module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 290,
"column": 31
} | {
"line": 290,
"column": 42
} | {
"line": 290,
"column": 43
} | [
{
"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",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 355,
"column": 8
} | {
"line": 355,
"column": 28
} | {
"line": 355,
"column": 29
} | [
{
"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",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants"... | [
"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 ∈ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 346,
"column": 23
} | {
"line": 346,
"column": 44
} | {
"line": 346,
"column": 45
} | [
{
"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) ((... | [
"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) ((fun x y ↦ ↑x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 349,
"column": 23
} | {
"line": 349,
"column": 44
} | {
"line": 349,
"column": 45
} | [
{
"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) ((... | [
"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) ((fun x y ↦ y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 362,
"column": 12
} | {
"line": 362,
"column": 41
} | {
"line": 362,
"column": 42
} | [
{
"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... | [
"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) I K)\n⊢ ¬↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 34
} | {
"line": 327,
"column": 6
} | [
{
"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... | [
"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 : ↥LieSubalgebra.... | 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": 447,
"column": 4
} | {
"line": 447,
"column": 15
} | {
"line": 447,
"column": 16
} | [
{
"pp": "case 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",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"case 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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.DedekindDomain | {
"line": 39,
"column": 2
} | {
"line": 57,
"column": 43
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : IsDedekindDomain R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nhI : I ≠ ⊥\nhM : Module.IsTorsionBySet R M ↑I\n⊢ DirectSum.IsInternal fun p ↦ torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (factors I))",
"ppTerm": "?m.38",
"assign... | [] | let P := factors I
have prime_of_mem := fun p (hp : p ∈ P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
· convert! hM
rw [← Finset.inf_eq_iInf, IsDedekindDomain.inf_pow_eq_prod_of_prime,
← Finset.prod_multiset_count, ← asso... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.DedekindDomain | {
"line": 39,
"column": 2
} | {
"line": 57,
"column": 43
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : IsDedekindDomain R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nhI : I ≠ ⊥\nhM : Module.IsTorsionBySet R M ↑I\n⊢ DirectSum.IsInternal fun p ↦ torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (factors I))",
"ppTerm": "?m.38",
"assign... | [] | let P := factors I
have prime_of_mem := fun p (hp : p ∈ P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
· convert! hM
rw [← Finset.inf_eq_iInf, IsDedekindDomain.inf_pow_eq_prod_of_prime,
← Finset.prod_multiset_count, ← asso... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 625,
"column": 23
} | {
"line": 625,
"column": 34
} | {
"line": 625,
"column": 35
} | [
{
"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 = ⊥",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.m... | [
"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⊢ y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 31
} | {
"line": 111,
"column": 32
} | [
{
"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",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"congrArg",
... | [
"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⊢ I ≤ P ∨ J ≤ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 134,
"column": 34
} | {
"line": 134,
"column": 81
} | {
"line": 134,
"column": 82
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\na : A\nha : a ≠ 0\n⊢ span {a} ≠ ⊥",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"_private.Mathlib.RingTheory.Dedekin... | [
"A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\na : A\nha : a ≠ 0\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | {
"line": 217,
"column": 14
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 28
} | {
"line": 282,
"column": 29
} | [
{
"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",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"iInf",
"Semiring.toModule",
"HMul.hMul",
"IsScala... | [
"A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI J K : Ideal A\n⊢ (⨅ i, I * bif i then J else K) = I * J ⊓ I * K"
] | mul_iInf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 329,
"column": 34
} | {
"line": 329,
"column": 63
} | {
"line": 329,
"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✝¹",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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✝¹ ⊔ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 330,
"column": 35
} | {
"line": 330,
"column": 64
} | {
"line": 330,
"column": 65
} | [
{
"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✝",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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✝ ≤ x✝¹ ⊔ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 41
} | {
"line": 92,
"column": 42
} | [
{
"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... | [
"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⊢ Submodule.span R ↑ι = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 71
} | {
"line": 93,
"column": 72
} | [
{
"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... | [
"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⊢ (Submodule.map (↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 130,
"column": 45
} | {
"line": 130,
"column": 56
} | {
"line": 130,
"column": 57
} | [
{
"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... | [
"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 := Free.chooseBa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 546,
"column": 56
} | {
"line": 552,
"column": 94
} | {
"line": 554,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nhR : ¬IsField R\nI : Ideal R\n⊢ I ≠ ⊤ ↔ ∃ P, I ≤ P.asIdeal",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MaximalSpectrum.asIdeal",
"Eq.mpr",
"Semiring.toModule",
"Equiv.instEquivLike",
... | [] | 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
} | {
"line": 556,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nP Q : HeightOneSpectrum R\nhPQ : P ≠ Q\n⊢ P.asIdeal ≠ Q.asIdeal",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsDedekindDomain.HeightOneSpectrum.asIdeal",
"id",
"Ne... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nP Q : HeightOneSpectrum R\nhPQ : P ≠ Q\n⊢ ¬P.asIdeal = Q.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 797,
"column": 26
} | {
"line": 797,
"column": 53
} | {
"line": 797,
"column": 54
} | [
{
"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)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiri... | [
"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 = 0 ∨ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 822,
"column": 4
} | {
"line": 822,
"column": 48
} | {
"line": 822,
"column": 49
} | [
{
"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)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"_private.Mathlib.RingT... | [
"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 J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Lattice | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 16
} | [
{
"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... | [
"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.FG\nhr : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 36
} | {
"line": 869,
"column": 37
} | [
{
"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",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
... | [
"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⊢ 0 ∣ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 398,
"column": 43
} | {
"line": 398,
"column": 54
} | {
"line": 398,
"column": 55
} | [
{
"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... | [
"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 : ↥LieSubalgebra.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 399,
"column": 25
} | {
"line": 399,
"column": 36
} | {
"line": 399,
"column": 37
} | [
{
"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... | [
"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 : ↥LieSubalgebra.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1010,
"column": 4
} | {
"line": 1010,
"column": 15
} | {
"line": 1010,
"column": 16
} | [
{
"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",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 39
} | {
"line": 273,
"column": 40
} | [
{
"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'... | [
"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' ∘ₗ v).HasNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | {
"line": 353,
"column": 59
} | [
{
"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... | [
"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\ninst✝ : Module.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1027,
"column": 6
} | {
"line": 1027,
"column": 43
} | {
"line": 1028,
"column": 4
} | [
{
"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",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"... | [] | exact hb (span_singleton_eq_bot.mp h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1044,
"column": 6
} | {
"line": 1044,
"column": 36
} | {
"line": 1045,
"column": 4
} | [
{
"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",
"ppTerm": "?pos.hk✝",
"assigned": true,
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
... | [] | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1044,
"column": 6
} | {
"line": 1044,
"column": 36
} | {
"line": 1045,
"column": 4
} | [
{
"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",
"ppTerm": "?pos.hk✝",
"assigned": true,
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
... | [] | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1044,
"column": 6
} | {
"line": 1044,
"column": 36
} | {
"line": 1045,
"column": 4
} | [
{
"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",
"ppTerm": "?pos.hk✝",
"assigned": true,
"usedConstants": [
"pow_multiplicity_dvd",
"CommSemiring.toSemiring",
... | [] | exact pow_multiplicity_dvd a b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1143,
"column": 2
} | {
"line": 1143,
"column": 36
} | {
"line": 1143,
"column": 37
} | [
{
"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)",
... | [
"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": 1174,
"column": 2
} | {
"line": 1174,
"column": 13
} | {
"line": 1174,
"column": 14
} | [
{
"pp": "A : 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",
"ppTerm": "?m.29... | [
"A : 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✝ ∈ normalizedFactors (map (algebraMap A B) p) ↔ x✝ ∈ p.primesOver B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 411,
"column": 25
} | {
"line": 411,
"column": 36
} | {
"line": 411,
"column": 37
} | [
{
"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... | [
"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 : ↥LieSubalgebra.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LocalizedModule.Away | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 38
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_4\nN : Type u_5\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : M →ₗ[R] N\nr : R\nh₁ : IsUnit ((algebraMap R (Module.End R N)) r)\nh₂ : ∀ (x : N), ∃ n y, r ^ n • x = f y\nh₃ : ∀ (x : M), f x = 0 → ... | [
"case h\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_4\nN : Type u_5\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : M →ₗ[R] N\nr : R\nh₁ : IsUnit ((algebraMap R (Module.End R N)) r)\nh₂ : ∀ (x : N), ∃ n y, r ^ n • x = f y\nh₃ : ∀ (x : M), f x = 0 → ∃ n, r ^ n •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 454,
"column": 11
} | {
"line": 454,
"column": 58
} | {
"line": 454,
"column": 59
} | [
{
"pp": "R : 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 u_2\nin... | [
"R : 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 u_2\ninst✝⁴ : AddCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Index | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : Module R M\ninst✝ : Module R N\n⊢ ↑(finrank R ↥⊤) - ↑(finrank R (N ⧸ ⊥)) = ↑(finrank R M) - ↑(finrank R N)",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"E... | [
"M : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : Module R M\ninst✝ : Module R N\n⊢ finrank R (N ⧸ ⊥) = finrank R N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Index | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 36
} | {
"line": 59,
"column": 37
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : Module R M\ninst✝¹ : Module R N\nf : M →ₗ[R] N\ninst✝ : Nontrivial R\nhf : Injective ⇑f\n⊢ f.index = -↑(finrank R (N ⧸ f.range))",
"ppTerm": "?m.45",
"assigned": true,
"used... | [
"M : Type u_1\nN : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : Module R M\ninst✝¹ : Module R N\nf : M →ₗ[R] N\ninst✝ : Nontrivial R\nhf : Injective ⇑f\n⊢ finrank R ↥f.ker = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LocalizedModule.Int | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 51
} | {
"line": 67,
"column": 2
} | [
{
"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),... | [
"case refine_1\nR : 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 : ... | refine ⟨∏ i ∈ s, (sec i).2, fun i hi => ⟨?_, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.LocalizedModule.Int | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 54
} | {
"line": 134,
"column": 2
} | [
{
"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... | [
"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 S f s\nhx₁✝... | 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.Lie.Weights.IsSimple | {
"line": 484,
"column": 27
} | {
"line": 484,
"column": 62
} | {
"line": 484,
"column": 63
} | [
{
"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... | [
"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 : ↥LieSubalgebra.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Supported | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | {
"line": 111,
"column": 16
} | [
{
"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",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 483,
"column": 4
} | {
"line": 484,
"column": 95
} | {
"line": 485,
"column": 4
} | [
{
"pp": "case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀... | [
"case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSu... | 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.Algebra.Module.FinitePresentation | {
"line": 584,
"column": 4
} | {
"line": 584,
"column": 15
} | {
"line": 584,
"column": 16
} | [
{
"pp": "case exists_of_eq\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.18\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 : IsLocalizedM... | [
"case exists_of_eq\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.18\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 : IsLocalizedModule S f✝\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 603,
"column": 66
} | {
"line": 603,
"column": 77
} | {
"line": 603,
"column": 78
} | [
{
"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 :... | [
"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⊢ ∃ a, a ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 617,
"column": 6
} | {
"line": 619,
"column": 33
} | {
"line": 619,
"column": 34
} | [
{
"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... | [
"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 : IsLocalizedModule... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Derivation | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"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)... | [
"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⊢ (AEval.of R M a) ((aeval a) (derivative f) • d a) = derivative ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 138,
"column": 58
} | {
"line": 138,
"column": 74
} | {
"line": 138,
"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.RingTheory.MvPolynomial.Localization | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 15
} | {
"line": 122,
"column": 16
} | [
{
"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",
"ppTerm": "?m.71",
"assigned": false,
"usedConstan... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 16
} | [
{
"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",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.PID | {
"line": 287,
"column": 22
} | {
"line": 287,
"column": 33
} | {
"line": 287,
"column": 34
} | [
{
"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) ... | [
"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) × ⨁ (i : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 381,
"column": 4
} | {
"line": 383,
"column": 32
} | {
"line": 384,
"column": 2
} | [
{
"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": 381,
"column": 4
} | {
"line": 383,
"column": 32
} | {
"line": 384,
"column": 2
} | [
{
"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": 408,
"column": 2
} | {
"line": 408,
"column": 13
} | {
"line": 408,
"column": 14
} | [
{
"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... | [
"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": 166,
"column": 24
} | {
"line": 166,
"column": 35
} | {
"line": 166,
"column": 36
} | [
{
"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",
"ppTerm": "?m.200",
"assigned": true,
... | [
"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 Finsupp.linearCombination S ⇑pres.cotangentSpaceBasis = pres.differentialsSolution.π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 21
} | {
"line": 136,
"column": 22
} | [
{
"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... | [
"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⊢ ∀ (a : P.Ring), (algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 23
} | {
"line": 141,
"column": 24
} | [
{
"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... | [
"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⊢ ∀ (a : P.Ring), (algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 464,
"column": 19
} | {
"line": 464,
"column": 30
} | {
"line": 464,
"column": 31
} | [
{
"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.16\nS' : Type ?u.18\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁰ : CommRing R''\ninst✝⁹... | [
"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.16\nS' : Type ?u.18\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁰ : CommRing R''\ninst✝⁹ : CommRing ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 21
} | {
"line": 433,
"column": 0
} | [
{
"pp": "R : 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 : Algebra S... | [] | simp [Hom.toAlgHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Basic | {
"line": 481,
"column": 28
} | {
"line": 481,
"column": 52
} | {
"line": 481,
"column": 53
} | [
{
"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.16\nS' : Type ?u.18\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁰ : CommRing R''\ninst✝⁹... | [
"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.16\nS' : Type ?u.18\ninst✝¹³ : CommRing R'\ninst✝¹² : CommRing S'\ninst✝¹¹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁰ : CommRing R''\ninst✝⁹ : CommRing ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 496,
"column": 69
} | {
"line": 500,
"column": 62
} | {
"line": 502,
"column": 0
} | [
{
"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
} | {
"line": 53,
"column": 20
} | [
{
"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",
"ppTerm": "?m.72",
"assigned": false,
"usedCon... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Presentation.Tautological | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 19
} | {
"line": 57,
"column": 20
} | [
{
"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",
"ppTerm": "?m.99",
"assigned": true,
"... | [
"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⊢ a • s.var m - s.var (a • m) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 535,
"column": 30
} | {
"line": 535,
"column": 64
} | {
"line": 535,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nP' : Extension R S\nf : P.Hom P'\nh : Function.Surjective ⇑f\neq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker\neq_map : P'.ker = Ideal.map f.toRingHom P.ker\nx : ↥P.ker\nhx : ... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nP' : Extension R S\nf : P.Hom P'\nh : Function.Surjective ⇑f\neq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker\neq_map : P'.ker = Ideal.map f.toRingHom P.ker\nx : ↥P.ker\nhx : mk x ∈ Submo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 628,
"column": 66
} | {
"line": 630,
"column": 5
} | {
"line": 632,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Algebra.Generators.ker",
"RingHom.ker.congr_simp",
"Eq.mp... | [] | by
simp only [ker, Extension.ker, toExtension_Ring, algebraMap_eq]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Basic | {
"line": 551,
"column": 6
} | {
"line": 551,
"column": 21
} | {
"line": 552,
"column": 6
} | [
{
"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.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹... | [
"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.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing... | intro r hr s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Extension.Basic | {
"line": 543,
"column": 74
} | {
"line": 561,
"column": 27
} | {
"line": 563,
"column": 0
} | [
{
"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.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing R''\ninst✝¹... | [] | 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.LinearAlgebra.TensorProduct.Vanishing | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 96
} | {
"line": 173,
"column": 6
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R... | [
"case refine_2\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R] M := linea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 30,
"column": 7
} | {
"line": 30,
"column": 52
} | {
"line": 32,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : ι → Ideal R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx✝ : M\n⊢ ((AlgebraTensorModule.curry (rTensor M (LinearMap.pi fun i ↦ Submodule.mkQ (I i)))) 1) x✝ =\n ((AlgebraTensorModule.curry... | [] | simp [LinearMap.pi, LinearEquiv.piCongrRight] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 13
} | {
"line": 43,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\ninst✝ : Finite ι\nthis :\n Surjective\n ⇑(↑(piLeft R M fun i ↦ R ⧸ I i).symm ∘ₗ\n (LinearMap.pi fun i ↦ (TensorProduct.mk R (R ⧸ I i) ... | [
"R : Type u_1\ninst✝³ : CommRing R\nι : Type u_2\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\ninst✝ : Finite ι\nthis :\n Surjective\n ⇑(↑(piLeft R M fun i ↦ R ⧸ I i).symm ∘ₗ\n (LinearMap.pi fun i ↦ (TensorProduct.mk R (R ⧸ I i) M) 1) ∘ₗ ↑(T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 299,
"column": 2
} | {
"line": 314,
"column": 86
} | {
"line": 316,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing S'\ninst✝⁵ : Algebra R' S'\nP' : Extension R' S'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ :... | [] | induction x using TensorProduct.induction_on with
| zero =>
simp only [map_zero]
| add =>
simp only [map_add, LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply, *]
| tmul x y =>
obtain ⟨y, rfl⟩ := KaehlerDifferential.tensorProductTo_surjective _ _ y
induction y with
| zero... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | {
"line": 96,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → R\nx : ι → M\nh : IsTrivialRelation f x\n⊢ ∑ i, f i • x i = 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → R\nx : ι → M\nh : IsTrivialRelation f x\n⊢ ∑ i, f i • x i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 70
} | {
"line": 401,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Algebra R S\nP✝ : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝²⁰ : CommRing R'\ninst✝¹⁹ : CommRing S'\ninst✝¹⁸ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁷ : Algebra R R'\ninst✝¹⁶ : Algebra S S'\ninst✝¹⁵ : Algebra R S'\... | [
"R : Type u\nS : Type v\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Algebra R S\nP✝ : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝²⁰ : CommRing R'\ninst✝¹⁹ : CommRing S'\ninst✝¹⁸ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁷ : Algebra R R'\ninst✝¹⁶ : Algebra S S'\ninst✝¹⁵ : Algebra R S'\ninst✝¹⁴ : I... | simp only [LinearMap.mem_ker, Submodule.restrictScalars_mem] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 30
} | {
"line": 91,
"column": 2
} | [
{
"pp": "case e'_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M)... | [] | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | {
"line": 93,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M)... | [] | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | {
"line": 93,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M)... | [] | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 30
} | {
"line": 93,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M)... | [] | exact Algebra.smul_def _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 105,
"column": 20
} | {
"line": 105,
"column": 31
} | {
"line": 105,
"column": 32
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nR' : Type u_3\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : Algebra R R'\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module R' N\ninst✝¹ : IsScalarTower R R' N\ninst✝ : IsLocalization M R'\nx : N\nhs' : x ∈ Submodule.span ... | [
"case h\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nR' : Type u_3\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : Algebra R R'\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module R' N\ninst✝¹ : IsScalarTower R R' N\ninst✝ : IsLocalization M R'\nx : N\nhs' : x ∈ Submodule.span R' ↑∅\n⊢ x =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 78
} | {
"line": 148,
"column": 79
} | [
{
"pp": "case h.left\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTri... | [
"case h.left\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "case h\nR : Type u_3\nS : Type u_4\nRₚ : Type u_1\nSₚ : Type u_2\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : CommSemiring Sₚ\ninst✝⁸ : Algebra R S\ninst✝⁷ : Algebra R Rₚ\ninst✝⁶ : Algebra R Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : Algebra Rₚ Sₚ\ninst✝³ : IsScalar... | [
"case h\nR : Type u_3\nS : Type u_4\nRₚ : Type u_1\nSₚ : Type u_2\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : CommSemiring Sₚ\ninst✝⁸ : Algebra R S\ninst✝⁷ : Algebra R Rₚ\ninst✝⁶ : Algebra R Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : Algebra Rₚ Sₚ\ninst✝³ : IsScalarTower R S Sₚ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
"pp": "case h\nR : Type u\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nRₚ : Type v\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : Algebra R Rₚ\ninst✝⁸ : IsLocalization S Rₚ\nM : Type w\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nMₚ : Type t\ninst✝⁵ : AddCommMonoid Mₚ\ninst✝⁴ : Module R Mₚ\ninst✝³ : Module Rₚ Mₚ\nins... | [
"case h\nR : Type u\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nRₚ : Type v\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : Algebra R Rₚ\ninst✝⁸ : IsLocalization S Rₚ\nM : Type w\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nMₚ : Type t\ninst✝⁵ : AddCommMonoid Mₚ\ninst✝⁴ : Module R Mₚ\ninst✝³ : Module Rₚ Mₚ\ninst✝² : IsScal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 56
} | {
"line": 556,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\n⊢ Module.FinitePresentation S Ω[S⁄R]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Algebra.Presentation.ofFinitePresentationVars",
"Algebra.Presen... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\nP : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))\n (Fin (Presentation.ofFinitePresentationRels R S)) :=\n Presentation.ofFinitePresentation R S\n⊢ Module.FinitePresentatio... | let P := Algebra.Presentation.ofFinitePresentation R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 156,
"column": 27
} | {
"line": 156,
"column": 85
} | {
"line": 156,
"column": 86
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x)... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔\n ∀ (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 554,
"column": 82
} | {
"line": 559,
"column": 82
} | {
"line": 561,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\n⊢ Module.FinitePresentation S Ω[S⁄R]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Function.Exact",
"Eq.mpr",
"Submodule",
"RingHomSur... | [] | by
let P := Algebra.Presentation.ofFinitePresentation R S
have : Algebra.FiniteType R P.toExtension.Ring := by simp [P]; infer_instance
refine Module.finitePresentation_of_surjective _ P.toExtension.toKaehler_surjective ?_
rw [LinearMap.exact_iff.mp P.toExtension.exact_cotangentComplex_toKaehler, ← Submodule.ma... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuotSMulTop | {
"line": 114,
"column": 63
} | {
"line": 115,
"column": 64
} | {
"line": 117,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\nr : R\nM : Type u_1\nM' : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf : Surjective ⇑f\nH₁ : Surjective (⇑(r • ⊤).mkQ ∘ ⇑f)\n⊢ Surjective (⇑((map r) f) ∘ ⇑(r • ⊤).mkQ)",
"ppTerm": "?m.73"... | [] | by
rwa [← LinearMap.coe_comp, map_comp_mkQ, LinearMap.coe_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 58
} | {
"line": 160,
"column": 8
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesT... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuotSMulTop | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 51
} | {
"line": 150,
"column": 52
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\nm : QuotSMulTop x M\nm' : M\nhm' : Submodule.Quotient.mk m' = m\n⊢ x • m = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAct... | [
"R : Type u_2\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\nm : QuotSMulTop x M\nm' : M\nhm' : Submodule.Quotient.mk m' = m\n⊢ x • m' ∈ x • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 625,
"column": 2
} | {
"line": 625,
"column": 56
} | {
"line": 626,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nι : Type w\nι' : Type u_1\nP : Generators R S ι\nS' : Type u_2\ninst✝⁹ : CommRing S'\ninst✝⁸ : Algebra R S'\nT : Type w\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S... | [
"R : Type u\nS : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nι : Type w\nι' : Type u_1\nP✝ : Generators R S ι\nS' : Type u_2\ninst✝⁹ : CommRing S'\ninst✝⁸ : Algebra R S'\nT : Type w\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S T\ninst✝³ ... | let P := Algebra.Presentation.ofFinitePresentation R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Support | {
"line": 112,
"column": 30
} | {
"line": 112,
"column": 41
} | {
"line": 112,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : m = 0\n⊢ 1 • m = 0",
"ppTerm": "?m.174",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congr... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : m = 0\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 17
} | {
"line": 116,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : ¬(R ∙ m).annihilator = ⊤\np : Ideal R\nhp : (R ∙ m).annihilator ≤ p\nhp' : p.IsPrime\n⊢ f ∈ p",
"ppTerm": "?m.221",
"assigned": f... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nH : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}\nm : M\nh : ¬(R ∙ m).annihilator = ⊤\np : Ideal R\nhp : (R ∙ m).annihilator ≤ p\nhp' : p.IsPrime\n⊢ f ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 56
} | {
"line": 143,
"column": 57
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\np : PrimeSpectrum R\nx✝ : ∃ m, ∀ r ∉ p.asIdeal, ¬r • m = 0\nx : R\nhx : x ∈ RingHom.ker (algebraMap R A)\nm : A\nhm : ∀ r ∉ p.asIdeal, ¬r • m = 0\nhx' : x ∉ p.asIdeal\n⊢ False",
"ppTerm": "?refine_... | [
"case refine_1\nR : Type u_1\ninst✝² : CommRing R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\np : PrimeSpectrum R\nx✝ : ∃ m, ∀ r ∉ p.asIdeal, ¬r • m = 0\nx : R\nhx : x ∈ RingHom.ker (algebraMap R A)\nm : A\nhm : ∀ r ∉ p.asIdeal, ¬r • m = 0\nhx' : x ∉ p.asIdeal\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 160,
"column": 28
} | {
"line": 160,
"column": 39
} | {
"line": 160,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\n⊢ r • f m ≠ 0",
... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\n⊢ ¬r • f m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 168,
"column": 37
} | {
"line": 168,
"column": 48
} | {
"line": 168,
"column": 49
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • f m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\ne : r • m = 0\n... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\nx : PrimeSpectrum R\nm : M\nhm : ∀ r ∉ x.asIdeal, r • f m ≠ 0\nr : R\nhr : r ∉ x.asIdeal\ne : r • m = 0\n⊢ r • f m = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 17
} | {
"line": 164,
"column": 18
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesT... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 230,
"column": 15
} | {
"line": 230,
"column": 26
} | {
"line": 230,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\np : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : Subsingleton (LocalizedModule p.primeCompl M)\nf : R\nhf : f ∈ ↑(Module.annihilator R M)\nhf' : { asIdeal := p, isPrime := ⋯ } ∈ (zeroLocus... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\np : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : Subsingleton (LocalizedModule p.primeCompl M)\nf : R\nhf : f ∈ ↑(Module.annihilator R M)\nhf' : { asIdeal := p, isPrime := ⋯ } ∈ (zeroLocus {f})ᶜ\n⊢ f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 49
} | {
"line": 227,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : Flat R M\nN : Type u_3\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : N\nx : N →ₗ[R] M\nh : x f = 0\ne : (Fin (Fintype.card (Free.ChooseBasisIndex R ... | [
"R : Type u_1\nM : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : Flat R M\nN : Type u_3\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : N\nx : N →ₗ[R] M\nh : x f = 0\ne : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) ≃ₗ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 239,
"column": 31
} | {
"line": 239,
"column": 42
} | {
"line": 239,
"column": 43
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh : x ∘ₗ f = 0\n⊢ ... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh : x ∘ₗ f = 0\n⊢ x (f k) = 0"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.