module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.ChainOfDivisors | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 28
} | [
{
"pp": "case succ.refine_2.succ\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\nq : Associates M\nhq : q ≠ 0\nn : ℕ\nhn : n + 1 ≠ 0\nc : Fin (n + 1 + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nn✝ : ℕ\nhi : n✝ + 1 < n + 1 + 1\nhb : c ⟨n✝ + 1, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 15
} | [
{
"pp": "case succ.refine_2.inr.refine_1\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\nq r : Associates M\nhr : r ∣ q\nn : ℕ\nhn : n + 1 ≠ 0\nc : Fin (n + 1 + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nj : Fin (n + 1)\nhp : Prime (c j.succ)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 42
} | [
{
"pp": "case succ.refine_2.inr.refine_2\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\nq r : Associates M\nhr : r ∣ q\nn : ℕ\nhn : n + 1 ≠ 0\nc : Fin (n + 1 + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nj : Fin (n + 1)\nhp : Prime (c j.succ)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 156,
"column": 51
} | {
"line": 156,
"column": 83
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nx : L\nhxI : x ∈ ↑↑(LieSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 59
} | [
{
"pp": "case neg.refine_2\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 18
} | [
{
"pp": "case neg.refine_3\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs... | rintro ⟨i, hr⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 34
} | [
{
"pp": "case neg.refine_3\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 86
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 419,
"column": 4
} | {
"line": 420,
"column": 69
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 182,
"column": 44
} | {
"line": 182,
"column": 80
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nx : L\nhxI : x ∈ ↑↑(LieSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 185,
"column": 36
} | {
"line": 185,
"column": 68
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nx : L\nhxI : x ∈ ↑↑(LieSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 54
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : FaithfulSMul R P\nI : FractionalIdeal S P\nreg : IsSMulRegular P I.den\nx✝¹ x✝ : ↥↑I\nhxy : ((DistribSMul.toLinearMap R P I.den).restrict ⋯) x✝¹ = ((DistribSMul.toLinearMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 259,
"column": 34
} | {
"line": 259,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI : Ideal R\n⊢ coeSubmodule P I ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nh_inj : Function.Injective ⇑(algebraMap R P)\n⊢ num 0 = 0",
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"FractionalIdeal.num",
"Submodule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 437,
"column": 20
} | {
"line": 437,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI : FractionalIdeal S P\nh : I = 0\nx : P\nhx : x ∈ I\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 22
} | [
{
"pp": "case a\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)\nχ : Wei... | exact y.property | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 573,
"column": 22
} | {
"line": 573,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI J J' : FractionalIdeal S P\nh : (fun x1 x2 ↦ x1 ≤ x2) J J'\n⊢ (fun x1 x2 ↦ x1 ≤ x2) ((fun x1 x2 ↦ x1 * x2) I J) ((fun x1 x2 ↦ x1 * x2) I J')",
"usedConstants": [
"Eq.mpr",
"Subm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 576,
"column": 22
} | {
"line": 576,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI J J' : FractionalIdeal S P\nh : (fun x1 x2 ↦ x1 ≤ x2) J J'\n⊢ (fun x1 x2 ↦ x1 ≤ x2) (Function.swap (fun x1 x2 ↦ x1 * x2) I J) (Function.swap (fun x1 x2 ↦ x1 * x2) I J')",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 54
} | [
{
"pp": "K : 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)\nχ : Weight K (↥... | exact q.add_mem h_chi_in_q (q.smul_mem (-1) hαq) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 66
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomainInv A\nI✝ J✝ I J : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nthis : I / J * J ≤ I\n⊢ I / J ≤ I * J⁻¹",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 94
} | [
{
"pp": "case maximalOfPrime\nA : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nh : IsDedekindDomainInv A\nthis : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := h.commGroupWithZero\nP : Ideal A\nP_ne : P ≠ ⊥\nhP : P.IsPrime\nM : Ideal A\nhM : P < M\nP'_ne : ↑P ≠ 0\n⊢ M = ⊤",
"usedConstants"... | have M'_ne : (M : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hM.ne_bot | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 22
} | [
{
"pp": "A : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1",
"usedConstants": [
"IsDedekindDomain.to... | by_cases hJ0 : J = ⊥ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 303,
"column": 35
} | {
"line": 305,
"column": 28
} | [
{
"pp": "A : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhJ0 : ¬↑I * (↑I)⁻¹ = 0\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nthis : x ∈ integralClosure A K\n⊢ x ∈ 1",
"usedConstants": [
"Su... | by
rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
← mem_one_iff] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 346,
"column": 23
} | {
"line": 346,
"column": 44
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK✝ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K✝\ninst✝² : Algebra A K✝\ninst✝¹ : IsFractionRing A K✝\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : { x // 0 < x }\nJ K : FractionalIdeal A⁰ K✝\nhJK : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 172,
"column": 24
} | {
"line": 176,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : Submodule R P\nhI : I.FG\n⊢ IsFractional S I",
"usedConstants": [
"IsLocalization.IsInteger",
"Eq.mpr",
"Submodule",
"instHSMul",
... | by
rcases hI with ⟨I, rfl⟩
rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩
rw [isFractional_span_iff]
exact ⟨s, hs1, hs⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 180,
"column": 47
} | {
"line": 180,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI J : FractionalIdeal S P\nx : P\nhx : x ∈ I * J\n⊢ x ∈ ↑I * ↑J",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 349,
"column": 23
} | {
"line": 349,
"column": 44
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK✝ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K✝\ninst✝² : Algebra A K✝\ninst✝¹ : IsFractionRing A K✝\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : { x // 0 < x }\nJ K : FractionalIdeal A⁰ K✝\nhJK : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ y * ↑x) I J) ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 362,
"column": 12
} | {
"line": 362,
"column": 41
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK✝ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K✝\ninst✝² : Algebra A K✝\ninst✝¹ : IsFractionRing A K✝\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : { x // 0 < x }\nJ K : Ideal A\ne : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((fun x y ↦ ↑x * y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 34
} | [
{
"pp": "K : 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 : ↥Li... | rw [hj, Weight.toLinear_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 232,
"column": 22
} | {
"line": 232,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P\nx : P'\n⊢ x ∈\n (mapEquiv\n (let __sr... | mapEquiv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 239,
"column": 67
} | {
"line": 239,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P'\nx : P\n⊢ x ∈\n (mapEquiv\n (let __sr... | mapEquiv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 238,
"column": 33
} | {
"line": 241,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P'\nx : P\n⊢ x ∈ (canonicalEquiv S P P').symm I ↔ x ∈ ... | by
rw [mem_canonicalEquiv_apply, canonicalEquiv, mapEquiv_symm, mapEquiv_apply,
mem_map]
exact ⟨fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩, fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 289,
"column": 29
} | {
"line": 289,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nI : FractionalIdeal R⁰ K\ninst✝ : Nontrivial R\nhI : I ≠ 0\n⊢ ?m.52 < ?m.53",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 290,
"column": 31
} | {
"line": 290,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nI : FractionalIdeal R⁰ K\ninst✝ : Nontrivial R\nhI : I ≠ 0\ny : K\ny_mem : y ∈ I\ny_notMem : y ∉ ⊥\n⊢ y ≠ 0",
"usedConstants": [
"id",
"Ne",
"Field.toSemifield",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : Ideal R\n⊢ ↑I = 1 ↔ I = 1",
"usedConstants": [
"Eq.mpr",
"Ideal.one_eq_top",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"nonZ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 355,
"column": 8
} | {
"line": 355,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁴ : CommRing P\ninst✝³ : Algebra R P\nR₁ : Type u_3\ninst✝² : CommRing R₁\nK : Type u_4\ninst✝¹ : Field K\ninst✝ : Algebra R₁ K\nh : 0 = 1\n⊢ (algebraMap R₁ K) 1 ∈ 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 370,
"column": 42
} | {
"line": 370,
"column": 58
} | [
{
"pp": "R₁ : Type u_3\ninst✝⁴ : CommRing R₁\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ b ∈ I, IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ b ∈ J, IsInteger R₁ (aJ • b)\nh : J ≠ 0\n⊢ 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 447,
"column": 4
} | {
"line": 447,
"column": 15
} | [
{
"pp": "case a.mp\nR₁ : Type u_3\ninst✝⁴ : CommRing R₁\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\nx✝ : K\nh : x✝ ∈ ⟨↑I / ↑1, ⋯⟩\n⊢ x✝ ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 625,
"column": 23
} | {
"line": 625,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\ny : P\nh : spanSingleton S y = 0\n⊢ R ∙ y = ⊥",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.mpr",
"Submodule",
"NonUnitalCommR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 398,
"column": 43
} | {
"line": 398,
"column": 54
} | [
{
"pp": "K : 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 : ↥Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 399,
"column": 25
} | {
"line": 399,
"column": 36
} | [
{
"pp": "K : 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 : ↥Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Lattice | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nK : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : Algebra R K\nV : Type u_3\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module K V\ninst✝⁴ : Module R V\ninst✝³ : IsScalarTower R K V\ninst✝² : IsDomain R\ninst✝¹ : Module.Finite K V\ninst✝ : IsFractionRing R K\nM : Submodule R V\nhfg : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 41
} | [
{
"pp": "case refine_1\nR : Type u\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nfp : FinitePresentation R M\nι : Finset M\nhι₁ : Submodule.span R ↑ι = ⊤\nhι₂ : (linearCombination R Subtype.val).ker.FG\n⊢ (linearCombination R Subtype.val ∘ₗ\n ↑(lcongr ι.equivFin (LinearEqui... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 71
} | [
{
"pp": "case refine_2\nR : Type u\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nfp : FinitePresentation R M\nι : Finset M\nhι₁ : Submodule.span R ↑ι = ⊤\nhι₂ : (linearCombination R Subtype.val).ker.FG\n⊢ (linearCombination R Subtype.val ∘ₗ\n ↑(lcongr ι.equivFin (LinearEqui... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 130,
"column": 45
} | {
"line": 130,
"column": 56
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\nhl' : l.ker.FG\nb : Basis (Free.ChooseBasisIndex R M) R M := F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 39
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu v : V →ₗ[K] V₂\nu' : V₂ →ₗ[K] V₃\nh : (u - v).HasNoetherianRange\n⊢ (u' ∘ₗ u - u'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 411,
"column": 25
} | {
"line": 411,
"column": 36
} | [
{
"pp": "K : 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 : ↥Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | [
{
"pp": "R : Type u_2\nM : Type u_4\nN : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module A N\ninst✝¹ : IsScalarTower R A N\nf : M →ₗ[R] N\nh : IsBaseChange A f\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 31
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nP : Ideal A\nhP : P ≠ ⊥\nh : P.IsPrime\nI J : Ideal A\nhIJ : P ∣ I * J\n⊢ P ∣ I ∨ P ∣ J",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 134,
"column": 34
} | {
"line": 134,
"column": 81
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\na : A\nha : a ≠ 0\n⊢ span {a} ≠ ⊥",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"_private.Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas.0.Ideal.prime_span_singl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LocalizedModule.Int | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nM' : Type u_3\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nf : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S f\nι : Type u_4\ns : Finset ι\ng : ι → M'\nsec : ι → M × ↥S\nhsec : ∀ (i : ι),... | refine ⟨∏ i ∈ s, (sec i).2, fun i hi => ⟨?_, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | [
{
"pp": "A : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : FractionalIdeal A⁰ K\nhI : I ≠ 0\nhJ : J ≠ 0\n⊢ I⁻¹ ≤ J ↔ J⁻¹ ≤ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LocalizedModule.Int | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\ninst✝ : DecidableEq M\nx : M\ns : Finset M'\ny : ↥S := commonDenomOfFinset... | rw [hx₁, ← f.map_smul, ← Submodule.map_span f] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 437,
"column": 11
} | {
"line": 437,
"column": 58
} | [
{
"pp": "case h\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nS : Submonoid R\nM' : Type u_1\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁵ : IsLocalizedModule S f\nN' : Type... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 28
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI J K : Ideal A\n⊢ (I * ⨅ b, bif b then J else K) = I * J ⊓ I * K",
"usedConstants": [
"cond",
"Eq.mpr",
"iInf",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"congrArg",
"CommS... | mul_iInf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 484,
"column": 27
} | {
"line": 484,
"column": 62
} | [
{
"pp": "K : 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 : ↥Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 329,
"column": 34
} | {
"line": 329,
"column": 63
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx✝¹ x✝ : Ideal A\n⊢ x✝¹ ⊔ x✝ ∣ x✝¹",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 483,
"column": 4
} | {
"line": 484,
"column": 95
} | [
{
"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 : ∀... | have hne (χ : Weight K H L) (hχ : ↑χ ∈ q) : (χ : H → K) ≠ ((α : Weight K H L) : H → K) :=
fun heq ↦ hα_not (by simpa [rootSystem_root_apply] using DFunLike.coe_injective heq ▸ hχ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 330,
"column": 35
} | {
"line": 330,
"column": 64
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx✝¹ x✝ : Ideal A\n⊢ x✝¹ ⊔ x✝ ∣ x✝",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 354,
"column": 39
} | {
"line": 354,
"column": 71
} | [
{
"pp": "case inl\nK : Type u_3\ninst✝ : Field K\n⊢ factors (span {0}) = Multiset.map (fun q ↦ span {q}) (factors 0)",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Eq.mpr",
"Submodule",
"UniqueFactorizationMonoid.factors_eq_normalizedFactors",
"Normalizati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 546,
"column": 56
} | {
"line": 552,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nhR : ¬IsField R\nI : Ideal R\n⊢ I ≠ ⊤ ↔ ∃ P, I ≤ P.asIdeal",
"usedConstants": [
"MaximalSpectrum.asIdeal",
"Eq.mpr",
"Semiring.toModule",
"Equiv.instEquivLike",
"congrArg",
"CommSemiring.toSemiring",
... | by
rw [Ideal.ne_top_iff_exists_maximal]
constructor
· rintro ⟨M, hMmax, hIM⟩
exact ⟨(equivMaximalSpectrum hR).symm ⟨M, hMmax⟩, hIM⟩
· rintro ⟨P, hP⟩
exact ⟨((equivMaximalSpectrum hR) P).asIdeal, ((equivMaximalSpectrum hR) P).isMaximal, hP⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 556,
"column": 8
} | {
"line": 556,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nP Q : HeightOneSpectrum R\nhPQ : P ≠ Q\n⊢ P.asIdeal ≠ Q.asIdeal",
"usedConstants": [
"CommSemiring.toSemiring",
"IsDedekindDomain.HeightOneSpectrum.asIdeal",
"id",
"Ne",
"Ideal",
"CommRing.toCommSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Supported | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 15
} | [
{
"pp": "case mp\nσ : Type u_1\nR : Type u\ninst✝¹ : CommSemiring R\ns t : Set σ\ninst✝ : Nontrivial R\nh : supported R s ≤ supported R t\ni : σ\n⊢ i ∈ s → i ∈ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 567,
"column": 4
} | {
"line": 567,
"column": 15
} | [
{
"pp": "case exists_of_eq.h.h\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.147698\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\ninst✝⁹ : AddCommGroup N'✝\ninst✝⁸ : Module R N'✝\nS : Submonoid R\nf✝ : N →ₗ[R] N'✝\ninst✝⁷ : IsLo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 586,
"column": 66
} | {
"line": 586,
"column": 77
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : Module.Finite R M\ninst✝ : Module.FinitePresentation R M'\n⊢ ∀ {x₁ x₂ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 15
} | [
{
"pp": "case refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type u_4\ninst✝² : CommRing B\ninst✝¹ : IsDedekindDomain B\nL : Ideal B\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nf_sur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 689,
"column": 4
} | {
"line": 689,
"column": 15
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type u_4\ninst✝² : CommRing B\ninst✝¹ : IsDedekindDomain B\nL : Ideal B\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nf_sur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 600,
"column": 6
} | {
"line": 602,
"column": 33
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : Module.Finite R M\ninst✝ : Module.FinitePresentation R M'\nthis : IsLoc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 797,
"column": 26
} | {
"line": 797,
"column": 53
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn✝ : ℕ\nh : a * b ∈ I ^ (n✝ + 1)\nhI0 : I = ⊥\n⊢ a ∈ I ∨ b ∈ I ^ (n✝ + 1)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"IsSca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 822,
"column": 4
} | {
"line": 822,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI J : Ideal R\nhJ : J.IsPrime\nhJ₀ : J ≠ ⊥\nhI : Associates.mk I ≠ 0\n⊢ Irreducible (Associates.mk J)",
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"_private.Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas.0.Ideal.cou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 36
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI J K : Ideal R\ncoprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime\nhJ : J ∣ I\nhK : K ∣ I\nhJ0 : J = 0\n⊢ J * K ∣ I",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1011,
"column": 4
} | {
"line": 1011,
"column": 15
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : Squarefree (span {a})\nx : R\nhx : span {x} * span {x} ∣ span {a}\n⊢ IsUnit x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1028,
"column": 6
} | {
"line": 1028,
"column": 43
} | [
{
"pp": "case neg.refine_2\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : NormalizationMonoid R\na b : R\nha : a ∈ normalizedFactors b\nhb : ¬b = 0\nthis : Prime (span {a})\nh : span {b} = 0\n⊢ False",
"usedConstants": [
"Semiring.toModule",
"Id... | exact hb (span_singleton_eq_bot.mp h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1045,
"column": 6
} | {
"line": 1045,
"column": 36
} | [
{
"pp": "case pos.hk\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : FiniteMultiplicity a b\n⊢ a ^ multiplicity a b ∣ b",
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
"CommRing.toCommSemiring",
"Semiring.t... | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1045,
"column": 6
} | {
"line": 1045,
"column": 36
} | [
{
"pp": "case pos.hk\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : FiniteMultiplicity a b\n⊢ a ^ multiplicity a b ∣ b",
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
"CommRing.toCommSemiring",
"Semiring.t... | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1045,
"column": 6
} | {
"line": 1045,
"column": 36
} | [
{
"pp": "case pos.hk\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : FiniteMultiplicity a b\n⊢ a ^ multiplicity a b ∣ b",
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
"CommRing.toCommSemiring",
"Semiring.t... | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1144,
"column": 2
} | {
"line": 1144,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : NormalizationMonoid R\ninst✝ : DecidableEq R\nr X : R\nhr : r ≠ 0\nhX₁ : normUnit X = 1\nhX : Prime X\n⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count X (normalizedFactors r)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1176,
"column": 2
} | {
"line": 1176,
"column": 13
} | [
{
"pp": "case h\nA : Type u_4\ninst✝⁵ : CommRing A\np : Ideal A\nhpb : p ≠ ⊥\nhpm : p.IsMaximal\nB : Type u_5\ninst✝⁴ : CommRing B\ninst✝³ : IsDedekindDomain B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain A\ninst✝ : IsTorsionFree A B\nx✝ : Ideal B\n⊢ x✝ ∈ ↑(primesOverFinset p B) ↔ x✝ ∈ p.primesOver B",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Derivation | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\na : A\nd : Derivation R A M\nf : R[X]\n⊢ (d.compAEval a) f = derivative f • (AEval.of R M a)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 123,
"column": 58
} | {
"line": 123,
"column": 74
} | [
{
"pp": "R : Type u\nσ : Type v\ninst✝¹ : CommSemiring R\nS : Type u_1\ninst✝ : CommSemiring S\nφ : R →+* S\nf : MvPolynomial σ R\ni : σ\np q : MvPolynomial σ R\nhp : (pderiv i) ((map φ) p) = (map φ) ((pderiv i) p)\nhq : (pderiv i) ((map φ) q) = (map φ) ((pderiv i) q)\n⊢ (pderiv i) ((map φ) (p + q)) = (map φ) (... | by simp [hp, hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PID | {
"line": 287,
"column": 22
} | {
"line": 287,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nι : Type u\nw✝ : Fintype ι\np : ι → R\nirr : ∀ (i : ι), Irreducible (p i)\nn : ι → ℕ\nx : R\nm : ℕ\ne : M ≃ₗ[R] (Fin (m + 1) →₀ R) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 15
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\nx : MvPolynomial Unit R ⧸ Ideal.span {C r * X () - 1}\n⊢ (auxInv S r) ((auxHom S r) x) = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 15
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\ns : S\n⊢ (auxHom S r) ((auxInv S r) s) = s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 380,
"column": 4
} | {
"line": 382,
"column": 32
} | [
{
"pp": "case add\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\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\np q : MvPolynomial ι' S\nhp : ∃ a, (Q.aux... | obtain ⟨a, rfl⟩ := hp
obtain ⟨b, rfl⟩ := hq
exact ⟨a + b, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 380,
"column": 4
} | {
"line": 382,
"column": 32
} | [
{
"pp": "case add\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\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\np q : MvPolynomial ι' S\nhp : ∃ a, (Q.aux... | obtain ⟨a, rfl⟩ := hp
obtain ⟨b, rfl⟩ := hq
exact ⟨a + b, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 13
} | [
{
"pp": "case hf\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\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\n⊢ Function.Bijective ⇑↑(sumAlgEquiv R ι' ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Presentation.Differentials | {
"line": 162,
"column": 24
} | {
"line": 162,
"column": 35
} | [
{
"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\n⊢ pres.toExtension.toKaehler ∘ₗ ↑pres.cotangentSpaceBasis.repr.symm = LinearMap.id ∘ₗ pres.differentialsSolution.π",
"usedConstants": [
"LinearMap.id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 21
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP✝ : Extension R S\nM : Submonoid S\nS' : Type u_1\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : IsLocalization M S'\ninst✝¹ : Algebra R S'\ninst✝ : IsScalarTower R S S'\nP : Extension R S\n⊢ ∀ (y : ↥(Submo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 23
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP✝ : Extension R S\nM : Submonoid S\nS' : Type u_1\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : IsLocalization M S'\ninst✝¹ : Algebra R S'\ninst✝ : IsScalarTower R S S'\nP : Extension R S\n⊢ ∀ (y : ↥(Submo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 54
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Extension R S\nR' : Type ?u.230553\nS' : Type ?u.230556\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.230680\nS'' : Type ?u.230683\ninst✝² : CommRing R'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 460,
"column": 19
} | {
"line": 460,
"column": 30
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Extension R S\nR' : Type ?u.303575\nS' : Type ?u.303578\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.303702\nS'' : Type ?u.303705\ninst✝¹⁰ : Comm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 477,
"column": 28
} | {
"line": 477,
"column": 52
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Extension R S\nR' : Type ?u.324610\nS' : Type ?u.324613\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.324737\nS'' : Type ?u.324740\ninst✝¹⁰ : Comm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 492,
"column": 69
} | {
"line": 496,
"column": 62
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type u_1\nS' : Type u_2\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type u_4\nS'' : Type u_5\ninst✝¹⁶ : CommRing R''\ninst✝¹⁵ : Comm... | by
ext x
obtain ⟨x, rfl⟩ := Cotangent.mk_surjective x
simp only [map_mk, Hom.toAlgHom_apply, Hom.comp_toRingHom, RingHom.coe_comp, Function.comp_apply,
val_mk, LinearMap.coe_comp, LinearMap.coe_restrictScalars] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Presentation.Tautological | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 19
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : (tautologicalRelations A M).Solution N\nm₁ m₂ : M\n⊢ s.var m₁ + s.var m₂ - s.var (m₁ + m₂) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Presentation.Tautological | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 19
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : (tautologicalRelations A M).Solution N\na : A\nm : M\n⊢ (RingHom.id A) a • s.var m - s.var (a • m) = 0",
"usedConstants": [
"AddGroup.toSubtracti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 515,
"column": 6
} | {
"line": 515,
"column": 21
} | [
{
"pp": "case refine_1.refine_1\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.346085\nS' : Type ?u.346088\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.346212\nS'' : Type ?... | intro r hr s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Extension.Generators | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 21
} | [
{
"pp": "case hf\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing S'\ninst✝⁵ : Algebra R' S'\nP' : Generators R' S' ι'\ninst✝⁴ : Algebra R R'\ninst✝³ : ... | simp [Hom.toAlgHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Basic | {
"line": 507,
"column": 74
} | {
"line": 525,
"column": 27
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.346085\nS' : Type ?u.346088\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.346212\nS'' : Type ?u.346215\ninst✝¹⁶ : Comm... | by
refine .ofBijective (Cotangent.mk.liftBaseChange _) ⟨?_, ?_⟩
· refine (injective_iff_map_eq_zero _).mpr fun x hx ↦ ?_
obtain ⟨x, rfl⟩ := TensorProduct.mk_surjective P.Ring P.ker S P.algebraMap_surjective x
simp only [mk_apply, LinearMap.liftBaseChange_tmul, one_smul, Cotangent.mk_eq_zero_iff,
pow_t... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Generators | {
"line": 617,
"column": 66
} | {
"line": 619,
"column": 5
} | [
{
"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\n⊢ P.ker = RingHom.ker (aeval P.val)",
"usedConstants": [
"Algebra.Generators.ker",
"RingHom.ker.congr_simp",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddM... | by
simp only [ker, Extension.ker, toExtension_Ring, algebraMap_eq]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
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