.mp P.toExtension.exact_cotangentComplex_toKaehler, ← Submodule.ma... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 609,
"column": 2
} | {
"line": 609,
"column": 56
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nι : Type w\nι' : Type u_1\nP : Generators R S ι\nS' : Type u_2\ninst✝⁹ : CommRing S'\ninst✝⁸ : Algebra R S'\nT : Type w\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S... | let P := Algebra.Presentation.ofFinitePresentation R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Flat R M\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → R\nx : ι → M\nh : ∑ i, f i • x i = 0\n⊢ ∑ i, (f ∘ ⇑(Fintype.equivFin ι).symm) i • (x ∘ ⇑(Fintype.equivFin ι).symm) i = 0",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 49
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : Flat R M\nN : Type u_3\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : N\nx : N →ₗ[R] M\nh : x f = 0\ne : (Fin (Fintype.card (Free.ChooseBasisIndex R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRankOperations | {
"line": 63,
"column": 58
} | {
"line": 63,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsLocalRing R\nfg : N.FG\nthis : Module.Finite R ↥N := Module.Finite.iff_fg.mpr fg\ns : Set (𝓀 ⊗[R] ↥N)\nhs₁ : Cardinal.mk ↑s = ⊤.spanRank\nhs₂ : span 𝓀 s = ⊤\n⊢ Cardinal.mk ↑s < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Parity | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | [
{
"pp": "case inr\nn : ℕ\nhn : Odd n\nk : Fin n\nthis : NeZero n\nhk : Odd ↑k\n⊢ Even k",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Parity | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | [
{
"pp": "case inl\nn : ℕ\ninst✝ : NeZero n\nhn : Odd n\nk : Fin n\nhk : Even ↑k\n⊢ Odd k",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 239,
"column": 31
} | {
"line": 239,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh : x ∘ₗ f = 0\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 17
} | [
{
"pp": "case singleton\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 264,
"column": 58
} | {
"line": 264,
"column": 82
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : Flat R M\nK : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommGroup K\ninst✝⁵ : Module R K\ninst✝⁴ : Module.Finite R K\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Modul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 57
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nN : Type u_4\nN₂ : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N₂\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module R N\ninst✝ : Module R N₂\nn : ℕ\nf : M →ₗ[R] M [⋀... | rcases exists_succAbove_eq hkj.symm with ⟨j, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 36
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → Module.Dual R M\ni j : Fin n\nhf : f i = f j\nhij : i ≠ j\nv : Fin n → M\nthis : (Matrix.of fun i j ↦ (f j) (v i)).det = 0\n⊢ (LinearMap.compAlternatingMap\n (((toTensorPower R M n).d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 102,
"column": 15
} | {
"line": 102,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₁ : ∀ (i : ι), (f i) (x i) = 1\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na : Fin n ↪o ι\ni j : Fin n\nhij : ¬i = j\n⊢ a j ≠ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 59,
"column": 21
} | {
"line": 59,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (ιMultiDual R n b s) ((ιMulti R n) (⇑b ∘ ⇑(ofFinEmbEquiv.symm s))) = 1",
"usedConstants": [
"Set.powersetCard.... | ιMultiDual_apply_ιMulti | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 73,
"column": 21
} | {
"line": 73,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ (ιMultiDual R n b s) ((ιMulti R n) (⇑b ∘ ⇑(ofFinEmbEquiv.symm t))) = 0",
"usedConstants": [
"Se... | ιMultiDual_apply_ιMulti | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 35
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\nx : ↥(⋀[R]^n M)\ns : ↑(powersetCard I n)\n⊢ ((Basis.exteriorPower n b).repr x) s = (ιMultiDual R n b s) x",
"usedConstants": [
"Finsupp.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ ((Basis.exteriorPower n b).repr (ιMulti_family R n (⇑b) s)) s = 1",
"usedConstants": [
"Finsupp.instFunLike",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ ((Basis.exteriorPower n b).repr (ιMulti_family R n (⇑b) s)) t = 0",
"usedConstants": [
"Finsupp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 13
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nN : Submodule R S\nn : ↥N\n⊢ N.lTensorOne' (1 ⊗ₜ[R] n) = n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM : Submodule R S\nm : ↥M\n⊢ M.rTensorOne' (m ⊗ₜ[R] 1) = m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 220,
"column": 2
} | {
"line": 223,
"column": 80
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nn : ↥M\nr : ↥⊥\n⊢ ((↑(TensorProduct.comm R ↥⊥ ↥M) ∘ₗ (TensorProduct.mk R ↥⊥ ↥M) 1) ∘ₗ M.rTensorOne') (n ⊗ₜ[R] r) =\n LinearMap.id (n ⊗ₜ[R] r)",
"usedConstants": [
"Subalgebra.ins... | change rTensorOne' M _ ⊗ₜ[R] 1 = n ⊗ₜ[R] r
obtain ⟨x, h⟩ := Algebra.mem_bot.1 r.2
replace h : algebraMap R _ x = r := Subtype.val_injective h
rw [← h, rTensorOne'_tmul, TensorProduct.smul_tmul, Algebra.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 220,
"column": 2
} | {
"line": 223,
"column": 80
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nn : ↥M\nr : ↥⊥\n⊢ ((↑(TensorProduct.comm R ↥⊥ ↥M) ∘ₗ (TensorProduct.mk R ↥⊥ ↥M) 1) ∘ₗ M.rTensorOne') (n ⊗ₜ[R] r) =\n LinearMap.id (n ⊗ₜ[R] r)",
"usedConstants": [
"Subalgebra.ins... | change rTensorOne' M _ ⊗ₜ[R] 1 = n ⊗ₜ[R] r
obtain ⟨x, h⟩ := Algebra.mem_bot.1 r.2
replace h : algebraMap R _ x = r := Subtype.val_injective h
rw [← h, rTensorOne'_tmul, TensorProduct.smul_tmul, Algebra.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Module | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 17
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsLocalRing R\nN₁ N₂ : Submodule R M\nh : N₁ ≤ N₂\nh' : N₂.FG\nhN : Submodule.map (𝔪 • N₂).mkQ N₁ = Submodule.map (𝔪 • N₂).mkQ N₂\n⊢ N₂ ≤ 𝔪 • N₂ ⊔ N₁",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 54
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn : ι\ns : Finset ι\nhn : n ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 536,
"column": 34
} | {
"line": 536,
"column": 68
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nhf : Flat R ↥M ∨ Flat R ↥N\nhc : ∀ (m n : ↥(M ⊓ N)), Commute ↑m ↑n\na✝ : Nontrivial R\ns : Finset ↥(M ⊓ N)\nh : LinearIndependent R fun i ↦ ↑i\nhs : 1 < Fintype.card ↥s\n⊢ Fal... | Fintype.one_lt_card_iff_nontrivial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 283,
"column": 47
} | {
"line": 283,
"column": 52
} | [
{
"pp": "case insert.specialize_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ... | ← hfv | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 84
} | [
{
"pp": "case right.refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsLocalRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nι : Type u\nv : ι → M\nh : Function.Bijective ⇑(linearCombination k (⇑((TensorProduct.mk R k M) 1) ∘ v))\n⊢ ⊤ ≤ Submo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 409,
"column": 47
} | {
"line": 409,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nn : ℕ\nrk : ∀ (P : MaximalSpectrum R), finrank (R ⧸ P.asIdeal) ((R ⧸ P.asIdeal) ⊗[R] M) = n\nthis : {R : Type u_1} → [inst : ... | LinearMap.coe_restrictScalars | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 53
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := ⋯\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 67
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 291,
"column": 94
} | {
"line": 294,
"column": 5
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nz : R\nf : (i : ι) → s i\n⊢ tprodCoeff R z f = z • (tprod R) f",
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.mpr",
"NonAssocSem... | by
have : z = z • (1 : R) := by simp only [mul_one, smul_eq_mul]
conv_lhs => rw [this]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 61,
"column": 43
} | {
"line": 61,
"column": 82
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\n⊢ f.range = map N₁.mkQ N₂",
"usedConstants": [
"Submodule",
"Submodule.... | simp [f, mapQ, range_liftQ, range_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 96
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\n⊢ N₁.IsQuotientEquivQuotientPrime N₂ ↔ ∃ x, (⊥.colon {N₁... | refine ⟨fun ⟨h, p, ⟨e⟩⟩ ↦ ?_, fun ⟨x, hx, hx'⟩ ↦ ⟨le_sup_left.trans_eq hx'.symm, ⟨_, hx⟩, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 234,
"column": 17
} | {
"line": 234,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ x ≠ 0",
"usedConstants": [
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"id"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 234,
"column": 42
} | {
"line": 234,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ r • x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 78
} | [
{
"pp": "case refine_2.refine_3\nA : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 21
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Nonempty (NormalizedGCDMonoid R)\nI : Ideal R\nhI : IsUnit ↑I\na : R\nK : Ideal R\nha0 : a ≠ 0\nh : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K\nhIK : I * K = span {a}\n⊢ Submodule.IsPrincipal (I * K... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 93
} | [
{
"pp": "A : Type u\ninst✝¹ : CommRing A\ninst✝ : IsNoetherianRing A\nI : Ideal A\nh : Disjoint ↑I ↑(nonZeroDivisors A)\nx : A\nhP : I ≤ ⊥.colon {x}\nprime : (⊥.colon {x}).IsPrime\n⊢ ∀ n ∈ I, x • n = 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ClassGroup | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 51
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ (toPrincipalIdeal R K) x = I ↔ spanSingleton R⁰ ↑x = ↑I",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Units.ext_iff",
"M... | simp only [toPrincipalIdeal]; exact Units.ext_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ClassGroup | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 51
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ (toPrincipalIdeal R K) x = I ↔ spanSingleton R⁰ ↑x = ↑I",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Units.ext_iff",
"M... | simp only [toPrincipalIdeal]; exact Units.ext_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ClassGroup | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI J : (FractionalIdeal R⁰ (FractionRing R))ˣ\nI' J' : Ideal R\nhI : ↑I = ↑I'\nhJ : ↑J = ↑J'\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\nh : Ideal.span {x} * I' = Ideal.span {y} * J'\n⊢ IsUnit (mk' (FractionRing R) x ⟨y, ⋯⟩)",
"usedConstants": [
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.TensorProduct | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nhRT : (algebraMap R T).SurjectiveOnStalks\np₁ p₂ : PrimeSpectrum (S ⊗[R] T)\nh : tensorProductTo R S T p₁ = tensorProductTo R S T p₂\ng : T →+* S ⊗[R] T :=... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ClassGroup | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 35
} | [
{
"pp": "case mp.refine_2\nR : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI J : ↥(Ideal R)⁰\nX : (FractionalIdeal R⁰ K)ˣ\nhx : ↑↑I * ↑X = ↑↑J\nx : K\nhX : spanSingleton R⁰ x = ↑X\n⊢ spanSingle... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ClassGroup | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 31
} | [
{
"pp": "case mpr\nR : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI J : ↥(Ideal R)⁰\nx : K\nhx : x ≠ 0\neq_J : spanSingleton R⁰ x * ↑↑I = ↑↑J\n⊢ ↑↑I * ↑(Units.mk0 (spanSingleton R⁰ x) ⋯) = ↑↑J... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Projective | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type uM\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nRₛ : Type u_4\nMₛ : Type u_5\ninst✝⁹ : AddCommGroup Mₛ\ninst✝⁸ : Module R Mₛ\ninst✝⁷ : CommRing Rₛ\ninst✝⁶ : Algebra R Rₛ\ninst✝⁵ : Module Rₛ Mₛ\ninst✝⁴ : IsScalarTower R Rₛ Mₛ\nS : Submonoid R\nf : M →ₗ[R]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ClassGroup | {
"line": 373,
"column": 8
} | {
"line": 373,
"column": 19
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : Subsingleton (ClassGroup R)\nI : Ideal R\nhI : IsUnit ↑I\nhsub : (↑↑I).IsPrincipal\n⊢ (coeSubmodule K I).IsPrincipal",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Projective | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 12
} | [
{
"pp": "case H\nR : Type u_1\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal ... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 20
} | [
{
"pp": "case h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\nhfs : Function.Surjective f.toFun\na b : LocallyConstant Y Z\nh : comap f a = comap f b\ny : Y\nx : X\nhx : f.toFun x = y\n⊢ a y = b y",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 575,
"column": 8
} | {
"line": 575,
"column": 40
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₀ C₁ C₂ : Set X\nh₀ : C₀ ⊆ C₁ ∪ C₂\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nf₁ : LocallyConstant (↑C₁) Z\nf₂ : LocallyConstant (↑C₂) Z\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\nhf : ∀ (x : X) (hx : x ∈ C₁ ∩ C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 17
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₀ C₁ C₂ : Set X\nh₀ : C₀ ⊆ C₁ ∪ C₂\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nf₁ : LocallyConstant (↑C₁) Z\nf₂ : LocallyConstant (↑C₂) Z\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\nhf : ∀ (x : X) (hx : x ∈ C₁ ∩ C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Int | {
"line": 33,
"column": 8
} | {
"line": 33,
"column": 78
} | [
{
"pp": "p : ℕ\nhp : _root_.Prime ↑p\na b : ℕ\n⊢ p ∣ a * b → p ∣ a ∨ p ∣ b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Int | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | [
{
"pp": "p q : ℕ\nhp : Prime p\nhq : Prime q\nm : ℕ\nhm : m + 1 ≠ 0\nn : ℕ\nhn : n + 1 ≠ 0\nh : p ^ (m + 1) = q ^ (n + 1)\n⊢ p = q ∧ m + 1 = n + 1",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAdd",
"Nat",
"in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Set.Ioc"... | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Set.Ioc"... | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Set.Ioc"... | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 159,
"column": 52
} | {
"line": 159,
"column": 75
} | [
{
"pp": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq : Fact (Nat.Prime q)\nhpq : p ≠ q\n⊢ ¬q ≡ 0 [MOD p]",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"id",
"instOfNatNat",
"Nat.ModEq",
"Nat.instDvd",
"Nat",
"propext",
"Nat.modEq_zero_iff_dvd",
... | Nat.modEq_zero_iff_dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 270,
"column": 2
} | {
"line": 280,
"column": 90
} | [
{
"pp": "R : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\n⊢ rankAtStalk ((i : ι) → M i) p = ∑ᶠ (i : ι), r... | cases nonempty_fintype ι
let f : (Π i, M i) →ₗ[R] Π i, LocalizedModule p.asIdeal.primeCompl (M i) :=
.pi (fun i ↦ mkLinearMap p.asIdeal.primeCompl (M i) ∘ₗ LinearMap.proj i)
let e : LocalizedModule p.asIdeal.primeCompl (Π i, M i) ≃ₗ[Localization.AtPrime p.asIdeal]
Π i, LocalizedModule p.asIdeal.primeCompl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 270,
"column": 2
} | {
"line": 280,
"column": 90
} | [
{
"pp": "R : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\n⊢ rankAtStalk ((i : ι) → M i) p = ∑ᶠ (i : ι), r... | cases nonempty_fintype ι
let f : (Π i, M i) →ₗ[R] Π i, LocalizedModule p.asIdeal.primeCompl (M i) :=
.pi (fun i ↦ mkLinearMap p.asIdeal.primeCompl (M i) ∘ₗ LinearMap.proj i)
let e : LocalizedModule p.asIdeal.primeCompl (Π i, M i) ≃ₗ[Localization.AtPrime p.asIdeal]
Π i, LocalizedModule p.asIdeal.primeCompl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Associated | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 45
} | [
{
"pp": "case mpr\na : ℤ\nu : ℤˣ\n⊢ a = a * ↑u ∨ a = -(a * ↑u)",
"usedConstants": [
"Int.units_eq_one_or"
]
}
] | obtain rfl | rfl := Int.units_eq_one_or u | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 56,
"column": 2
} | {
"line": 58,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : S ⊗[R] p.ResidueField\nr : R\na : S\ny : R\nt : ↥p.primeCompl\ne :\n 1 ⊗ₜ[R]\n (r •\n (IsLocalRing.residue (Localization.AtPrime p))\n ((fun x ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 73,
"column": 23
} | {
"line": 73,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : p.Fiber S\nr : R\nhr : r ∉ p\ns : S\ne : r • (Algebra.TensorProduct.comm R p.ResidueField S) x = s ⊗ₜ[R] 1\n⊢ r • x = 1 ⊗ₜ[R] s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 17
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np✝ : Ideal R\ninst✝ : p✝.IsPrime\np : PrimeSpectrum R\nq₁ q₂ : PrimeSpectrum (p.asIdeal.Fiber S)\n⊢ (preimageEquivFiber R S p) ((preimageEquivFiber R S p).symm q₁) ≤\n (preimageEquivFiber R S p) ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 134,
"column": 44
} | {
"line": 134,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np✝ : Ideal R\ninst✝ : p✝.IsPrime\np : PrimeSpectrum R\nq₁ q₂ : ↑(comap (algebraMap R S) ⁻¹' {p})\nH : q₁ ≤ q₂\nx : p.asIdeal.Fiber S\nr : R\nhr : r ∉ p.asIdeal\ns : S\nhx : 1 ⊗ₜ[R] s ∈ ((preimageEquivFiber R S p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 19
} | [
{
"pp": "case mpr\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np✝ : Ideal R\ninst✝ : p✝.IsPrime\np : PrimeSpectrum R\nq₁ q₂ : ↑(comap (algebraMap R S) ⁻¹' {p})\nH : q₁ ≤ q₂\nx : p.asIdeal.Fiber S\nr : R\nhr : r ∉ p.asIdeal\ns : S\ne : r • x = 1 ⊗ₜ[R] s\nhx : s ∈ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Quotient.Pi | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 17
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nMs : ι → Type u_3\ninst✝⁵ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁴ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nNs : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommGroup (Ns i)\ninst✝ : (i : ι) → Module R (Ns i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 26
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\n⊢ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 20
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\ne : M ⊗[R] N ≃ₗ[R] R\nthis :\n curry ↑e =\n ↑(LinearEquiv.congrLeft (M ⊗[R] N) R (TensorProduct.lid R N) ≪≫ₗ e.congrRight) ∘ₗ\n rTensorHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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