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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 12
} | {
"line": 720,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\n⊢ I ≠ ⊥",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants"... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\nhI : I = ⊥\n⊢ False"
] | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1114,
"column": 4
} | {
"line": 1114,
"column": 84
} | {
"line": 1114,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : NormalizationMonoid R\nr : R\nhr : r ≠ 0\nI : ↑{I | I ∈ normalizedFactors (span {r})}\na : R\nha : a ∈ {d | d ∈ normalizedFactors r}\nhx : (normalizedFactorsEquivSpanNormalizedFactors hr) ⟨a, ha⟩ = I\n⊢ emu... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : NormalizationMonoid R\nr : R\nhr : r ≠ 0\nI : ↑{I | I ∈ normalizedFactors (span {r})}\na : R\nha : a ∈ {d | d ∈ normalizedFactors r}\nhx : (normalizedFactorsEquivSpanNormalizedFactors hr) ⟨a, ha⟩ = I\n⊢ emultiplicity (... | emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_emultiplicity hr ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 363,
"column": 49
} | {
"line": 406,
"column": 6
} | {
"line": 408,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nN' : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\nh : FinitePresentatio... | [] | by
obtain ⟨σ, hσ, τ, hτ⟩ := h
let π := Finsupp.linearCombination R ((↑) : σ → M)
have hπ : Function.Surjective π :=
LinearMap.range_eq_top.mp
(by rw [range_linearCombination, Subtype.range_val, ← hσ])
classical
choose s hs using IsLocalizedModule.surj S f
let i : σ → N :=
fun x ↦ (∏ j ∈ σ.eras... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 417,
"column": 21
} | {
"line": 417,
"column": 44
} | {
"line": 417,
"column": 45
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\nN' : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\ng₁ g₂ : M →ₗ[... | [
"case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\nN' : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\ng₁ g₂ : M →ₗ[R] N\nh : f ... | ← LinearMap.ker_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.PID | {
"line": 214,
"column": 10
} | {
"line": 214,
"column": 35
} | {
"line": 215,
"column": 10
} | [
{
"pp": "case h.succ.refine_3.refine_2.refine_2\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\np : R\nhp : Irreducible p\nd : ℕ\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhM : IsTorsion' M ↥(Submonoid.powers p)\ns : Fin (d + 1) → M\nhs : Submodule.span R (... | [
"case h.succ.refine_3.refine_2.refine_2\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\np : R\nhp : Irreducible p\nd : ℕ\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhM : IsTorsion' M ↥(Submonoid.powers p)\ns : Fin (d + 1) → M\nhs : Submodule.span R (Set.range s)... | rw [LinearMap.comp_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.PID | {
"line": 238,
"column": 2
} | {
"line": 244,
"column": 79
} | {
"line": 245,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\nh' : Module.Finite R M\nhM : IsTorsion R M\nI : Type u\nfI : Fintype I\nw✝ : DecidableEq I\np : I → R\nhp : ∀ (i : I), Irreducible (p i)\ne : I → ℕ\nh : Direct... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\nh' : Module.Finite R M\nhM : IsTorsion R M\nI : Type u\nfI : Fintype I\nw✝ : DecidableEq I\np : I → R\nhp : ∀ (i : I), Irreducible (p i)\ne : I → ℕ\nh : DirectSum.IsIntern... | have :
∀ i,
∃ (d : ℕ) (k : Fin d → ℕ),
Nonempty <| torsionBy R M (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
exact fun i =>
torsion_by_prime_power_decomposition.{u, v} (hp i)
((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 16
} | {
"line": 481,
"column": 4
} | [
{
"pp": "case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀... | [
"case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSu... | intro hα_mem | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 16
} | {
"line": 481,
"column": 4
} | [
{
"pp": "case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀... | [
"case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSu... | intro hα_mem | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 22
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case a\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nT : Type u_1\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nP : Presentation R S ι σ\nx : MvPolynomial ι T\ny : σ\nhy : (fun i ↦ (MvPolynomial.map (algebraMap R T)) (P.relation i)) y = x\... | [
"case a\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nT : Type u_1\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nP : Presentation R S ι σ\nx : MvPolynomial ι T\ny : σ\nhy : (fun i ↦ (MvPolynomial.map (algebraMap R T)) (P.relation i)) y = x\nZ : TensorP... | rw [map_zero] at Z | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Basic | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 59
} | {
"line": 413,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nr : R\nx : P.Cotangent\n⊢ (r • x).val = r • x.val",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Algebra.Extension.commRing",
... | [] | rw [← algebraMap_smul P.Ring, val_smul', algebraMap_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Basic | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 59
} | {
"line": 413,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nr : R\nx : P.Cotangent\n⊢ (r • x).val = r • x.val",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Algebra.Extension.commRing",
... | [] | rw [← algebraMap_smul P.Ring, val_smul', algebraMap_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Basic | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 59
} | {
"line": 413,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nr : R\nx : P.Cotangent\n⊢ (r • x).val = r • x.val",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Algebra.Extension.commRing",
... | [] | rw [← algebraMap_smul P.Ring, val_smul', algebraMap_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Differentials | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 30
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\nφ : (σ →₀ S) →ₗ[pres.Ring] pres.toExtension.Cotangent := { toFun := ⇑(hom₁ pres), map_add' := ⋯, map_smul' := ⋯ }\n⊢ Function.Surjective ⇑(hom₁ pres)",
"ppTerm... | [
"R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\nφ : (σ →₀ S) →ₗ[pres.Ring] pres.toExtension.Cotangent := { toFun := ⇑(hom₁ pres), map_add' := ⋯, map_smul' := ⋯ }\n⊢ Function.Surjective ⇑φ"
] | change Function.Surjective φ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Extension.Generators | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 30
} | {
"line": 165,
"column": 31
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S ι\ns : Set S\nhs : adjoin R s = ⊤\n⊢ adjoin R (Set.range Subtype.val) = ⊤",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatt... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S ι\ns : Set S\nhs : adjoin R s = ⊤\n⊢ adjoin R {x | x ∈ s} = ⊤"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 368,
"column": 16
} | {
"line": 368,
"column": 96
} | {
"line": 369,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPol... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPolynomial σ R ... | rw [← hs x, ← Ideal.Quotient.mkₐ_eq_mk R, aeval_unique (Ideal.Quotient.mkₐ _ I)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.RingTheory.Extension.Generators | {
"line": 368,
"column": 16
} | {
"line": 368,
"column": 96
} | {
"line": 369,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPol... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPolynomial σ R ... | rw [← hs x, ← Ideal.Quotient.mkₐ_eq_mk R, aeval_unique (Ideal.Quotient.mkₐ _ I)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.RingTheory.Extension.Generators | {
"line": 368,
"column": 16
} | {
"line": 368,
"column": 96
} | {
"line": 369,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPol... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nι' : Type u_1\nT : Type ?u.18\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nσ : Type u_2\nI : Ideal (MvPolynomial σ R)\ns✝ : MvPolynomial σ R ⧸ I → MvPolynomial σ R\nhs✝ : ∀ (x : MvPolynomial σ R ... | rw [← hs x, ← Ideal.Quotient.mkₐ_eq_mk R, aeval_unique (Ideal.Quotient.mkₐ _ I)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 159,
"column": 4
} | {
"line": 170,
"column": 9
} | {
"line": 171,
"column": 2
} | [
{
"pp": "case refine_1\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... | [] | apply_fun finsuppScalarLeft R N ι at hkn
apply_fun (· i) at hkn
symm at hkn
simp only [map_sum, finsuppScalarLeft_apply_tmul, zero_smul, Finsupp.single_zero,
Finsupp.sum_single_index, one_smul, Finsupp.finsetSum_apply, Finsupp.single_apply,
Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, rTenso... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 159,
"column": 4
} | {
"line": 170,
"column": 9
} | {
"line": 171,
"column": 2
} | [
{
"pp": "case refine_1\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... | [] | apply_fun finsuppScalarLeft R N ι at hkn
apply_fun (· i) at hkn
symm at hkn
simp only [map_sum, finsuppScalarLeft_apply_tmul, zero_smul, Finsupp.single_zero,
Finsupp.sum_single_index, one_smul, Finsupp.finsetSum_apply, Finsupp.single_apply,
Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, rTenso... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Support | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 41
} | {
"line": 251,
"column": 6
} | [
{
"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⊢ I ≤ annihilator R (M ⧸ I • ⊤)",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"... | [
"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⊢ I ≤ (I • ⊤).colon Set.univ"
] | rw [Submodule.annihilator_quotient] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.SpanRankOperations | {
"line": 49,
"column": 7
} | {
"line": 49,
"column": 34
} | {
"line": 49,
"column": 35
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ ⊤.spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ (baseChange A ⊤).spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank"
] | ← Submodule.baseChange_top, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Algebra.Module.SpanRankOperations | {
"line": 53,
"column": 7
} | {
"line": 53,
"column": 34
} | {
"line": 53,
"column": 35
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nfg : N.FG\n⊢ ⊤.spanFinrank ≤ N.spanFinrank",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nfg : N.FG\n⊢ (baseChange A ⊤).spanFinrank ≤ N.spanFinrank"
] | ← Submodule.baseChange_top, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 160,
"column": 71
} | {
"line": 165,
"column": 74
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Nontrivial R\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\n⊢ finrank R ↥(⋀[R]^n M) = (finrank R M).choose n",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | by
classical
let : LinearOrder (Module.Free.ChooseBasisIndex R M) := linearOrderOfSTO WellOrderingRel
let B := (Module.Free.chooseBasis R M).exteriorPower n
rw [Module.finrank_eq_card_basis (Module.Free.chooseBasis R M), Module.finrank_eq_card_basis B,
Fintype.card_eq_nat_card, powersetCard.card, Fintype.ca... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian | {
"line": 27,
"column": 30
} | {
"line": 41,
"column": 55
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nhR : IsNoetherianRing R\nI : Ideal R\n⊢ I.minimalPrimes.Finite",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian.0.Ideal.finite_minimalPrimes_of_isNoetherianRing._simp_1_1",
"... | [] | by
by_contra hI
obtain ⟨I : Ideal R, hI : ¬ I.minimalPrimes.Finite, hmax⟩ :=
set_has_maximal_iff_noetherian.mpr hR {I : Ideal R | ¬ I.minimalPrimes.Finite} ⟨I, hI⟩
simp only [Set.mem_setOf_eq, not_imp_not] at hmax
have h1 : ¬ I.IsPrime := by contrapose hI; simp [minimalPrimes_eq_subsingleton_self]
have h2... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.Module | {
"line": 203,
"column": 4
} | {
"line": 207,
"column": 81
} | {
"line": 208,
"column": 4
} | [
{
"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✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1)... | [
"case refine_1\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✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M)... | refine lTensor_injective_of_exact_of_exact_of_rTensor_injective
(N₁ := LinearMap.ker i) (N₂ := ι →₀ R) (N₃ := M)
(f₁ := (𝔪).subtype) (f₂ := Submodule.mkQ 𝔪)
(g₁ := (LinearMap.ker i).subtype) (g₂ := i) (LinearMap.exact_subtype_mkQ 𝔪)
(Submodule.mkQ_surjective _) (LinearMap.exact_subtype_ker_ma... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 24
} | {
"line": 39,
"column": 2
} | [
{
"pp": "case neg\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\nhI : I ∈ (N.colon {x}).minimalPrimes\nhx : x ∉ N\n⊢ ∃ x', I = N.colon {x'}",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\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 := N.colon {x}\nhI : I ∈ ann.minimalPrimes\n⊢ ∃ x', I = N.colon {x'}"
] | (Mathlib.Tactic.setTactic
"set"
[]
(Mathlib.Tactic.setArgsRest
(Lean.binderIdent `ann)
[]
":="
(Term.app
`colon
[`N
(choice
(«term{_}» "{" [`x] "}")
(Term.structInst
"{"
[]
(Term.structInstFields [(Term.structInstField (Term.structInstLVal `x []) [])])
(Term.optEll... | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.LocalRing.Module | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 37
} | {
"line": 296,
"column": 2
} | [
{
"pp": "R : 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⊢ Function.Bijective ⇑(linearCom... | [
"case right\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⊢ Surjective ⇑(linearCombination... | use linearIndependent_of_flat _ h.1 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 68
} | {
"line": 107,
"column": 0
} | [
{
"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 [hIK] using (inferInstance : (Ideal.span {a}).IsPrincipal) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.TensorProduct | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 14
} | {
"line": 48,
"column": 2
} | [
{
"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 :=... | [
"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 := Algebra.Ten... | intro x hxp₁ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 10
} | {
"line": 360,
"column": 2
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (↑↑I).IsPrincipal → ∃ x, spanSingleton R⁰ ↑x = ↑I",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Units.v... | [
"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 10
} | {
"line": 360,
"column": 2
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (↑↑I).IsPrincipal → ∃ x, spanSingleton R⁰ ↑x = ↑I",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Units.v... | [
"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I"
] | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Localization.Free | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 45
} | {
"line": 86,
"column": 2
} | [
{
"pp": "R : Type u_4\nM : Type u_5\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝¹⁰ : AddCommGroup M'\ninst✝⁹ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁸ : IsLocalizedModule S f\nRₛ : Type u_3\ninst✝⁷ : CommRing Rₛ\ninst✝⁶ : Algebra R Rₛ\ninst✝⁵ : Module ... | [
"R : Type u_4\nM : Type u_5\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝¹⁰ : AddCommGroup M'\ninst✝⁹ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁸ : IsLocalizedModule S f\nRₛ : Type u_3\ninst✝⁷ : CommRing Rₛ\ninst✝⁶ : Algebra R Rₛ\ninst✝⁵ : Module Rₛ M'\ninst✝... | let I := Module.Free.ChooseBasisIndex Rₛ M' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 84,
"column": 10
} | {
"line": 84,
"column": 41
} | {
"line": 84,
"column": 42
} | [
{
"pp": "case h1\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\ny : ℤ\nh : x = ↑y ∧ y * 2 ∈ Set.Ioc (-↑n) ↑n\n⊢ ↑n ∣ x.valMinAbs - y",
"ppTerm": "?h1",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Dvd.dvd",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiri... | [
"case h1\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\ny : ℤ\nh : x = ↑y ∧ y * 2 ∈ Set.Ioc (-↑n) ↑n\n⊢ ↑(x.valMinAbs - y) = 0"
] | ← intCast_zmod_eq_zero_iff_dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 116,
"column": 21
} | {
"line": 116,
"column": 52
} | {
"line": 116,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : Fintype Kˣ\na : Kˣ\n⊢ a ∈ univ.erase (-1) → a⁻¹ ∈ univ.erase (-1)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"CommRing.toNon... | [] | simp [@inv_eq_iff_eq_inv _ _ a] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Finite.Basic | {
"line": 116,
"column": 21
} | {
"line": 116,
"column": 52
} | {
"line": 116,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : Fintype Kˣ\na : Kˣ\n⊢ a ∈ univ.erase (-1) → a⁻¹ ∈ univ.erase (-1)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"CommRing.toNon... | [] | simp [@inv_eq_iff_eq_inv _ _ a] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 116,
"column": 21
} | {
"line": 116,
"column": 52
} | {
"line": 116,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : Fintype Kˣ\na : Kˣ\n⊢ a ∈ univ.erase (-1) → a⁻¹ ∈ univ.erase (-1)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"CommRing.toNon... | [] | simp [@inv_eq_iff_eq_inv _ _ a] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Associated | {
"line": 24,
"column": 2
} | {
"line": 24,
"column": 42
} | {
"line": 25,
"column": 2
} | [
{
"pp": "a b : ℤ\n⊢ a.natAbs = b.natAbs ↔ Associated a b",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Int.natAbs_eq_natAbs_iff",
"Int.instNegInt",
"Int",
"Int.instMonoid",
"Associated",
"Nat",
"Int.natAbs",
"Iff.trans",
"Or",
"... | [
"a b : ℤ\n⊢ a = b ∨ a = -b ↔ Associated a b"
] | refine Int.natAbs_eq_natAbs_iff.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Int.Associated | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 28
} | {
"line": 33,
"column": 0
} | [
{
"pp": "case mpr.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOn... | [] | exact Or.inr (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Int.Associated | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 28
} | {
"line": 33,
"column": 0
} | [
{
"pp": "case mpr.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOn... | [] | exact Or.inr (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Associated | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 28
} | {
"line": 33,
"column": 0
} | [
{
"pp": "case mpr.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOn... | [] | exact Or.inr (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Int.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 45
} | {
"line": 116,
"column": 2
} | [
{
"pp": "a : ℤ\n⊢ Ideal.span {↑a.natAbs} = Ideal.span {a}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.span_singleton_eq_span_singleton",
"congrArg",
"CommSemiring.toSemiring",
"Set.instSingletonSet",
"id",
"Int",
"Ideal",... | [
"a : ℤ\n⊢ Associated (↑a.natAbs) a"
] | rw [Ideal.span_singleton_eq_span_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 193,
"column": 8
} | {
"line": 194,
"column": 25
} | {
"line": 195,
"column": 6
} | [
{
"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\nhx : 1 ⊗ₜ[R] s ∈ ((preimageEquivF... | [] | replace hx : s ∈ q₁.1.asIdeal := by simpa using! hx
simpa using! H hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 193,
"column": 8
} | {
"line": 194,
"column": 25
} | {
"line": 195,
"column": 6
} | [
{
"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\nhx : 1 ⊗ₜ[R] s ∈ ((preimageEquivF... | [] | replace hx : s ∈ q₁.1.asIdeal := by simpa using! hx
simpa using! H hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod | {
"line": 40,
"column": 68
} | {
"line": 40,
"column": 83
} | {
"line": 40,
"column": 83
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nf g : R →+* ZMod n\nh : RingHom.ker f = RingHom.ker g\nthis : ((f.liftOfRightInverse cast ⋯) ⟨g, ⋯⟩).comp f = g\n⊢ f = RingHom.comp ?m.65 f",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZMod.commRing",
"congrArg",... | [
"n : ℕ\nR : Type u_1\ninst✝ : Ring R\nf g : R →+* ZMod n\nh : RingHom.ker f = RingHom.ker g\nthis : ((f.liftOfRightInverse cast ⋯) ⟨g, ⋯⟩).comp f = g\n⊢ f = f"
] | RingHom.id_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 49
} | {
"line": 528,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = ↑x\n⊢ ↑a.valMinAbs.natAbs ^ 2 + ↑b.valMinAbs.natAbs ^ 2 ≡ x [ZMOD ↑p]",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Int.cast",
"ZMod.commRing",
"congrArg",
"ZMod.valMinAbs",
... | [
"p : ℕ\ninst✝ : Fact (Prime p)\nx : ℤ\na b : ZMod p\nhx : ↑a.valMinAbs ^ 2 + ↑b.valMinAbs ^ 2 = ↑x\n⊢ ↑a.valMinAbs.natAbs ^ 2 + ↑b.valMinAbs.natAbs ^ 2 ≡ x [ZMOD ↑p]"
] | rw [← a.coe_valMinAbs, ← b.coe_valMinAbs] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PicardGroup | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 29
} | {
"line": 558,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nS : Type u_9\nT : Type u_10\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : R →+* S\ng : S →+* T\n⊢ (mapRingHom g).comp (mapRingHom f) = mapRingHom (g.comp f)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [
"R : Type u\ninst✝² : CommSemiring R\nS : Type u_9\nT : Type u_10\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : R →+* S\ng : S →+* T\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalarTower R S T := IsSc... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.LinearAlgebra.FreeModule.Finite.Quotient | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 69
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case mpr\nι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : Finite ι\nN : Submodule R M\nb : Basis ι R M\nh : finrank R ↥N = finrank R M\nthis✝ : Fintype ι\na : ι → R := smithNormalF... | [] | exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Finite.Basic | {
"line": 694,
"column": 6
} | {
"line": 694,
"column": 13
} | {
"line": 694,
"column": 14
} | [
{
"pp": "case succ.zero\np a d k : ℕ\nhk : k ∈ range p\nhd : p * a = d\n⊢ p ∣ (p ^ 0 * d) ^ k * ↑(p.choose (k + 1))",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"Dvd.dvd",
"Nat.choose",
"HMu... | [
"case succ.zero.zero\np a d : ℕ\nhd : p * a = d\nhk : 0 ∈ range p\n⊢ p ∣ (p ^ 0 * d) ^ 0 * ↑(p.choose (0 + 1))",
"case succ.zero.succ\np a d : ℕ\nhd : p * a = d\nn✝ : ℕ\nhk : n✝ + 1 ∈ range p\n⊢ p ∣ (p ^ 0 * d) ^ (n✝ + 1) * ↑(p.choose (n✝ + 1 + 1))"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.FieldTheory.Finite.Basic | {
"line": 695,
"column": 6
} | {
"line": 695,
"column": 13
} | {
"line": 695,
"column": 14
} | [
{
"pp": "case succ.succ\np a d k : ℕ\nhk : k ∈ range p\nn✝ : ℕ\nhd : (1 + p * a) ^ p ^ (n✝ + 1) = p ^ (n✝ + 1) * d + 1\n⊢ p ∣ (p ^ (n✝ + 1) * d) ^ k * ↑(p.choose (k + 1))",
"ppTerm": "?succ.succ",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.... | [
"case succ.succ.zero\np a d n✝ : ℕ\nhd : (1 + p * a) ^ p ^ (n✝ + 1) = p ^ (n✝ + 1) * d + 1\nhk : 0 ∈ range p\n⊢ p ∣ (p ^ (n✝ + 1) * d) ^ 0 * ↑(p.choose (0 + 1))",
"case succ.succ.succ\np a d n✝¹ : ℕ\nhd : (1 + p * a) ^ p ^ (n✝¹ + 1) = p ^ (n✝¹ + 1) * d + 1\nn✝ : ℕ\nhk : n✝ + 1 ∈ range p\n⊢ p ∣ (p ^ (n✝¹ + 1) * d)... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.Norm.Basic | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 18
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case hfg\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Ring S\ninst✝² : Algebra R S\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra R F\npb : PowerBasis R S\nhE : (Polynomial.map (algebraMap R F) (minpoly R pb.gen)).Splits\nhfx : IsSeparable R pb.gen\nthis : DecidableEq F := Classical.dec... | [] | intro x; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Norm.Basic | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 18
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case hfg\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Ring S\ninst✝² : Algebra R S\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra R F\npb : PowerBasis R S\nhE : (Polynomial.map (algebraMap R F) (minpoly R pb.gen)).Splits\nhfx : IsSeparable R pb.gen\nthis : DecidableEq F := Classical.dec... | [] | intro x; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 144,
"column": 2
} | {
"line": 146,
"column": 98
} | {
"line": 148,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDedekindDomain S\ninst✝⁵ : Free ℤ S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Free ℤ R\ninst✝² : Algebra S R\ninst✝¹ : Module.Finite S R\nP : Ideal R\np : Ideal S\ninst✝ : P.LiesOver p\nhp : p.IsPrime\nhp_ne_bot : p ≠ ⊥\n⊢ absN... | [] | have : p.IsMaximal := hp.isMaximal hp_ne_bot
let _ : Field (S ⧸ p) := Quotient.field p
simpa [absNorm_apply, Submodule.cardQuot_apply] using Module.natCard_eq_pow_finrank (K := S ⧸ p) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 144,
"column": 2
} | {
"line": 146,
"column": 98
} | {
"line": 148,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDedekindDomain S\ninst✝⁵ : Free ℤ S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Free ℤ R\ninst✝² : Algebra S R\ninst✝¹ : Module.Finite S R\nP : Ideal R\np : Ideal S\ninst✝ : P.LiesOver p\nhp : p.IsPrime\nhp_ne_bot : p ≠ ⊥\n⊢ absN... | [] | have : p.IsMaximal := hp.isMaximal hp_ne_bot
let _ : Field (S ⧸ p) := Quotient.field p
simpa [absNorm_apply, Submodule.cardQuot_apply] using Module.natCard_eq_pow_finrank (K := S ⧸ p) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 74
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ (span {p}).inertiaDeg P",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Int... | [] | simpa using absNorm_eq_pow_inertiaDeg_of_liesOver P (span {p})
(by rwa [span_singleton_prime hp.ne_zero]) (by simpa using hp.ne_zero) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 74
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ (span {p}).inertiaDeg P",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Int... | [] | simpa using absNorm_eq_pow_inertiaDeg_of_liesOver P (span {p})
(by rwa [span_singleton_prime hp.ne_zero]) (by simpa using hp.ne_zero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 74
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ (span {p}).inertiaDeg P",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Int... | [] | simpa using absNorm_eq_pow_inertiaDeg_of_liesOver P (span {p})
(by rwa [span_singleton_prime hp.ne_zero]) (by simpa using hp.ne_zero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 198,
"column": 52
} | {
"line": 198,
"column": 68
} | {
"line": 198,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : IsDedekindDomain S\nh₁ : map f p ≠ ⊤\nh₂ : map f p ≠ ⊥\nh : map f p ≤ map f p ^ (1 + 1)\nthis : map f p ^ 1 = map f p ^ 2\n⊢ map f p ≠ 1",
"ppTerm": "?m.128",
"assigned": true,
"used... | [] | rwa [one_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 198,
"column": 52
} | {
"line": 198,
"column": 68
} | {
"line": 198,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : IsDedekindDomain S\nh₁ : map f p ≠ ⊤\nh₂ : map f p ≠ ⊥\nh : map f p ≤ map f p ^ (1 + 1)\nthis : map f p ^ 1 = map f p ^ 2\n⊢ map f p ≠ 1",
"ppTerm": "?m.128",
"assigned": true,
"used... | [] | rwa [one_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 198,
"column": 52
} | {
"line": 198,
"column": 68
} | {
"line": 198,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : IsDedekindDomain S\nh₁ : map f p ≠ ⊤\nh₂ : map f p ≠ ⊥\nh : map f p ≤ map f p ^ (1 + 1)\nthis : map f p ^ 1 = map f p ^ 2\n⊢ map f p ≠ 1",
"ppTerm": "?m.128",
"assigned": true,
"used... | [] | rwa [one_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 12
} | {
"line": 331,
"column": 4
} | [
{
"pp": "case mp\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\n⊢ absNorm I = 0 → I = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Ideal.absNorm",
"CommSemiring.toS... | [
"case mp\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\nhI : absNorm I = 0\n⊢ I = ⊥"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 12
} | {
"line": 331,
"column": 4
} | [
{
"pp": "case mp\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\n⊢ absNorm I = 0 → I = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Ideal.absNorm",
"CommSemiring.toS... | [
"case mp\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\nhI : absNorm I = 0\n⊢ I = ⊥"
] | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 88
} | {
"line": 223,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\n⊢ MeasurableSet {x | p x ↔ q x}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"setOf",
"Membership.mem"... | [
"α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\nh_eq : {x | p x ↔ q x} = {x | p x → q x} ∩ {x | q x → p x}\n⊢ MeasurableSet {x | p x ↔ q x}"
] | have h_eq : {x | p x ↔ q x} = {x | p x → q x} ∩ {x | q x → p x} := by ext; simp; grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 12
} | {
"line": 212,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nv : Associates (Ideal R)\nhv : Irreducible v\nJ : Ideal R\nhJv : Associates.mk J = v\n⊢ v.count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors = v.count (Associates.mk I).factors"... | [
"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nv : Associates (Ideal R)\nJ : Ideal R\nhv : Irreducible (Associates.mk J)\nhJv : Associates.mk J = v\n⊢ v.count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors = v.count (Associates.mk I).fact... | ← hJv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 226,
"column": 6
} | {
"line": 228,
"column": 50
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nh0 : I ≠ 0\nx : R\n⊢ x ∈ ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I → x ∈ ⨅ i, i.maxPowDividing I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"MulOne.... | [] | simp only [Finite.mem_toFinset, mem_mulSupport, one_eq_top, ne_eq, Submodule.mem_iInf]
intro h i
by_cases i.maxPowDividing I = ⊤ <;> simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 226,
"column": 6
} | {
"line": 228,
"column": 50
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nh0 : I ≠ 0\nx : R\n⊢ x ∈ ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I → x ∈ ⨅ i, i.maxPowDividing I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"MulOne.... | [] | simp only [Finite.mem_toFinset, mem_mulSupport, one_eq_top, ne_eq, Submodule.mem_iInf]
intro h i
by_cases i.maxPowDividing I = ⊤ <;> simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 58
} | {
"line": 118,
"column": 2
} | [
{
"pp": "A : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nhD : Disjoint N₁ N₂\nx : M\n⊢ x ∈ Submodule.map (N₁ ⊔ N₂).subtype (primaryComponent (↥(N₁ ⊔ N₂)) I) ↔\n x ∈ Submodule.map N₁.subtype (primaryComponent (↥N₁) I) ⊔ Submodul... | [
"A : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nhD : Disjoint N₁ N₂\nx : M\n⊢ ((∃ n, ∀ a ∈ I ^ n, a • x = 0) ∧ ∃ y ∈ N₁, ∃ z ∈ N₂, y + z = x) ↔\n ∃ y, ((∃ n, ∀ a ∈ I ^ n, a • y = 0) ∧ y ∈ N₁) ∧ ∃ z, ((∃ n, ∀ a ∈ I ^ n, a • z = 0) ... | simp_all only [mem_map, primaryComponent_mem, mem_torsionBySet_iff, SetLike.coe_sort_coe,
Subtype.forall, subtype_apply, Subtype.exists, SetLike.mk_smul_mk, mk_eq_zero, exists_and_left,
exists_prop, exists_eq_right_right, Submodule.mem_sup] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 120,
"column": 2
} | {
"line": 122,
"column": 37
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nhp : ∀ (x : α), ∃ N, p x N\nhm : ∀ (k : ℕ), MeasurableSet {x | p x k}\n⊢ Measurable fun x ↦ Nat.find ⋯",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"measurable_t... | [] | refine measurable_to_nat fun x => ?_
rw [preimage_find_eq_disjointed (fun k => {x | p x k})]
exact MeasurableSet.disjointed hm _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 120,
"column": 2
} | {
"line": 122,
"column": 37
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nhp : ∀ (x : α), ∃ N, p x N\nhm : ∀ (k : ℕ), MeasurableSet {x | p x k}\n⊢ Measurable fun x ↦ Nat.find ⋯",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"measurable_t... | [] | refine measurable_to_nat fun x => ?_
rw [preimage_find_eq_disjointed (fun k => {x | p x k})]
exact MeasurableSet.disjointed hm _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 199,
"column": 56
} | {
"line": 200,
"column": 89
} | {
"line": 201,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDedekindDomain A\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nhM₁ : IsTorsion A M₁\nP : HeightOneSpectrum A\nφ : M₁ →ₗ[A] M₂\nhf : Surjective ⇑φ\nb : M₁\nhy : φ b ∈ primaryComp... | [] | by
simp only [← mem_iSup_iff_exists_dfinsupp, iSup_primaryComponent_eq_top hM₁, mem_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 774,
"column": 4
} | {
"line": 775,
"column": 32
} | {
"line": 776,
"column": 4
} | [
{
"pp": "case pos\nδ : Type u_4\nX : δ → Type u_6\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\ninst✝ : DecidableEq δ\ni : δ\nis : List δ\nj : δ\nhj : j ∈ i :: is\nhji : j = i\n⊢ Measurable fun v ↦ v.elim hj",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.TProd.eli... | [
"case neg\nδ : Type u_4\nX : δ → Type u_6\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\ninst✝ : DecidableEq δ\ni : δ\nis : List δ\nj : δ\nhj : j ∈ i :: is\nhji : ¬j = i\n⊢ Measurable fun v ↦ v.elim hj"
] | · subst hji
simpa using measurable_fst | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 262,
"column": 33
} | {
"line": 262,
"column": 53
} | {
"line": 263,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝¹⁴ : CommRing R\nS : Type v\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : Algebra R K\nL : Type u_2\ninst✝⁹ : Field L\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsFractionRing S L\nhRK : IsFractionRing R K\ninst✝⁶ : IsDomain S... | [
"case refine_1\nR : Type u\ninst✝¹⁴ : CommRing R\nS : Type v\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : Algebra R K\nL : Type u_2\ninst✝⁹ : Field L\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsFractionRing S L\nhRK : IsFractionRing R K\ninst✝⁶ : IsDomain S\ninst✝⁵ : I... | rw [b_eq_b'] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 42
} | {
"line": 329,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set α\n⊢ ⇑e ⁻¹' ⇑e '' s = s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"MeasurableEquiv.toEquiv",
"Equiv.instE... | [] | rw [← coe_toEquiv, Equiv.preimage_image] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 42
} | {
"line": 329,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set α\n⊢ ⇑e ⁻¹' ⇑e '' s = s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"MeasurableEquiv.toEquiv",
"Equiv.instE... | [] | rw [← coe_toEquiv, Equiv.preimage_image] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 42
} | {
"line": 329,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set α\n⊢ ⇑e ⁻¹' ⇑e '' s = s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"MeasurableEquiv.toEquiv",
"Equiv.instE... | [] | rw [← coe_toEquiv, Equiv.preimage_image] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 446,
"column": 12
} | {
"line": 446,
"column": 52
} | {
"line": 446,
"column": 52
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α✝\ninst✝⁵ : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ✝\nα : Type ?u.23\nβ : Type ?u.25\nγ : Type ?u.102\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : ... | [] | rintro ⟨a | b, c⟩ <;> simp [Set.prod_eq] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 446,
"column": 12
} | {
"line": 446,
"column": 52
} | {
"line": 446,
"column": 52
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α✝\ninst✝⁵ : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ✝\nα : Type ?u.23\nβ : Type ?u.25\nγ : Type ?u.102\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : ... | [] | rintro ⟨a | b, c⟩ <;> simp [Set.prod_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 446,
"column": 12
} | {
"line": 446,
"column": 52
} | {
"line": 446,
"column": 52
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α✝\ninst✝⁵ : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ✝\nα : Type ?u.23\nβ : Type ?u.25\nγ : Type ?u.102\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : ... | [] | rintro ⟨a | b, c⟩ <;> simp [Set.prod_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 10
} | {
"line": 748,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf✝ : α → β\ng✝ : β → α\nf : α → β\ng : β → α\nhf : MeasurableEmbedding f\nhg : MeasurableEmbedding g\nF : Set α →... | [
"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf✝ : α → β\ng✝ : β → α\nf : α → β\ng : β → α\nhf : MeasurableEmbedding f\nhg : MeasurableEmbedding g\nF : Set α → Set α := ⋯\... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 633,
"column": 4
} | {
"line": 633,
"column": 15
} | {
"line": 634,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\n⊢ ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal",
"ppTerm": "?m.180",
"assigned": true,
"usedConstants":... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\np : HeightOneSpectrum R\nhps : p ∈ s\n⊢ J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal"
] | intro p hps | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 479,
"column": 4
} | {
"line": 479,
"column": 38
} | {
"line": 480,
"column": 4
} | [
{
"pp": "case pos\nR : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\ninst✝² : IsDedekindDomain S\nhP0 : P ≠ ⊥\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhe : e ≠ 0\nthis✝¹ : NeZero e := { out := he }\nthis✝ : Algebra (R ⧸ p) (S ⧸ P) := Quotient.alge... | [
"case pos\nR : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\ninst✝² : IsDedekindDomain S\nhP0 : P ≠ ⊥\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhe : e ≠ 0\nthis✝¹ : NeZero e := ⋯\nthis✝ : Algebra (R ⧸ p) (S ⧸ P) := ⋯\nhdim : Module.rank (R ⧸ p) (S ⧸ P ... | apply @Nat.cast_injective Cardinal | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.ENNReal.Real | {
"line": 203,
"column": 62
} | {
"line": 204,
"column": 46
} | {
"line": 206,
"column": 0
} | [
{
"pp": "p : ℝ\n⊢ ENNReal.ofReal p < 1 ↔ p < 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Real",
"Preorder.toLT",
"Nat.instOne",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"AddGroupWithOne.toAddMonoi... | [] | by
exact mod_cast ofReal_lt_natCast one_ne_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Real | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case inl\nq : ℝ≥0∞\nhpq : 0 ≤ q\n⊢ 0 = 0 ∧ q = 0 ∨\n 0 = 0 ∧ q = ∞ ∨\n 0 = 0 ∧ 0 < q.toReal ∨\n 0 = ∞ ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ 0 < q.toReal ∧ ENNReal.toReal 0 ≤ q.toReal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa using q.trichotomy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.ENNReal.Real | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case inl\nq : ℝ≥0∞\nhpq : 0 ≤ q\n⊢ 0 = 0 ∧ q = 0 ∨\n 0 = 0 ∧ q = ∞ ∨\n 0 = 0 ∧ 0 < q.toReal ∨\n 0 = ∞ ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ 0 < q.toReal ∧ ENNReal.toReal 0 ≤ q.toReal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa using q.trichotomy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Real | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case inl\nq : ℝ≥0∞\nhpq : 0 ≤ q\n⊢ 0 = 0 ∧ q = 0 ∨\n 0 = 0 ∧ q = ∞ ∨\n 0 = 0 ∧ 0 < q.toReal ∨\n 0 = ∞ ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ 0 < q.toReal ∧ ENNReal.toReal 0 ≤ q.toReal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa using q.trichotomy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNReal.Defs | {
"line": 972,
"column": 2
} | {
"line": 980,
"column": 19
} | {
"line": 981,
"column": 2
} | [
{
"pp": "Γ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\ng : Γ₀ˣ\nhg1 : g ≠ 1\nu : Γ₀ˣ := if g < 1 then g else g⁻¹\nhu : u = if g < 1 then g else g⁻¹\n⊢ ∃ d, f ↑d < r",
"ppTerm": "?m.54",
"assigned": true,
"usedCo... | [
"Γ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\ng : Γ₀ˣ\nhg1 : g ≠ 1\nu : Γ₀ˣ := if g < 1 then g else g⁻¹\nhu : u = if g < 1 then g else g⁻¹\nhfu : f ↑u < 1\n⊢ ∃ d, f ↑d < r"
] | have hfu : f u < 1 := by
rw [hu]
split_ifs with hu1
· rw [← map_one f]; exact hf hu1
· have hfg0 : f g ≠ 0 :=
fun h0 ↦ (Units.ne_zero g) ((map_eq_zero f).mp h0)
have hg1' : 1 < g := lt_of_le_of_ne (not_lt.mp hu1) hg1.symm
rw [Units.val_inv_eq_inv_val, map_inv₀, inv_lt_one_iff hfg0, ←... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Operations | {
"line": 288,
"column": 50
} | {
"line": 288,
"column": 94
} | {
"line": 290,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\nha : a ≠ ∞\n⊢ ∞ - a = ∞",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.canLift",
"ENNReal.top_sub_coe",
"ENNReal.ofNNReal",
"HSub.hSub",
"Exists",
"NNReal",
"Ne",
"instHSub",
"Exists.casesOn",
"ENNReal... | [] | by lift a to ℝ≥0 using ha; exact top_sub_coe | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 337,
"column": 71
} | {
"line": 338,
"column": 11
} | {
"line": 340,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nhb : b ≠ ∞\n⊢ a + b - b = a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"False",
"ENNReal.instOrderedSub",
"eq_false",
"ENNReal.instAddCommMonoid",
"congrArg",
"HSub.hSub",
"Ne",
"AddLECancell... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 429,
"column": 80
} | {
"line": 430,
"column": 58
} | {
"line": 432,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nhb : b ≠ ∞\n⊢ (a - b).toNNReal = a.toNNReal - b.toNNReal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"ENNReal.canLift",
"False",
"ENNReal.ofNNReal",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSu... | [] | by
lift b to ℝ≥0 using hb; induction a <;> simp [← coe_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 63
} | {
"line": 620,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ∞\n⊢ (iSup f).toReal = ⨆ i, (f i).toReal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"NNReal.coe_iSup",
"congrArg",
"iSup",
"Real.instSupSet",
"ENNReal.toNNReal_iSup",
"NNReal",... | [] | simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Operations | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 63
} | {
"line": 620,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ∞\n⊢ (iSup f).toReal = ⨆ i, (f i).toReal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"NNReal.coe_iSup",
"congrArg",
"iSup",
"Real.instSupSet",
"ENNReal.toNNReal_iSup",
"NNReal",... | [] | simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 63
} | {
"line": 620,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ∞\n⊢ (iSup f).toReal = ⨆ i, (f i).toReal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"NNReal.coe_iSup",
"congrArg",
"iSup",
"Real.instSupSet",
"ENNReal.toNNReal_iSup",
"NNReal",... | [] | simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Pointwise | {
"line": 83,
"column": 8
} | {
"line": 83,
"column": 74
} | {
"line": 84,
"column": 8
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Module α ℝ\ninst✝ : IsOrderedModule α ℝ\na : α\nha : a ≤ 0\ns : Set ℝ\nhs : s.Nonempty\nha' : a < 0\nh : ¬BddAbove s\n⊢ sInf (a • s) = a • sSup s",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\nα : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Module α ℝ\ninst✝ : IsOrderedModule α ℝ\na : α\nha : a ≤ 0\ns : Set ℝ\nhs : s.Nonempty\nha' : a < 0\nh : ¬BddAbove s\n⊢ 0 = a • sSup s"
] | Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_neg ha').1 h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 94,
"column": 19
} | {
"line": 94,
"column": 33
} | {
"line": 95,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (∞ ^ (n + 1))⁻¹ = ∞⁻¹ ^ (n + 1)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",
"one_ne_zero._simp_1",
"in... | [] | simp [top_pow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Inv | {
"line": 94,
"column": 19
} | {
"line": 94,
"column": 33
} | {
"line": 95,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (∞ ^ (n + 1))⁻¹ = ∞⁻¹ ^ (n + 1)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",
"one_ne_zero._simp_1",
"in... | [] | simp [top_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 94,
"column": 19
} | {
"line": 94,
"column": 33
} | {
"line": 95,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (∞ ^ (n + 1))⁻¹ = ∞⁻¹ ^ (n + 1)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",
"one_ne_zero._simp_1",
"in... | [] | simp [top_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 20
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case inl\nn : ℕ\n⊢ (↑0 ^ (n + 1))⁻¹ = (↑0)⁻¹ ^ (n + 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ENNReal.ofNNReal",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",... | [] | simp [top_pow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Inv | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 20
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case inl\nn : ℕ\n⊢ (↑0 ^ (n + 1))⁻¹ = (↑0)⁻¹ ^ (n + 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ENNReal.ofNNReal",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",... | [] | simp [top_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 20
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case inl\nn : ℕ\n⊢ (↑0 ^ (n + 1))⁻¹ = (↑0)⁻¹ ^ (n + 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ENNReal.ofNNReal",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.add_eq_zero_iff._simp_1",... | [] | simp [top_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 54,
"column": 47
} | {
"line": 56,
"column": 21
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\na₁ a₂ : α\nhf : HasProd f a₁ L\nhg : HasProd g a₂ L\n⊢ HasProd (fun b ↦ f b / g b) (a₁ / a₂) L",
"ppTerm": "?m.24",
"assigned": true,
"usedConstan... | [] | by
simp only [div_eq_mul_inv]
exact hf.mul hg.inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 21
} | {
"line": 217,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ Tendsto (fun x ↦ (∏ b ∈ x.2, f b) / ∏ b ∈ x.1, f b) atTop (𝓝 1) ↔\n ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e",
"ppTerm": "?m.37",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ (∀ s ∈ 𝓝 1, ∃ a, ∀ (b : Finset β × Finset β), a ≤ b → (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s) ↔\n ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e"
] | rw [tendsto_atTop'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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