module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.FractionalIdeal.Inverse | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 38
} | {
"line": 140,
"column": 39
} | [
{
"pp": "K : Type u_3\ninst✝⁴ : Field K\nR₁ : Type u_6\ninst✝³ : CommRing R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsLocalization R₁⁰ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nh : I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I) * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1",
"ppTerm": "?m.90",
"assigned":... | [
"K : Type u_3\ninst✝⁴ : Field K\nR₁ : Type u_6\ninst✝³ : CommRing R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsLocalization R₁⁰ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nh : I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I * (generator ↑I)⁻¹) = 1"
] | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 19
} | {
"line": 147,
"column": 20
} | [
{
"pp": "A : 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\nM'_ne : ↑M ≠ 0\nthis : (↑M)⁻¹ * ↑P ≤ ↑P\n⊢ M = ⊤"... | [
"A : 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\nM'_ne : ↑M ≠ 0\nthis : (↑M)⁻¹ * ↑P ≤ ↑P\n⊢ ⊤ ≤ M"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 264,
"column": 10
} | {
"line": 264,
"column": 28
} | {
"line": 264,
"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\nhNF : ¬IsField A\nI : Ideal A\nhI0✝ : I ≠ ⊥\nhI1 : I ≠ ⊤\nhM : I.IsMaximal\nhI0 : ⊥ < I\na : A\nhaI : a ∈ I\nha0 : a ≠ 0\nJ : Ideal A := ⋯\nhJ0... | [
"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\nhNF : ¬IsField A\nI : Ideal A\nhI0✝ : I ≠ ⊥\nhI1 : I ≠ ⊤\nhM : I.IsMaximal\nhI0 : ⊥ < I\na : A\nhaI : a ∈ I\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 38
} | {
"line": 337,
"column": 39
} | [
{
"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\na : A\nJ : Ideal A\nha : a ≠ 0\nhI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥\n⊢ spanS... | [
"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\na : A\nJ : Ideal A\nha : a ≠ 0\nhI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥\n⊢ spanSingleton A⁰ ... | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 611,
"column": 21
} | {
"line": 611,
"column": 42
} | {
"line": 611,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝³ : CommRing P\ninst✝² : Algebra R P\ninst✝¹ : IsLocalization S P\nI : FractionalIdeal S P\ninst✝ : (↑I).IsPrincipal\n⊢ I = ⟨R ∙ generator ↑I, ⋯⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝³ : CommRing P\ninst✝² : Algebra R P\ninst✝¹ : IsLocalization S P\nI : FractionalIdeal S P\ninst✝ : (↑I).IsPrincipal\n⊢ ↑I = ↑⟨R ∙ generator ↑I, ⋯⟩"
] | ← coeToSubmodule_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 648,
"column": 35
} | {
"line": 648,
"column": 67
} | {
"line": 648,
"column": 68
} | [
{
"pp": "case succ\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✝ : IsLocalization S P\nx : P\nn : ℕ\nhn : spanSingleton S x ^ n = spanSingleton S (x ^ n)\n⊢ spanSingleton S (x ^ n) * spanSingleton S x = spanSingleton S (x ^ (n + 1))",
"pp... | [
"case succ\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✝ : IsLocalization S P\nx : P\nn : ℕ\nhn : spanSingleton S x ^ n = spanSingleton S (x ^ n)\n⊢ spanSingleton S (x ^ n * x) = spanSingleton S (x ^ (n + 1))"
] | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 158,
"column": 18
} | {
"line": 158,
"column": 35
} | {
"line": 158,
"column": 36
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\np : Ideal R\nh : p.IsMaximal\ninst✝² : Algebra R A\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R A\nhp : p ≠ ⊥\nP : Ideal A\n⊢ P ∈ {P | P.IsPrime ∧ P.LiesOver p} ↔ P ∈ normalizedFactors (map (algebraMap R... | [
"R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\np : Ideal R\nh : p.IsMaximal\ninst✝² : Algebra R A\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R A\nhp : p ≠ ⊥\nP : Ideal A\n⊢ P.IsPrime ∧ P.LiesOver p ↔ P ∈ normalizedFactors (map (algebraMap R A) p)"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 707,
"column": 8
} | {
"line": 707,
"column": 40
} | {
"line": 707,
"column": 41
} | [
{
"pp": "R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\n⊢ spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1",
"ppTerm": "?m.102",
"assigned": true,
"usedC... | [
"R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\n⊢ spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩ * (algebraMap R₁ K) y) = 1"
] | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 713,
"column": 15
} | {
"line": 713,
"column": 47
} | {
"line": 713,
"column": 48
} | [
{
"pp": "R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\nthis : spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1\ny' : (FractionalIdeal R₁⁰ K)ˣ :=\n Units.mkOfMulEqO... | [
"R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\nthis : spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1\ny' : (FractionalIdeal R₁⁰ K)ˣ :=\n Units.mkOfMulEqOne (spanSing... | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 713,
"column": 61
} | {
"line": 713,
"column": 93
} | {
"line": 714,
"column": 4
} | [
{
"pp": "R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\nthis : spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1\ny' : (FractionalIdeal R₁⁰ K)ˣ :=\n Units.mkOfMulEqO... | [
"R₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nx y : R₁\nhy : y ∈ R₁⁰\nthis : spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1\ny' : (FractionalIdeal R₁⁰ K)ˣ :=\n Units.mkOfMulEqOne (spanSing... | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 28
} | {
"line": 343,
"column": 2
} | [
{
"pp": "case h_sub\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 ... | [] | exact this h_minus_bot | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 745,
"column": 6
} | {
"line": 745,
"column": 38
} | {
"line": 745,
"column": 39
} | [
{
"pp": "case a\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\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J * (spanSingleton R₁⁰ d⁻¹ * spanSingleton R₁⁰ d) ≤ J",
"ppTerm"... | [
"case a\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\nJ : FractionalIdeal R₁⁰ K\nd : K\nhd : ¬d = 0\nh_spand : spanSingleton R₁⁰ d ≠ 0\n⊢ J * spanSingleton R₁⁰ (d⁻¹ * d) ≤ J"
] | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 21
} | {
"line": 258,
"column": 2
} | [
{
"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\nJ : Ideal A\nhJ : J ≠ ⊤\nι : Type u_4\ns : Finset ι\nf : ι → K\nj : ι\nhjs : j ∈ s\nhjf : f j ≠ 0\nI : FractionalIdeal A⁰ K := spanFinset A s f\nhI0 : I ≠ 0\... | [
"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\nJ : Ideal A\nhJ : J ≠ ⊤\nι : Type u_4\ns : Finset ι\nf : ι → K\nj : ι\nhjs : j ∈ s\nhjf : f j ≠ 0\nI : FractionalIdeal A⁰ K := spanFinset A s f\nhI0 : I ≠ 0\n⊢ ↑J * I⁻¹ ... | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 320,
"column": 22
} | {
"line": 320,
"column": 33
} | {
"line": 320,
"column": 34
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI J : Ideal A\nthis : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)\nhgcd : gcd I J = I ⊔ J\n⊢ lcm I J ≤ I ⊓ J",
"ppTerm": "?m.155",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI J : Ideal A\nthis : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)\nhgcd : gcd I J = I ⊔ J\n⊢ lcm I J ≤ I ∧ lcm I J ≤ J"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 362,
"column": 10
} | {
"line": 362,
"column": 17
} | {
"line": 362,
"column": 18
} | [
{
"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.... | hi_val, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 873,
"column": 24
} | {
"line": 873,
"column": 56
} | {
"line": 873,
"column": 57
} | [
{
"pp": "case h\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\nx : R₁\nI : FractionalIdeal R₁⁰ K\nhI : ∀ J ≤ I, (↑J).FG\nhx : ¬x = 0\nh_gx : (algebraMap R₁ K) x ≠ 0\nh_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x)... | [
"case h\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\nx : R₁\nI : FractionalIdeal R₁⁰ K\nhI : ∀ J ≤ I, (↑J).FG\nhx : ¬x = 0\nh_gx : (algebraMap R₁ K) x ≠ 0\nh_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0\nJ : Fr... | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 639,
"column": 4
} | {
"line": 641,
"column": 65
} | {
"line": 642,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\nX : Ideal R\nhX : X ∣ I\nY : Ideal R\nhY : Y ∣ I\nh : X ≤ Y\n⊢ comap (Ideal.Quotient.mk J) (map f... | [
"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\nX : Ideal R\nhX : X ∣ I\nY : Ideal R\nhY : Y ∣ I\nh : X ≤ Y\n⊢ map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quoti... | rw [Subtype.coe_mk, comap_le_comap_iff_of_surjective (Ideal.Quotient.mk J)
Ideal.Quotient.mk_surjective, map_le_iff_le_comap, Subtype.coe_mk,
comap_map_of_surjective _ hf (map (Ideal.Quotient.mk I) Y)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 40
} | {
"line": 764,
"column": 41
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\n⊢ emultiplicity (↑((normalizedFactorsEquivOfQuotEquiv f hI hJ)... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\n⊢ emultiplicity\n (↑({ toFun := fun j ↦ ⟨↑((idealFactorsEquivOfQuotEq... | normalizedFactorsEquivOfQuotEquiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Lattice | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 17
} | {
"line": 132,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nK : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra R K\nM N : Type u\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsFractionRing R K\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : Module K N\ninst✝¹ : IsScalarTower R K N\ninst✝... | [
"R : Type u_1\ninst✝¹¹ : CommRing R\nK : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Algebra R K\nM N : Type u\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsFractionRing R K\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : Module K N\ninst✝¹ : IsScalarTower R K N\ninst✝ : Module.Fi... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Lattice | {
"line": 144,
"column": 65
} | {
"line": 144,
"column": 86
} | {
"line": 144,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nK : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : Algebra R K\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : Module R V\ninst✝² : IsScalarTower R K V\ninst✝¹ : IsFractionRing R K\nκ : Type u_4\nM : Submodule R V\ninst✝ : IsLattice K M\nb : Basis κ R ↥M\n... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nK : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : Algebra R K\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : Module R V\ninst✝² : IsScalarTower R K V\ninst✝¹ : IsFractionRing R K\nκ : Type u_4\nM : Submodule R V\ninst✝ : IsLattice K M\nb : Basis κ R ↥M\ns : Finset κ... | Submodule.coe_eq_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 385,
"column": 30
} | {
"line": 385,
"column": 46
} | {
"line": 386,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ v ∘ₗ id ≈ v ∘ₗ u' ∘ₗ v'",
"ppTerm... | [] | grw [h'.1.equiv] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 385,
"column": 30
} | {
"line": 385,
"column": 46
} | {
"line": 386,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ v ∘ₗ id ≈ v ∘ₗ u' ∘ₗ v'",
"ppTerm... | [] | grw [h'.1.equiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 385,
"column": 30
} | {
"line": 385,
"column": 46
} | {
"line": 386,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ v ∘ₗ id ≈ v ∘ₗ u' ∘ₗ v'",
"ppTerm... | [] | grw [h'.1.equiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 388,
"column": 24
} | {
"line": 388,
"column": 39
} | {
"line": 389,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ (v ∘ₗ u) ∘ₗ v' ≈ id ∘ₗ v'",
"ppTe... | [] | grw [h.2.equiv] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 388,
"column": 24
} | {
"line": 388,
"column": 39
} | {
"line": 389,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ (v ∘ₗ u) ∘ₗ v' ≈ id ∘ₗ v'",
"ppTe... | [] | grw [h.2.equiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 388,
"column": 24
} | {
"line": 388,
"column": 39
} | {
"line": 389,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₂ : Type u_4\nV₃ : Type u_6\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nu u' : V₃ →ₗ[K] V₂\nv v' : V₂ →ₗ[K] V₃\nh : u.IsQuasiInverse v\nh' : u'.IsQuasiInverse v'\nhu : u ≈ u'\n⊢ (v ∘ₗ u) ∘ₗ v' ≈ id ∘ₗ v'",
"ppTe... | [] | grw [h.2.equiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1046,
"column": 6
} | {
"line": 1046,
"column": 22
} | {
"line": 1047,
"column": 2
} | [
{
"pp": "case hsucc\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : FiniteMultiplicity a b\n⊢ multiplicity a b < multiplicity a b + 1",
"ppTerm": "?hsucc",
"assigned": true,
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"Nat.instOn... | [] | apply lt_add_one | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1041,
"column": 2
} | {
"line": 1046,
"column": 22
} | {
"line": 1047,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : FiniteMultiplicity a b\n⊢ emultiplicity (span {a}) (span {b}) = emultiplicity a b",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"pow_multiplicity_dvd",
"Eq.... | [
"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : ¬FiniteMultiplicity a b\n⊢ emultiplicity (span {a}) (span {b}) = emultiplicity a b"
] | · rw [h.emultiplicity_eq_multiplicity]
apply emultiplicity_eq_of_dvd_of_not_dvd <;>
rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
· exact pow_multiplicity_dvd a b
· apply h.not_pow_dvd_of_multiplicity_lt
apply lt_add_one | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.PID | {
"line": 96,
"column": 2
} | {
"line": 99,
"column": 75
} | {
"line": 101,
"column": 0
} | [
{
"pp": "case refine_2\nR : 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\nhM : Module.IsTorsion R M\n⊢ ∀ (i : ↥(factors ⊤.annihilator).toFinset), Irreducible (IsPrincipal.generator ↑i)",
... | [] | · rintro ⟨p, hp⟩
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime_of_prime hP
exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.PID | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 73
} | {
"line": 140,
"column": 6
} | [
{
"pp": "case ha\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nhM : IsTorsion' M ↥(Submonoid.powers p)\ndec : (x : M) → Decidable (x = 0)\nx y : M\nk : ℕ\nhM' : IsTorsionBy R M (p ^ pOrd... | [] | · exact mem_nonZeroDivisors_of_ne_zero (pow_ne_zero _ hp.ne_zero) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.PID | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 10
} | [
{
"pp": "case pos.refine_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nhM : IsTorsion' M ↥(Submonoid.powers p)\ndec : (x : M) → Decidable (x = 0)\nx y : M\nk : ℕ\nhM' : IsTorsionBy R M... | [
"case pos.refine_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nhM : IsTorsion' M ↥(Submonoid.powers p)\ndec : (x : M) → Decidable (x = 0)\nx y : M\nk : ℕ\nhM' : IsTorsionBy R M (p ^ pOrder... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 8
} | {
"line": 183,
"column": 4
} | [
{
"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\nP : Presentation R S ι σ\nh : Function.Bijective ⇑(algebraMap R S)\n⊢ ⊥ = (Generators.ofSurjectiveAlgebraMap ⋯).ker",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Presentation R S ι σ\nh : Function.Bijective ⇑(algebraMap R S)\n⊢ (Generators.ofSurjectiveAlgebraMap ⋯).ker = ⊥"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 181,
"column": 4
} | {
"line": 186,
"column": 21
} | {
"line": 188,
"column": 0
} | [
{
"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\nP : Presentation R S ι σ\nh : Function.Bijective ⇑(algebraMap R S)\n⊢ Ideal.span (Set.range PEmpty.elim) = (Generators.ofSurjectiveAlgebraMap ⋯).ker",
"ppTerm": "?m.33",
"assigned": tr... | [] | simp only [Set.range_eq_empty, Ideal.span_empty]
symm
rw [← RingHom.injective_iff_ker_eq_bot]
change Function.Injective (aeval PEmpty.elim)
rw [aeval_injective_iff_of_isEmpty]
exact h.injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 181,
"column": 4
} | {
"line": 186,
"column": 21
} | {
"line": 188,
"column": 0
} | [
{
"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\nP : Presentation R S ι σ\nh : Function.Bijective ⇑(algebraMap R S)\n⊢ Ideal.span (Set.range PEmpty.elim) = (Generators.ofSurjectiveAlgebraMap ⋯).ker",
"ppTerm": "?m.33",
"assigned": tr... | [] | simp only [Set.range_eq_empty, Ideal.span_empty]
symm
rw [← RingHom.injective_iff_ker_eq_bot]
change Function.Injective (aeval PEmpty.elim)
rw [aeval_injective_iff_of_isEmpty]
exact h.injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.DirectSum | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 12
} | {
"line": 64,
"column": 8
} | [
{
"pp": "A : Type u\ninst✝⁵ : Ring A\nι : Type w\ninst✝⁴ : DecidableEq ι\nrelations : ι → Relations A\nM : ι → Type v\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module A (M i)\nN : Type v\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : (directSum relations).Solution N\ni : ι\nr : (relations i)... | [
"A : Type u\ninst✝⁵ : Ring A\nι : Type w\ninst✝⁴ : DecidableEq ι\nrelations : ι → Relations A\nM : ι → Type v\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module A (M i)\nN : Type v\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : (directSum relations).Solution N\ni : ι\nr : (relations i).R\n⊢ (Finsu... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Extension.Basic | {
"line": 280,
"column": 32
} | {
"line": 280,
"column": 78
} | {
"line": 280,
"column": 78
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹¹ : CommRing R'\ninst✝¹⁰ : CommRing S'\ninst✝⁹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝⁸ : CommRing R''\ninst✝⁷ :... | [] | simp [show algebraMap P.Ring S x = 0 from x.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Basic | {
"line": 280,
"column": 32
} | {
"line": 280,
"column": 78
} | {
"line": 280,
"column": 78
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹¹ : CommRing R'\ninst✝¹⁰ : CommRing S'\ninst✝⁹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝⁸ : CommRing R''\ninst✝⁷ :... | [] | simp [show algebraMap P.Ring S x = 0 from x.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Basic | {
"line": 280,
"column": 32
} | {
"line": 280,
"column": 78
} | {
"line": 280,
"column": 78
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹¹ : CommRing R'\ninst✝¹⁰ : CommRing S'\ninst✝⁹ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝⁸ : CommRing R''\ninst✝⁷ :... | [] | simp [show algebraMap P.Ring S x = 0 from x.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Tautological | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 12
} | {
"line": 52,
"column": 8
} | [
{
"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 + m₂) = s.var m₁ + s.var m₂",
"ppTerm": "?m.63",
"assigned": true,
"usedConstant... | [
"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)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Module.Presentation.Tautological | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 12
} | {
"line": 56,
"column": 8
} | [
{
"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⊢ s.var (a • m) = (RingHom.id A) a • s.var m",
"ppTerm": "?m.90",
"assigned": true,
"used... | [
"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)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Extension.Generators | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 40
} | {
"line": 542,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nv₁ : ι' →₀ ℕ\nv₂ : ι →₀ ℕ\na : R\n⊢... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nv₁ : ι' →₀ ℕ\nv₂ : ι →₀ ℕ\na : R\n⊢ (a • (v₁.su... | simp only [ofComp_val, aeval_monomial] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Generators | {
"line": 702,
"column": 8
} | {
"line": 702,
"column": 41
} | {
"line": 703,
"column": 8
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ) := Cla... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ) := Classical.decEq... | obtain ⟨j, rfl⟩ := e.surjective j | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 6
} | {
"line": 244,
"column": 2
} | [
{
"pp": "case e'_2\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n\nM' : Submodule R M\ns : Finset (↥M' × N... | [
"case e'_2\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n\nM' : Submodule R M\ns : Finset (↥M' × N)\ne : Fin (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 125,
"column": 11
} | {
"line": 125,
"column": 33
} | {
"line": 125,
"column": 33
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nR' : Type u_3\ninst✝⁵ : CommSemiring R'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra R' S\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower R R' S\ninst✝ : IsLocalization M R'\ns : Set S\nx : S\nhx : x ∈ Subalgebra.t... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nR' : Type u_3\ninst✝⁵ : CommSemiring R'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra R' S\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower R R' S\ninst✝ : IsLocalization M R'\ns : Set S\nx : S\nhx : x ∈ Submodule.span R' ↑(Subm... | Algebra.adjoin_eq_span | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 31
} | {
"line": 316,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nt : Set R\nht : span t = ⊤\nH : ∀ (g : ↑t), (map (algebraMap R (Localization.Away ↑g)) I).FG\n⊢ I.FG",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"CommSemiring.toSemiring",
... | [
"R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nt : Set R\nht : span t = ⊤\nH : ∀ (g : ↑t), (map (algebraMap R (Localization.Away ↑g)) I).FG\n⊢ Module.Finite R ↥I"
] | apply Module.Finite.iff_fg.mp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Support | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 34
} | {
"line": 227,
"column": 35
} | [
{
"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)\nthis : { asIdeal := p, isPrime := ⋯ } ∈ (Module.support R M)ᶜ\n⊢ ∃ f ∉ p, Subsingleton (Loca... | [
"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)\nthis : { asIdeal := p, isPrime := ⋯ } ∈ (zeroLocus ↑(Module.annihilator R M))ᶜ\n⊢ ∃ f ∉ p, Subsingleton ... | Module.support_eq_zeroLocus, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 34
} | {
"line": 188,
"column": 2
} | [
{
"pp": "case e_a.e_a.e_a.e_6\nR : Type u_1\nM : Type u_2\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nn : ℕ\nf : M →ₗ[R] M →ₗ[R] M [⋀^Fin n]→ₗ[R] N\nv : Fin (n + 2) → M\ni j : Fin (n + 1)\nhj : i ≤ j\nH₁ : i.castSucc.removeNth v ... | [
"case e_a.e_a.e_a.e_6\nR : Type u_1\nM : Type u_2\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nn : ℕ\nf : M →ₗ[R] M →ₗ[R] M [⋀^Fin n]→ₗ[R] N\nv : Fin (n + 2) → M\ni j : Fin (n + 1)\nhj : i ≤ j\nH₁ : i.castSucc.removeNth v j = v j.succ... | rw [removeNth_removeNth_eq_swap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 41
} | {
"line": 52,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → Module.Dual R M\ni j : Fin n\nhf : f i = f j\nhij : i ≠ j\nv : Fin n → M\n⊢ (LinearMap.compAlternatingMap\n (((toTensorPower R M n).dualMap.compMultilinearMap (TensorPower.multilinear... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → Module.Dual R M\ni j : Fin n\nhf : f i = f j\nhij : i ≠ j\nv : Fin n → M\n⊢ (Matrix.of fun i j ↦ (f j) (v i)).det = 0"
] | suffices Matrix.det (n := Fin n) (.of (fun i j ↦ f j (v i))) = 0 by
simpa [Matrix.det_apply] using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 97,
"column": 79
} | {
"line": 97,
"column": 90
} | {
"line": 97,
"column": 90
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nhc : ∀ (m : ↥M) (n : ↥N), Commute ↑m ↑n\nn : ↥N\nm : ↥M\n⊢ (N.mulMap M) (n ⊗ₜ[R] m) = (M.mulMap N) (m ⊗ₜ[R] n)",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Sub... | [
"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nhc : ∀ (m : ↥M) (n : ↥N), Commute ↑m ↑n\nn : ↥N\nm : ↥M\n⊢ ↑n * ↑m = ↑m * ↑n"
] | mulMap_tmul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 119,
"column": 2
} | {
"line": 124,
"column": 54
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\n⊢ (M.mulMap N).range = M * N",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule.mulMap",
"Eq.mpr",
"Submodule",
"RingHom... | [] | refine le_antisymm ?_ (mul_le.2 fun m hm n hn ↦ ⟨⟨m, hm⟩ ⊗ₜ[R] ⟨n, hn⟩, rfl⟩)
rintro _ ⟨x, rfl⟩
induction x with
| zero => rw [map_zero]; exact zero_mem _
| tmul a b => exact mul_mem_mul a.2 b.2
| add a b ha hb => rw [map_add]; exact add_mem ha hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 119,
"column": 2
} | {
"line": 124,
"column": 54
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\n⊢ (M.mulMap N).range = M * N",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule.mulMap",
"Eq.mpr",
"Submodule",
"RingHom... | [] | refine le_antisymm ?_ (mul_le.2 fun m hm n hn ↦ ⟨⟨m, hm⟩ ⊗ₜ[R] ⟨n, hn⟩, rfl⟩)
rintro _ ⟨x, rfl⟩
induction x with
| zero => rw [map_zero]; exact zero_mem _
| tmul a b => exact mul_mem_mul a.2 b.2
| add a b ha hb => rw [map_add]; exact add_mem ha hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 70
} | {
"line": 182,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb nc : ℕ\na : ⨂[R]^na M\nb : ⨂[R]^nb M\nc : ⨂[R]^nc M\nmul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=\n fun n m ↦ (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv\ne : ⨂... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb nc : ℕ\na : ⨂[R]^na M\nb : ⨂[R]^nb M\nc : ⨂[R]^nc M\nmul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=\n fun n m ↦ (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv\ne : ⨂[R]^(na + nb... | have rhs_eq : ∀ a b c, rhs a b c = a ₜ* (b ₜ* c) := fun _ _ _ => rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LocalRing.Module | {
"line": 228,
"column": 48
} | {
"line": 228,
"column": 59
} | {
"line": 228,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nH : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))\nι : Type u\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nthis : Submodule.s... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nH : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))\nι : Type u\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nthis : ⊤ ≤ Submodule.span R (S... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 240,
"column": 54
} | {
"line": 240,
"column": 93
} | {
"line": 240,
"column": 93
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹⁴ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹³ : (i : ι) → AddCommMonoid (s i)\ninst✝¹² : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nE : Type u_9\ninst✝⁹ : AddCommM... | [] | simp only [smul_tprodCoeff', smul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 240,
"column": 54
} | {
"line": 240,
"column": 93
} | {
"line": 240,
"column": 93
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹⁴ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹³ : (i : ι) → AddCommMonoid (s i)\ninst✝¹² : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nE : Type u_9\ninst✝⁹ : AddCommM... | [] | simp only [smul_tprodCoeff', smul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 240,
"column": 54
} | {
"line": 240,
"column": 93
} | {
"line": 240,
"column": 93
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹⁴ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹³ : (i : ι) → AddCommMonoid (s i)\ninst✝¹² : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nE : Type u_9\ninst✝⁹ : AddCommM... | [] | simp only [smul_tprodCoeff', smul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Module | {
"line": 275,
"column": 13
} | {
"line": 275,
"column": 18
} | {
"line": 275,
"column": 19
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v i = 0 → ∀ i ∈... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s, f i = 0\... | a_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 275,
"column": 80
} | {
"line": 275,
"column": 85
} | {
"line": 275,
"column": 85
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v i = 0 → ∀ i ∈... | [] | n_def | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 52
} | {
"line": 360,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nmotive : (⨂[R] (i : ι), s i) → Prop\nz : ⨂[R] (i : ι), s i\nsmul_tprod : ∀ (r : R) (f : (i : ι) → s i), motive (r • (tprod R) f)\nadd : ∀ (x y : ⨂[R] (i : ι),... | [
"ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nmotive : (⨂[R] (i : ι), s i) → Prop\nz : ⨂[R] (i : ι), s i\nadd : ∀ (x y : ⨂[R] (i : ι), s i), motive x → motive y → motive (x + y)\nsmul_tprod : ∀ (r : R) (f : (i : ι) →... | simp_rw [← tprodCoeff_eq_smul_tprod] at smul_tprod | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 648,
"column": 35
} | {
"line": 649,
"column": 49
} | {
"line": 651,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : CommSemiring R\ns : ι → Type u_7\ninst✝⁵ : (i : ι) → AddCommMonoid (s i)\ninst✝⁴ : (i : ι) → Module R (s i)\nt : ι → Type u_11\nt' : ι → Type u_12\ninst✝³ : (i : ι) → AddCommMonoid (t i)\ninst✝² : (i : ι) → Module R (t i)\ninst✝¹ : (i : ι) → AddCommMonoid (t' i)\nin... | [] | simp only [map_smul, LinearMap.compMultilinearMap_apply,
lift.tprod, smul_apply, LinearMap.smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 72
} | {
"line": 234,
"column": 78
} | {
"line": 234,
"column": 78
} | [
{
"pp": "⊢ Function.LeftInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.LeftInverse",
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"inst... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 72
} | {
"line": 234,
"column": 78
} | {
"line": 234,
"column": 78
} | [
{
"pp": "⊢ Function.LeftInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.LeftInverse",
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"inst... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 72
} | {
"line": 234,
"column": 78
} | {
"line": 234,
"column": 78
} | [
{
"pp": "⊢ Function.LeftInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.LeftInverse",
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"inst... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 83
} | {
"line": 234,
"column": 89
} | {
"line": 234,
"column": 89
} | [
{
"pp": "⊢ Function.RightInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"instOfNatNat",
"Fintype.decidableRightInverseFintype",
"Fin.fintyp... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 83
} | {
"line": 234,
"column": 89
} | {
"line": 234,
"column": 89
} | [
{
"pp": "⊢ Function.RightInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"instOfNatNat",
"Fintype.decidableRightInverseFintype",
"Fin.fintyp... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 234,
"column": 83
} | {
"line": 234,
"column": 89
} | {
"line": 234,
"column": 89
} | [
{
"pp": "⊢ Function.RightInverse ![2, 0, 1, 3, 4] ![1, 2, 0, 3, 4]",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id",
"Fin.instOfNat",
"instOfNatNat",
"Fintype.decidableRightInverseFintype",
"Fin.fintyp... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 72
} | {
"line": 235,
"column": 78
} | {
"line": 235,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.Left... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 72
} | {
"line": 235,
"column": 78
} | {
"line": 235,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.Left... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 72
} | {
"line": 235,
"column": 78
} | {
"line": 235,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Fintype.decidableLeftInverseFintype",
"Function.Left... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 83
} | {
"line": 235,
"column": 89
} | {
"line": 235,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 83
} | {
"line": 235,
"column": 89
} | {
"line": 235,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 235,
"column": 83
} | {
"line": 235,
"column": 89
} | {
"line": 235,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![3, 4, 2, 0, 1] ![3, 4, 2, 0, 1]",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instDecidableEqFin",
"id... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 72
} | {
"line": 236,
"column": 78
} | {
"line": 236,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.153",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 72
} | {
"line": 236,
"column": 78
} | {
"line": 236,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.153",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 72
} | {
"line": 236,
"column": 78
} | {
"line": 236,
"column": 78
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.LeftInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.153",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 83
} | {
"line": 236,
"column": 89
} | {
"line": 236,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.154",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 83
} | {
"line": 236,
"column": 89
} | {
"line": 236,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.154",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 236,
"column": 83
} | {
"line": 236,
"column": 89
} | {
"line": 236,
"column": 89
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\n⊢ Function.RightInverse ![0, 3, 2, 1, 4] ![0, 3, 2, 1, 4]",
"ppTerm": "?m.154",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Solvable | {
"line": 237,
"column": 64
} | {
"line": 237,
"column": 70
} | {
"line": 238,
"column": 2
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\nz : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_i... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 238,
"column": 58
} | {
"line": 238,
"column": 64
} | {
"line": 238,
"column": 64
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\nz : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_i... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.Solvable | {
"line": 238,
"column": 58
} | {
"line": 238,
"column": 64
} | {
"line": 238,
"column": 64
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\nz : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_i... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 238,
"column": 58
} | {
"line": 238,
"column": 64
} | {
"line": 238,
"column": 64
} | [
{
"pp": "x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }\ny : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }\nz : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_i... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Normal.Basic | {
"line": 43,
"column": 31
} | {
"line": 43,
"column": 42
} | {
"line": 43,
"column": 43
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\nh : Normal F K\ninst✝ : FiniteDimensional F K\ns : Module.Basis (↑(Module.Basis.ofVectorSpaceIndex F K)) F K := Module.Basis.ofVectorSpace F K\n⊢ Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x, minpoly F (s x)).rootSe... | [
"F : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\nh : Normal F K\ninst✝ : FiniteDimensional F K\ns : Module.Basis (↑(Module.Basis.ofVectorSpaceIndex F K)) F K := Module.Basis.ofVectorSpace F K\n⊢ ⊤ ≤ Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x, minpoly F (s x)).rootSet K))"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 7
} | {
"line": 62,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite E\nα : Eˣ\nhα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α\n⊢ ∃ α, F⟮α⟯ = ⊤",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Units.val",
"Lattice.toSemilatticeSup",
"Comp... | [
"case h\nF : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite E\nα : Eˣ\nhα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α\n⊢ F⟮↑α⟯ = ⊤"
] | use α | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.Normal.Basic | {
"line": 149,
"column": 10
} | {
"line": 149,
"column": 91
} | {
"line": 149,
"column": 91
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Algebra F K\nι : Type u_3\nhι : Nonempty ι\nt : ι → IntermediateField F K\nh : ∀ (i : ι), Normal F ↥(t i)\nx : ↥(⨅ i, t i)\ni : ι\n⊢ (Polynomial.map (algebraMap F ↥(t i)) (minpoly F x)).Splits",
"ppTerm": "?m.77",
"assigned... | [
"F : Type u_1\nK : Type u_2\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Algebra F K\nι : Type u_3\nhι : Nonempty ι\nt : ι → IntermediateField F K\nh : ∀ (i : ι), Normal F ↥(t i)\nx : ↥(⨅ i, t i)\ni : ι\n⊢ (Polynomial.map (algebraMap F ↥(t i)) (minpoly F ((inclusion ⋯) x))).Splits"
] | ← minpoly.algHom_eq (inclusion (iInf_le t i)) (inclusion (iInf_le t i)).injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 134,
"column": 79
} | {
"line": 139,
"column": 46
} | {
"line": 140,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | [] | by
have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
simp [map_div₀, map_neg, ← coeff_map, ← coeff_map, p_linear,
mul_sub, coeff_C, mul_div_cancel_left₀ β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 37
} | {
"line": 57,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : S ⊗[R] p.ResidueField\nr : R\na : S\ny : R\nt : ↥p.primeCompl\ne :\n 1 ⊗ₜ[R]\n (r •\n (IsLocalRing.residue (Localization.AtPrime p))\n ((fun x ↦\n ... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : S ⊗[R] p.ResidueField\nr : R\na : S\ny : R\nt : ↥p.primeCompl\ne :\n 1 ⊗ₜ[R]\n (r •\n (IsLocalRing.residue (Localization.AtPrime p))\n ((fun x ↦\n ... | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 34
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np✝ : Ideal R\ninst✝ : p✝.IsPrime\np : PrimeSpectrum R\nq : PrimeSpectrum (p.asIdeal.Fiber S)\nx : p.asIdeal.Fiber S\nr : R\nhr : r ∉ p.asIdeal\ns : S\ne : r • x = 1 ⊗ₜ[R] s\n⊢ x ∈\n ((fun q ↦\n {... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np✝ : Ideal R\ninst✝ : p✝.IsPrime\np : PrimeSpectrum R\nq : PrimeSpectrum (p.asIdeal.Fiber S)\nx : p.asIdeal.Fiber S\nr : R\nhr : r ∉ p.asIdeal\ns : S\ne : r • x = 1 ⊗ₜ[R] s\nthis : ∀ {R : Type ?u.120} [inst : CommSemiring R... | have := @PrimeSpectrum.isPrime | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 18
} | {
"line": 459,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\nh : B.Nonempty\n⊢ M.RankPos",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Matroid.rankPos_iff",
"Eq.mpr",
"congrArg",
"Matroid.IsBase",
"id",
"Matroid.RankPos",
"propext",
"S... | [
"α : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\nh : B.Nonempty\n⊢ ¬M.IsBase ∅"
] | rw [rankPos_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 516,
"column": 35
} | {
"line": 516,
"column": 69
} | {
"line": 516,
"column": 70
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : B ⊆ M.E\nh : M.IsBase (M.E \\ B)\nI : Set α\nhI : I ⊆ M.E\nB' : Set α\nx✝ : M.IsBase B' ∧ Disjoint I B'\nhBI : B ⊆ I\nhB' : M.IsBase B'\nhIB' : Disjoint I (M.E \\ B)\n⊢ I ⊆ B",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstant... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : B ⊆ M.E\nh : M.IsBase (M.E \\ B)\nI : Set α\nhI : I ⊆ M.E\nB' : Set α\nx✝ : M.IsBase B' ∧ Disjoint I B'\nhBI : B ⊆ I\nhB' : M.IsBase B'\nhIB' : I ⊆ (M.E \\ B)ᶜ\n⊢ I ⊆ B",
"case refine_1\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : B ⊆ M.E\nh : M.IsBase... | ← subset_compl_iff_disjoint_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 43
} | {
"line": 706,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\nhB : M.Indep B ∧ ∀ ⦃t : Set α⦄, M.Indep t → B ⊆ t → B = t\nhInotmax : ∃ x, M.Indep x ∧ I ⊆ x ∧ ¬I = x\nhIb : M.IsBase I\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"False",
"Exist... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\nhB : M.Indep B ∧ ∀ ⦃t : Set α⦄, M.Indep t → B ⊆ t → B = t\nhIb : M.IsBase I\nI' : Set α\nhII' : M.Indep I'\nhI' : I ⊆ I'\nhne : ¬I = I'\n⊢ False"
] | obtain ⟨I', hII', hI', hne⟩ := hInotmax | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 945,
"column": 2
} | {
"line": 945,
"column": 28
} | {
"line": 947,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.Indep I\nhIX : I ⊆ X\nhX : X ⊆ M.E\nJ : Set α\nhJ : I ⊆ J\nhJmax : Maximal (fun K ↦ M.Indep K ∧ K ⊆ X) J\n⊢ ∃ J, M.IsBasis J X ∧ I ⊆ J",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Matroid.E",
"Matroid.Indep",
"... | [] | exact ⟨J, ⟨hJmax, hX⟩, hJ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1097,
"column": 51
} | {
"line": 1097,
"column": 84
} | {
"line": 1097,
"column": 84
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\nhXY : X ⊆ Y\nhY : Y ⊆ M.E\nI I' : Set α\nhI : M.IsBasis I X\nhI' : M.IsBasis I' Y\nhII' : I ⊆ I'\n⊢ M.IsBasis (I ∪ I') Y",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
... | [
"α : Type u_1\nM : Matroid α\nX Y : Set α\nhXY : X ⊆ Y\nhY : Y ⊆ M.E\nI I' : Set α\nhI : M.IsBasis I X\nhI' : M.IsBasis I' Y\nhII' : I ⊆ I'\n⊢ M.IsBasis I' Y"
] | union_eq_self_of_subset_left hII' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 48
} | {
"line": 75,
"column": 4
} | [
{
"pp": "α : Type u_1\nM✝ : Matroid α\nI B✝ X✝ : Set α\nM : Matroid α\nX I' : Set α\nhI'E : I' ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhI'B : Disjoint I' B\nhI'X : I' ⊆ X\n⊢ ∃ J, I' ⊆ J ∧ Maximal (fun K ↦ (fun I ↦ I ⊆ M.E ∧ ∃ B, M.IsBase B ∧ Disjoint I B) K ∧ K ⊆ X) J",
"ppTerm": "?m.449",
"assigned": true,
... | [
"α : Type u_1\nM✝ : Matroid α\nI✝ B✝ X✝ : Set α\nM : Matroid α\nX I' : Set α\nhI'E : I' ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhI'B : Disjoint I' B\nhI'X : I' ⊆ X\nI : Set α\nhI : M.IsBasis I (M.E \\ X)\n⊢ ∃ J, I' ⊆ J ∧ Maximal (fun K ↦ (fun I ↦ I ⊆ M.E ∧ ∃ B, M.IsBase B ∧ Disjoint I B) K ∧ K ⊆ X) J"
] | obtain ⟨I, hI⟩ := M.exists_isBasis (M.E \ X) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 81,
"column": 28
} | {
"line": 81,
"column": 46
} | {
"line": 81,
"column": 47
} | [
{
"pp": "case refine_1\nα : Type u_1\nM✝ : Matroid α\nI B✝ X✝ : Set α\nM : Matroid α\nX I' : Set α\nhI'E : I' ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhI'B : Disjoint I' B\nhI'X : I' ⊆ X\nB' : Set α\nhB' : M.IsBase B'\nhI : M.IsBasis (B' \\ X) (M.E \\ X)\nhIB' : B' \\ X ⊆ B'\nhB'IB : B' ⊆ B' \\ X ∪ B\n⊢ I' ⊆ X \\ B' ... | [
"case refine_1\nα : Type u_1\nM✝ : Matroid α\nI B✝ X✝ : Set α\nM : Matroid α\nX I' : Set α\nhI'E : I' ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhI'B : Disjoint I' B\nhI'X : I' ⊆ X\nB' : Set α\nhB' : M.IsBase B'\nhI : M.IsBasis (B' \\ X) (M.E \\ X)\nhIB' : B' \\ X ⊆ B'\nhB'IB : B' ⊆ B' \\ X ∪ B\n⊢ I' ⊆ X \\ B'"
] | and_iff_left hI'E, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 33
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ (emptyOn α)✶ = emptyOn α",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"Eq.refl",
"EmptyCo... | [] | rw [← ground_eq_empty_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 33
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ (emptyOn α)✶ = emptyOn α",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"Eq.refl",
"EmptyCo... | [] | rw [← ground_eq_empty_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 142,
"column": 34
} | {
"line": 142,
"column": 70
} | {
"line": 142,
"column": 70
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set α\nh : B ⊆ M.E\n⊢ M✶.IsBase B ↔ M.IsBase (M.E \\ B) ∧ B ⊆ M.E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.dual",
"Matroid.IsBas... | [] | rw [dual_isBase_iff, and_iff_left h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 142,
"column": 34
} | {
"line": 142,
"column": 70
} | {
"line": 142,
"column": 70
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set α\nh : B ⊆ M.E\n⊢ M✶.IsBase B ↔ M.IsBase (M.E \\ B) ∧ B ⊆ M.E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.dual",
"Matroid.IsBas... | [] | rw [dual_isBase_iff, and_iff_left h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 142,
"column": 34
} | {
"line": 142,
"column": 70
} | {
"line": 142,
"column": 70
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set α\nh : B ⊆ M.E\n⊢ M✶.IsBase B ↔ M.IsBase (M.E \\ B) ∧ B ⊆ M.E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.dual",
"Matroid.IsBas... | [] | rw [dual_isBase_iff, and_iff_left h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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