module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.LocalRing.Length | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 35
} | {
"line": 72,
"column": 36
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : IsLocalRing A\ninst✝⁶ : IsLocalRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsLocalHom (algebraMap A B)\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\ninst✝¹ : Module B M\ninst✝ : IsScalarTower A B M\nh : length B M ... | [
"A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : IsLocalRing A\ninst✝⁶ : IsLocalRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsLocalHom (algebraMap A B)\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\ninst✝¹ : Module B M\ninst✝ : IsScalarTower A B M\nh : length B M = ⊤\nthis : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 116,
"column": 19
} | {
"line": 116,
"column": 48
} | {
"line": 116,
"column": 49
} | [
{
"pp": "case pos.succ\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\ns : Comp... | [
"case pos.succ\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\ns : CompositionSerie... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 53
} | {
"line": 125,
"column": 54
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : length A M = ⊤\nthis : length B (B ⊗[A] M) ... | [
"A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : length A M = ⊤\nthis : length B (B ⊗[A] M) = ⊤\n⊢ ¬Idea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 120,
"column": 38
} | {
"line": 120,
"column": 49
} | {
"line": 120,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nh : ¬map f p ≤ P\n⊢ ¬map f p ≤ P ^ (0 + 1)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
... | [
"R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nh : ¬map f p ≤ P\n⊢ ¬map f p ≤ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 37
} | [
{
"pp": "R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : S... | [
"R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : Small.{w, u} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 97
} | {
"line": 100,
"column": 98
} | [
{
"pp": "R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : S... | [
"R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : Small.{w, u} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 18
} | [
{
"pp": "case mpr.refine_2\nR : Type v\ninst✝³ : CommRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Small.{w, u} A\nH :\n ∀ ⦃B : Type u⦄ [inst : CommRing B] [Small.{w, u} B] [inst_2 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nh : Nont... | [
"case mpr.refine_2\nR : Type v\ninst✝³ : CommRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Small.{w, u} A\nH :\n ∀ ⦃B : Type u⦄ [inst : CommRing B] [Small.{w, u} B] [inst_2 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nh : Nontrivial Ω[A⁄R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RamificationInertia.Inertia | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 44
} | {
"line": 195,
"column": 45
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nq : Ideal S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : q.IsPrime\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : IsDedekindDomain S\ninst✝² : Module.Free ℤ R\ninst✝¹ : Module.Free ℤ S\ninst✝ : q.LiesOver ⊥\n⊢ absNor... | [
"case pos\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nq : Ideal S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : q.IsPrime\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : IsDedekindDomain S\ninst✝² : Module.Free ℤ R\ninst✝¹ : Module.Free ℤ S\ninst✝ : q.LiesOver ⊥\n⊢ ¬⊥.inertiaDeg R = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 372,
"column": 29
} | {
"line": 372,
"column": 40
} | {
"line": 372,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI : Ideal R\nhI : IsUnit I... | [
"R : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI : Ideal R\nhI : IsUnit I\nh : I ≠ ⊥\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RamificationInertia.Inertia | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Module.Free ℤ R\ninst✝² : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver (span {p})\n⊢ p.natAbs ^ P.inertiaDeg ℤ = absNorm P",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
... | [
"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Module.Free ℤ R\ninst✝² : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver (span {p})\n⊢ p.natAbs ^ P.inertiaDeg ℤ = absNorm P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 17
} | {
"line": 179,
"column": 18
} | [
{
"pp": "case neg\nR : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nI : Ideal B\ninst✝ : FormallyUnramified R A\nhI : ⨅ i, I ^ i = ⊥\ng₁ g₂ : A →ₐ[R] B\nH : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk ... | [
"case neg\nR : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nI : Ideal B\ninst✝ : FormallyUnramified R A\nhI : ⨅ i, I ^ i = ⊥\ng₁ g₂ : A →ₐ[R] B\nH : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 77
} | {
"line": 410,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsDedekindDomain S\ninst✝ : IsDedekindDomain T\np : Ideal R\nP : Ideal S\nQ : Ideal T\nhpm : ... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsDedekindDomain S\ninst✝ : IsDedekindDomain T\np : Ideal R\nP : Ideal S\nQ : Ideal T\nhpm : P.IsPrime\nh... | IsDedekindDomain.ramificationIdx'_eq_normalizedFactors_count hfg hqm hq0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 43
} | {
"line": 238,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\n... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\ne : I ^ 2 = ... | refine FormallyUnramified.ext I ⟨2, e⟩ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Unramified.Finite | {
"line": 76,
"column": 8
} | {
"line": 77,
"column": 57
} | {
"line": 77,
"column": 58
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nthis✝ : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S\nt e : S ⊗[R] S\nht₁ : ∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * (1 - e) = 0\nht₂ : e ∈ Ideal... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nthis✝ : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S\nt e : S ⊗[R] S\nht₁ : ∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * (1 - e) = 0\nht₂ : e ∈ Ideal.span (Set.r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Finite | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 34
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\ni : I\n⊢ ∀ (a : I) (b₁ b₂ : R), (b₁ + b₂) • b a ⊗ₜ[R] b i = b₁ • b a ⊗ₜ[R] b i + b... | [] | · intros; simp only [add_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Unramified.Finite | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 67
} | {
"line": 114,
"column": 68
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\nh₁ :\n ∀ (k : I),\n ((f.sum fun i y ↦ (a i) k • b.repr y).sum fun j z ↦ z • b j ⊗ₜ[R] b k)... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\nh₁ :\n ∀ (k : I),\n ((f.sum fun i y ↦ (a i) k • b.repr y).sum fun j z ↦ z • b j ⊗ₜ[R] b k) =\n f.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 408,
"column": 30
} | {
"line": 408,
"column": 45
} | {
"line": 408,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\ninst✝ : QuasiFiniteAt R Q\nf : Localization.AtPrime Q →ₐ[R] Localization.AtPrime P := IsLocalization.liftAlgHom ⋯\nx s : S\nhs : s ∈ P.primeComp... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\ninst✝ : QuasiFiniteAt R Q\nf : Localization.AtPrime Q →ₐ[R] Localization.AtPrime P := IsLocalization.liftAlgHom ⋯\nx s : S\nhs : s ∈ P.primeCompl\n⊢ IsUnit ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 445,
"column": 36
} | {
"line": 445,
"column": 47
} | {
"line": 445,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : IsArtinianRing R\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R p\n⊢ ∀ {x₁ x₂ : Localization.AtPrime p}, LinearMap.id x₁ = LinearMap.id x₂ → ∃ c, c • x₁ = c • x₂",... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : IsArtinianRing R\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R p\n⊢ ∃ a, a ∉ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Basic | {
"line": 219,
"column": 8
} | {
"line": 219,
"column": 23
} | {
"line": 220,
"column": 8
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst... | [
"R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst✝ : Formally... | intro r hr s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 535,
"column": 6
} | {
"line": 535,
"column": 17
} | {
"line": 535,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isP... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 536,
"column": 6
} | {
"line": 536,
"column": 83
} | {
"line": 536,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isP... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Basic | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 70
} | {
"line": 439,
"column": 70
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\nA : Type u_6\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nh : Function.Surjective ⇑(algebraMap R A)\n⊢ FormallySmooth R A ↔ IsIdempotentElem (RingHom.ker (algebraMap R A))",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingH... | [
"R : Type u_4\ninst✝² : CommRing R\nA : Type u_6\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nh : Function.Surjective ⇑(algebraMap R A)\n⊢ (∃ g, (ofId R A).kerSquareLift.comp g = AlgHom.id R A) ↔ IsIdempotentElem (RingHom.ker (algebraMap R A))"
] | Algebra.FormallySmooth.iff_split_surjection (Algebra.ofId R A) h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | {
"line": 63,
"column": 47
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R ... | [
"R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R S\nx : P\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 52
} | {
"line": 69,
"column": 4
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R ... | [
"R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R S\nx : R\ny ... | simp only [Algebra.smul_def, map_mul, AlgHom.commutes,
RingHom.id_apply, Submodule.coe_smul_of_tower] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 50
} | {
"line": 76,
"column": 51
} | [
{
"pp": "case hr\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = Alg... | [
"case hr\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R S\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 50
} | {
"line": 77,
"column": 51
} | [
{
"pp": "case hs\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = Alg... | [
"case hs\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R S\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 44
} | {
"line": 100,
"column": 45
} | [
{
"pp": "case e_a\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.toAlgHom R P S... | [
"case e_a\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.toAlgHom R P S).comp g = A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 17
} | {
"line": 143,
"column": 6
} | [
{
"pp": "case h2.refine_2\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Rin... | [
"case h2.refine_2\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Ring\nhx' : x ∈... | rw [map_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 114,
"column": 2
} | {
"line": 143,
"column": 9
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case h2\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ ∀ (x : (Q.comp P).toExten... | [] | · intro x hx
obtain ⟨⟨x : (Q.comp P).Ring, hx'⟩, rfl⟩ := Extension.Cotangent.mk_surjective x
replace hx : (Q.ofComp P).toAlgHom x ∈ Q.ker ^ 2 := by
simpa only [map_mk, val_mk, val_zero, Ideal.toCotangent_eq_zero] using! congr(($hx).val)
rw [pow_two, ← map_ofComp_ker (P := P), ← Ideal.map_mul, Ideal.me... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 170,
"column": 18
} | {
"line": 172,
"column": 61
} | {
"line": 173,
"column": 2
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\nl : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))\nhl : l ∘ₗ kerToTensor R P S = LinearMap.id\nσ : S → P\nhσ : ∀ (x : S), (algebraMap P S) (σ x) = x... | [] | by
simp only [sectionOfRetractionKerToTensorAux_prop l hl (σ (a + b)) (σ a + σ b) (by simp [hσ]),
map_add, tmul_add, Submodule.coe_add, add_sub_add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 182,
"column": 2
} | {
"line": 194,
"column": 98
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | [] | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 182,
"column": 2
} | {
"line": 194,
"column": 98
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | [] | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 182,
"column": 2
} | {
"line": 194,
"column": 98
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | [] | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Field | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 53
} | {
"line": 49,
"column": 54
} | [
{
"pp": "K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra K B\nI : Ideal B\nhI : I ^ 2 = ⊥\nf₁ f₂ : L →ₐ[K] B\ne : (Ideal.Quotient.mkₐ K I).comp f₁ = (Ideal.Quotient.mkₐ K I).comp f₂\nx : L\... | [
"K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra K B\nI : Ideal B\nhI : I ^ 2 = ⊥\nf₁ f₂ : L →ₐ[K] B\ne : (Ideal.Quotient.mkₐ K I).comp f₁ = (Ideal.Quotient.mkₐ K I).comp f₂\nx : L\n⊢ f₁ x - f₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Field | {
"line": 64,
"column": 2
} | {
"line": 109,
"column": 40
} | {
"line": 111,
"column": 0
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\n⊢ Function.Bijective ⇑(algebraMap K A)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants":... | [] | have := finite_of_free (R := K) (S := A)
have : IsArtinianRing A := isArtinian_of_tower K inferInstance
have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by
rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]
· exact IsArtinianRing.isNilpotent_jacobson_bot
· exact bot_ne_top
let e : K ≃ₐ[K] A ⧸ IsLocalR... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Field | {
"line": 64,
"column": 2
} | {
"line": 109,
"column": 40
} | {
"line": 111,
"column": 0
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\n⊢ Function.Bijective ⇑(algebraMap K A)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants":... | [] | have := finite_of_free (R := K) (S := A)
have : IsArtinianRing A := isArtinian_of_tower K inferInstance
have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by
rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]
· exact IsArtinianRing.isNilpotent_jacobson_bot
· exact bot_ne_top
let e : K ≃ₐ[K] A ⧸ IsLocalR... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 399,
"column": 6
} | {
"line": 400,
"column": 45
} | {
"line": 400,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : ↥P.ker\nhx : P.toInfinitesimal.toRingHom ↑x ∈ ⊥\n⊢ ↑x ∈ P.ker ^ 2",
"ppTerm": "?m.211",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : ↥P.ker\nhx : P.toInfinitesimal.toRingHom ↑x ∈ ⊥\n⊢ ↑x ∈ P.ker ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 418,
"column": 4
} | {
"line": 419,
"column": 89
} | {
"line": 419,
"column": 90
} | [
{
"pp": "case right\nR : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.Cotangent\nhx : (Cotangent.map P.toInfinitesimal) x ∈ P.infinitesimal.cotangentComplex.ker\n⊢ x ∈ P.cotangentComplex.ker",
"ppTerm": "?right",
"assigned": true,
"u... | [
"case right\nR : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.Cotangent\nhx : (Cotangent.map P.toInfinitesimal) x ∈ P.infinitesimal.cotangentComplex.ker\n⊢ P.cotangentComplex x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 57,
"column": 2
} | {
"line": 58,
"column": 61
} | {
"line": 59,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : Algebra.FormallyEtale S T\n⊢ ⇑(lift (↑S ((Algebra.linearMap T (Module.End T (Ω[S⁄R] →ₗ[S] Ω[T⁄... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : Algebra.FormallyEtale S T\n⊢ ⇑(lift (↑S ((Algebra.linearMap T (Module.End T (Ω[S⁄R] →ₗ[S] Ω[T⁄R]))).flip (... | change _ = ((tensorKaehlerEquivOfFormallyEtale
R S T).toLinearMap.restrictScalars S : T ⊗[S] Ω[S⁄R] → _) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 84
} | {
"line": 108,
"column": 85
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing R\ninst✝³ : IsLocalRing S\ninst✝² : IsLocalHom (algebraMap R S)\ninst✝¹ : EssFiniteType R S\ninst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)\nH : Ideal.map (algebraMap R S) (ma... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing R\ninst✝³ : IsLocalRing S\ninst✝² : IsLocalHom (algebraMap R S)\ninst✝¹ : EssFiniteType R S\ninst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)\nH : Ideal.map (algebraMap R S) (maximalIdeal R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 79
} | {
"line": 174,
"column": 80
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝¹ : Algebra.IsIntegral R S\ninst✝ : FaithfulSMul R S\nx : R\ns : ↥p.primeCompl\nhx : (localRingHom p q (algebraMa... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝¹ : Algebra.IsIntegral R S\ninst✝ : FaithfulSMul R S\nx : R\ns : ↥p.primeCompl\nhx : (localRingHom p q (algebraMap R S) ⋯) (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 485,
"column": 21
} | {
"line": 485,
"column": 32
} | {
"line": 485,
"column": 33
} | [
{
"pp": "case add\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx y : T ⊗[S] P.toExtension... | [
"case add\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx y : T ⊗[S] P.toExtension.H1Cotangent... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 59
} | {
"line": 498,
"column": 60
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nT : Type u₃\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\ninst✝ : Flat S T\nx : (Q.comp P).to... | [
"R : Type u₁\nS : Type u₂\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nT : Type u₃\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\ninst✝ : Flat S T\nx : (Q.comp P).toExtension.Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RamificationInertia.Basic | {
"line": 62,
"column": 58
} | {
"line": 62,
"column": 69
} | {
"line": 62,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsPrime\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.QuasiFinite R S\ninst✝ : Fintype ↑(p.primesOver S)\nthis✝¹ : Fintype (PrimeSpectrum (p.Fiber S)) := Fintype.ofFinite (PrimeSpectrum (p.Fiber S))\nq : PrimeSpectr... | [
"R : Type u_1\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsPrime\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.QuasiFinite R S\ninst✝ : Fintype ↑(p.primesOver S)\nthis✝¹ : Fintype (PrimeSpectrum (p.Fiber S)) := Fintype.ofFinite (PrimeSpectrum (p.Fiber S))\nq : PrimeSpectrum (p.Fiber ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 286,
"column": 7
} | {
"line": 286,
"column": 18
} | {
"line": 286,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nP : Extension R S := (Generators.self R S).toExtension\nM... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nP : Extension R S := (Generators.self R S).toExtension\nM' : Submonoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 299,
"column": 57
} | {
"line": 299,
"column": 89
} | {
"line": 299,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\np : Ideal R\ninst✝⁷ : p.IsPrime\nq : Ideal S\ninst✝⁶ : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : q.LiesOver p\ninst✝² : Algebra.IsUnramifiedAt R q\nin... | [] | by simpa using! Submodule.fg_bot | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 359,
"column": 7
} | {
"line": 359,
"column": 18
} | {
"line": 359,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nx : H1Cotangent R S\nP : Extension R S := (Generators.sel... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nx : H1Cotangent R S\nP : Extension R S := (Generators.self R S).toExt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 92,
"column": 57
} | {
"line": 92,
"column": 68
} | {
"line": 92,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nh_fin : Fintype { x // x ∣ I }\nv w : { x // x.asIdeal ∣ I }\nhvw : (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) v = (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) w\n⊢ ((fun a ↦ ↑a) v).asIdeal = ((fun a ↦ ↑a) w).asIdeal",
"ppTerm": "?m.63",
"... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nh_fin : Fintype { x // x ∣ I }\nv w : { x // x.asIdeal ∣ I }\nhvw : (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) v = (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) w\n⊢ (↑v).asIdeal = (↑w).asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.biUnion S)) = ∑ t, (-1) ^ (#↑t + 1) * ↑(#((↑t).inf' ⋯ S))",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Int.instAddCommMonoid",
"HMul.hMul",
"Finset.univ",
"Finset... | [
"ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.biUnion S)) = ∑ t ∈ {x ∈ s.powerset | x.Nonempty}.attach, (-1) ^ (#↑t + 1) * ↑(#((↑t).inf' ⋯ S))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 17
} | {
"line": 58,
"column": 18
} | [
{
"pp": "case mp\nA : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A M\nx : M\na : ∃ i, x ∈ torsionBySet A M ↑(I ^ i)\n⊢ ∃ n, ∀ a ∈ I ^ n, a • x = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"case mp\nA : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A M\nx : M\na : ∃ i, x ∈ torsionBySet A M ↑(I ^ i)\n⊢ ∃ n, ∀ a ∈ I ^ n, a • x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.inf fun i ↦ (S i)ᶜ)) = ∑ t ∈ s.powerset, (-1) ^ #t * ↑(#(t.inf S))",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.inf fun i ↦ (S i)ᶜ)) = ∑ t ∈ s.powerset, (-1) ^ #t * ↑(#(t.inf S))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 73,
"column": 15
} | {
"line": 73,
"column": 26
} | {
"line": 73,
"column": 27
} | [
{
"pp": "s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ s\nhy : ↑(f n) ≤ y * ↑n\n⊢ ↑(↑(f n) / ↑n) ≤ y",... | [
"s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ s\nhy : ↑(f n) ≤ y * ↑n\n⊢ ↑(f n) / ↑n ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 78,
"column": 7
} | {
"line": 78,
"column": 53
} | {
"line": 78,
"column": 54
} | [
{
"pp": "A : Type u_1\nM : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : CommRing A\nI : Ideal A\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module A M\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nφ : M₁ →ₗ[A] M₂\nc : ↥(primaryComponent M₁ I)\n⊢ (φ.domRestrict (pr... | [
"A : Type u_1\nM : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : CommRing A\nI : Ideal A\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module A M\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nφ : M₁ →ₗ[A] M₂\nc : ↥(primaryComponent M₁ I)\n⊢ φ ↑c ∈ primaryComponent M₂ I"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 147,
"column": 32
} | {
"line": 147,
"column": 43
} | {
"line": 147,
"column": 44
} | [
{
"pp": "A : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx : M\na : A\nha : a ∈ A⁰\nhmem : x ∈ torsionBySet A M ↑(span {a})\n⊢ span {a} ≠ ⊥",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"S... | [
"A : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx : M\na : A\nha : a ∈ A⁰\nhmem : x ∈ torsionBySet A M ↑(span {a})\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 15
} | {
"line": 93,
"column": 16
} | [
{
"pp": "s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nhf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑... | [
"s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nhf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 29
} | {
"line": 555,
"column": 30
} | [
{
"pp": "r : ℝ\n⊢ r.toNNReal = 0 ↔ r ≤ 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\n⊢ r.toNNReal = 0 ↔ r ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 13
} | {
"line": 582,
"column": 14
} | [
{
"pp": "r : ℝ\n⊢ r.toNNReal ≤ 1 ↔ r ≤ 1",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\n⊢ r.toNNReal ≤ 1 ↔ r ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 27
} | {
"line": 586,
"column": 28
} | [
{
"pp": "r : ℝ\n⊢ 1 < r.toNNReal ↔ 1 < r",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\n⊢ 1 < r.toNNReal ↔ 1 < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 13
} | {
"line": 590,
"column": 14
} | [
{
"pp": "r : ℝ\nn : ℕ\n⊢ r.toNNReal ≤ ↑n ↔ r ≤ ↑n",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\nn : ℕ\n⊢ r.toNNReal ≤ ↑n ↔ r ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 27
} | {
"line": 594,
"column": 28
} | [
{
"pp": "r : ℝ\nn : ℕ\n⊢ ↑n < r.toNNReal ↔ ↑n < r",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\nn : ℕ\n⊢ ↑n < r.toNNReal ↔ ↑n < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 13
} | {
"line": 633,
"column": 14
} | [
{
"pp": "r : ℝ\n⊢ 1 ≤ r.toNNReal ↔ 1 ≤ r",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\n⊢ 1 ≤ r.toNNReal ↔ 1 ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 40
} | {
"line": 640,
"column": 41
} | [
{
"pp": "n : ℕ\nr : ℝ\n⊢ ↑n ≤ r.toNNReal ↔ ↑n ≤ r ∨ n = 0",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nr : ℝ\n⊢ ↑n ≤ r.toNNReal ↔ ↑n ≤ r ∨ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 31
} | {
"line": 644,
"column": 32
} | [
{
"pp": "n : ℕ\nr : ℝ\n⊢ r.toNNReal < ↑n ↔ r < ↑n ∧ n ≠ 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Real.instLT",
"id",
"AddMonoidWithOne.toNatCast... | [
"n : ℕ\nr : ℝ\n⊢ r.toNNReal < ↑n ↔ r < ↑n ∧ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 49
} | {
"line": 725,
"column": 4
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : b < 1\n⊢ ∃ n, b ^ n < a",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0\nha : 0 < a\nhb : b < 1\n⊢ ∃ n, b ^ n < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 818,
"column": 6
} | {
"line": 818,
"column": 16
} | {
"line": 818,
"column": 17
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\n⊢ x⁻¹ < 1 ↔ 1 < x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"Preorder.toLT",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"Monoid.toMulOneClass",
"congrArg",
... | [
"x : ℝ≥0\nhx : x ≠ 0\n⊢ 1 / x < 1 ↔ 1 < x"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 935,
"column": 42
} | {
"line": 935,
"column": 53
} | {
"line": 935,
"column": 54
} | [
{
"pp": "r : ℝ\n⊢ 0 ≤ r ∨ 0 ≤ -r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"id",
"Real.instAddGroup",
"SubtractionMonoid.toSubNegZeroMon... | [
"r : ℝ\n⊢ 0 ≤ r ∨ r ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 272,
"column": 14
} | {
"line": 272,
"column": 42
} | {
"line": 272,
"column": 43
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x_1 ↦ x_1 x) ∘ Subtype... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove ((fun x_1 ↦ x_1 x) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 272,
"column": 14
} | {
"line": 272,
"column": 42
} | {
"line": 272,
"column": 43
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x ↦ x y) ∘ Subtype.val... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove ((fun x ↦ x y) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 13
} | {
"line": 374,
"column": 14
} | [
{
"pp": "x✝ : ℝ\n⊢ x✝ ∈ {x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x ↦ ↑x) → a ≤ x} ↔ x✝ ∈ ∅",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congr... | [
"x✝ : ℝ\n⊢ ∃ x, x✝ < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 13
} | {
"line": 380,
"column": 14
} | [
{
"pp": "x✝ : ℝ\n⊢ x✝ ∈ {x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x ↦ ↑x) → x ≤ a} ↔ x✝ ∈ ∅",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congr... | [
"x✝ : ℝ\n⊢ ∃ x, ↑x < x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 413,
"column": 8
} | {
"line": 413,
"column": 71
} | {
"line": 413,
"column": 72
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝ : E\np q : GroupSeminorm E\nx : E\n⊢ p 1 + q (x / 1) ≤ q x",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.par... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝ : E\np q : GroupSeminorm E\nx : E\n⊢ q x ≤ q x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 467,
"column": 4
} | {
"line": 467,
"column": 84
} | {
"line": 468,
"column": 6
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : AddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : AddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 65
} | {
"line": 337,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\nh_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J\na₁ : R := choose ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\nh_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J\na₁ : R := choose ⋯\nJ₁ : Idea... | have h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁ :=
(choose_spec (choose_spec (exists_eq_spanSingleton_mul I))).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Basic | {
"line": 567,
"column": 2
} | {
"line": 567,
"column": 13
} | {
"line": 567,
"column": 14
} | [
{
"pp": "b a : ℝ≥0\nh : ↑a ≤ ↑b\n⊢ (↑a).toReal ≤ ↑b",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.coe_le_coe._simp_1",
"id",
"NNR... | [
"b a : ℝ≥0\nh : ↑a ≤ ↑b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 547,
"column": 14
} | {
"line": 547,
"column": 42
} | {
"line": 547,
"column": 43
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x_1 ↦ x_1 x) ∘ Subtype.val))",
"ppTerm": "?... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove ((fun x_1 ↦ x_1 x) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 547,
"column": 14
} | {
"line": 547,
"column": 42
} | {
"line": 547,
"column": 43
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x ↦ x y) ∘ Subtype.val))",
"ppTerm": "?m.15... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove ((fun x ↦ x y) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Basic | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 13
} | {
"line": 672,
"column": 14
} | [
{
"pp": "b a : ℝ≥0\nh : ↑a = ∞ → ↑b = ∞\nh_nnreal : ↑a ≠ ∞ → ↑b ≠ ∞ → (↑a).toNNReal ≤ (↑b).toNNReal\nhlt : ↑b < ↑a\n⊢ ↑a ≤ ↑b",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
... | [
"b a : ℝ≥0\nh : ↑a = ∞ → ↑b = ∞\nh_nnreal : ↑a ≠ ∞ → ↑b ≠ ∞ → (↑a).toNNReal ≤ (↑b).toNNReal\nhlt : ↑b < ↑a\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 654,
"column": 4
} | {
"line": 654,
"column": 84
} | {
"line": 655,
"column": 6
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : Group E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : GroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"R : Type u_1\nE : Type u_3\ninst✝³ : Group E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : GroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 352,
"column": 56
} | {
"line": 352,
"column": 82
} | {
"line": 353,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑J₁ / spanSingleton R... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑J₁ * spanSingleton R⁰ ((algebraM... | div_eq_div_iff h_a₁' h_a', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Real | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 46
} | {
"line": 265,
"column": 47
} | [
{
"pp": "a : ℝ\nb : ℝ≥0\n⊢ ENNReal.ofReal a ≤ ↑b ↔ a ≤ (↑b).toReal",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"a : ℝ\nb : ℝ≥0\n⊢ a.toNNReal ≤ b ↔ a ≤ ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 46
} | {
"line": 270,
"column": 47
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\nb : ℝ≥0\n⊢ ENNReal.ofReal a < ↑b ↔ a < (↑b).toReal",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Re... | [
"a : ℝ\nha : 0 ≤ a\nb : ℝ≥0\n⊢ a.toNNReal < b ↔ a < ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 46
} | {
"line": 281,
"column": 47
} | [
{
"pp": "b : ℝ\nhb : 0 ≤ b\na : ℝ≥0\n⊢ ↑a ≤ ENNReal.ofReal b ↔ (↑a).toReal ≤ b",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Preo... | [
"b : ℝ\nhb : 0 ≤ b\na : ℝ≥0\n⊢ a ≤ b.toNNReal ↔ ↑a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 46
} | {
"line": 291,
"column": 47
} | [
{
"pp": "b : ℝ\na : ℝ≥0\n⊢ ↑a < ENNReal.ofReal b ↔ (↑a).toReal < b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
... | [
"b : ℝ\na : ℝ≥0\n⊢ a < b.toNNReal ↔ ↑a < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 84
} | {
"line": 709,
"column": 6
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : NonarchAddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : NonarchAddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 40
} | {
"line": 358,
"column": 41
} | [
{
"pp": "p : ℝ≥0∞\n⊢ p = 0 ∨ p = ∞ ∨ 0 < p.toReal",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"id",
"_private.Mathlib.Data.ENNReal.Real.0.ENNReal.trichotomy._simp_1_1",
"ENNRea... | [
"p : ℝ≥0∞\n⊢ ¬p = 0 → ¬p = ∞ → 0 < p.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 15
} | {
"line": 367,
"column": 16
} | [
{
"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.... | [
"case inl\nq : ℝ≥0∞\nhpq : 0 ≤ q\n⊢ q = 0 ∨ q = ∞ ∨ 0 < q.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 15
} | {
"line": 369,
"column": 16
} | [
{
"pp": "case inr.inl\np : ℝ≥0∞\nhp : 0 < p\nhpq : p ≤ ∞\n⊢ p = 0 ∧ ∞ = 0 ∨\n p = 0 ∧ ∞ = ∞ ∨\n p = 0 ∧ 0 < ∞.toReal ∨ p = ∞ ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ 0 < ∞.toReal ∧ p.toReal ≤ ∞.toReal",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case inr.inl\np : ℝ≥0∞\nhp : 0 < p\nhpq : p ≤ ∞\n⊢ p = 0 ∨ p = ∞ ∨ 0 < p.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 15
} | {
"line": 376,
"column": 16
} | [
{
"pp": "p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\n⊢ p = ∞ ∨ 0 < p.toReal ∧ 1 ≤ p.toReal",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\n⊢ p = ∞ ∨ 0 < p.toReal ∧ 1 ≤ p.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 58
} | {
"line": 209,
"column": 59
} | [
{
"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 ∈ primaryCom... | [
"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 ∈ primaryComponent M₂ P.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 14
} | [
{
"pp": "a b : ℝ≥0∞\nha₀ : a ≠ 0\nha : a ≠ ∞\n⊢ a * b = a ↔ b = 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\nha₀ : a ≠ 0\nha : a ≠ ∞\n⊢ a * b = a ↔ b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 14
} | [
{
"pp": "a b : ℝ≥0∞\nhb₀ : b ≠ 0\nhb : b ≠ ∞\n⊢ a * b = b ↔ a = 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\nhb₀ : b ≠ 0\nhb : b ≠ ∞\n⊢ a * b = b ↔ a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 36
} | {
"line": 103,
"column": 37
} | [
{
"pp": "a : ℝ≥0∞\n⊢ a ≠ 0 → ∀ (n : ℕ), a ^ n ≠ 0",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℝ≥0∞\n⊢ a ≠ 0 → ∀ (n : ℕ), a ^ n ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 39
} | {
"line": 165,
"column": 40
} | [
{
"pp": "a b c : ℝ≥0∞\nhle : a ≤ b + c\nhb : b = ∞ → a = ∞\nhc : c = ∞ → a = ∞\n⊢ b + c = ∞ → a = ∞",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"id",
"instHAdd",
"And",
"HAdd.hAdd",
"ENNReal.add_eq_top._simp_1",
... | [
"a b c : ℝ≥0∞\nhle : a ≤ b + c\nhb : b = ∞ → a = ∞\nhc : c = ∞ → a = ∞\n⊢ (b = ∞ → a = ∞) ∧ (c = ∞ → a = ∞)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 176,
"column": 53
} | {
"line": 176,
"column": 89
} | {
"line": 176,
"column": 90
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\n⊢ a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 495,
"column": 76
} | {
"line": 495,
"column": 90
} | {
"line": 497,
"column": 0
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoc x y = uIoc ↑x ↑y",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Set.uIoc",
"SemilatticeInf.toPartialOrder",
... | [] | by simp [uIoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 511,
"column": 4
} | {
"line": 511,
"column": 38
} | {
"line": 513,
"column": 0
} | [
{
"pp": "case inr\nι : Sort u_1\nh✝ : Nonempty ι\nf : ι → ℝ≥0\n⊢ (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i ↦ ↑(f i)) i).toNNReal",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"iInf",
"congrArg",
"id",
"_private.Mathlib.Data.ENN... | [] | simp_rw [← coe_iInf, toNNReal_coe] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.ENNReal.Operations | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 13
} | {
"line": 597,
"column": 14
} | [
{
"pp": "x y z : ℝ≥0∞\nh : ∀ y' > y, ∀ z' > z, x ≤ y' + z'\n⊢ x ≤ y + z",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℝ≥0∞\nh : ∀ y' > y, ∀ z' > z, x ≤ y' + z'\n⊢ x ≤ y + z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Field | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 24
} | {
"line": 30,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulAction 𝕜 G\ninst✝ : PosSMulMono 𝕜 G\n_a : 𝕜\nha : 0 < _a\nb₁ b₂ : G\nh : _a • b₁ ≤ _a • b₂\n⊢ b₁ ≤ b₂",
"ppTerm": "?m.23",
... | [
"𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulAction 𝕜 G\ninst✝ : PosSMulMono 𝕜 G\n_a : 𝕜\nha : 0 < _a\nb₁ b₂ : G\nh : _a • b₁ ≤ _a • b₂\n⊢ b₁ ≤ b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Field | {
"line": 35,
"column": 32
} | {
"line": 35,
"column": 52
} | {
"line": 35,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulActionWithZero 𝕜 G\ninst✝ : PosSMulStrictMono 𝕜 G\na : 𝕜\nha : 0 < a\nb₁ b₂ : G\nh : a • b₁ < a • b₂\n⊢ b₁ < b₂",
"ppTerm": ... | [
"𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulActionWithZero 𝕜 G\ninst✝ : PosSMulStrictMono 𝕜 G\na : 𝕜\nha : 0 < a\nb₁ b₂ : G\nh : a • b₁ < a • b₂\n⊢ b₁ < b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 712,
"column": 2
} | {
"line": 712,
"column": 13
} | {
"line": 712,
"column": 14
} | [
{
"pp": "x y z : ℝ≥0∞\nhy : y ≠ 0\nhz : z ≠ 0\nh : ∀ y' < y, ∀ z' < z, y' + z' ≤ x\n⊢ y + z ≤ x",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℝ≥0∞\nhy : y ≠ 0\nhz : z ≠ 0\nh : ∀ y' < y, ∀ z' < z, y' + z' ≤ x\n⊢ y + z ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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