module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.Abelian | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 52
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\n⊢ N ≤ maxTrivSubmodule R L M ↔ ⁅⊤, N⁆ = ⊥",
"usedConstants": [
"le_b... | refine ⟨fun h => ?_, fun h m hm => ?_⟩
· rw [← le_bot_iff, ← ideal_oper_maxTrivSubmodule_eq_bot R L M ⊤]
exact LieSubmodule.mono_lie_right ⊤ h
· rw [mem_maxTrivSubmodule]
rw [LieSubmodule.lie_eq_bot_iff] at h
exact fun x => h x (LieSubmodule.mem_top x) m hm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Abelian | {
"line": 250,
"column": 12
} | {
"line": 250,
"column": 29
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieMo... | LieHom.lie_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"usedConstants": [
"Eq.mpr",
"Polyno... | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"usedConstants": [
"Eq.mpr",
"Polyno... | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"usedConstants": [
"Eq.mpr",
"Polyno... | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 174,
"column": 63
} | {
"line": 174,
"column": 87
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (x + (algebraMap A B) a)) q = 0\n⊢ (Polynomial.aeval x) (q.comp (X + C a)) = 0",
"usedConstants": [
"Eq.mpr",
"Polyn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 43
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : ¬IsIntegral A x\nh : IsIntegral A (x + (algebraMap A B) a)\n⊢ IsIntegral A x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\n⊢ minpoly A (x - (algebraMap A B) a) = (minpoly A x).comp (X + C a)",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Algebra.algebraMap",
"AddGroupWithOne.toAddGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 32
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (-x)) q = 0\n⊢ (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (-X)) = 0",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 70
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 0 = X",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 49
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 1 = X - 1",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 43
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nh : (minpoly A x).coeff 0 = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 15
} | [
{
"pp": "case refine_2\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L ≃ₐ[K] L\nhσ : IsOfFinOrder σ\nq : K[X]\nhq : q.Monic\nH : q.natDegree < orderOf σ\nhs : ∑ x, q.coeff ↑x • (σ ^ ↑x).toLinearMap = 0\n⊢ q = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 51
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : ↑U ⊆ (y - x) +ᵥ ↑U\n⊢ x - y ∈ U",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 51
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\nz : M\nhz : z ∈ ↑U\n⊢ z ∈ (y - x) +ᵥ ↑U",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\n⊢ x +ᵥ ↑U ⊆ y +ᵥ ↑U",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Submodule",
"Se... | rw [Set.vadd_set_subset_iff_subset_neg_vadd_set, vadd_vadd, neg_add_eq_sub]
intro z hz
simpa [Set.mem_vadd_set_iff_neg_vadd_mem] using U.add_mem h hz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\n⊢ x +ᵥ ↑U ⊆ y +ᵥ ↑U",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Submodule",
"Se... | rw [Set.vadd_set_subset_iff_subset_neg_vadd_set, vadd_vadd, neg_add_eq_sub]
intro z hz
simpa [Set.mem_vadd_set_iff_neg_vadd_mem] using U.add_mem h hz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 75,
"column": 31
} | {
"line": 75,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\ny a✝ b✝ : ↥R[x]\nha✝ : a✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\nhb✝ : b✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\na : R[X]\nha : ∀ (i : ℕ), a.coeff i ∈ I\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 102,
"column": 47
} | {
"line": 102,
"column": 58
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nI : Ideal R\nhI : I ≠ ⊤\ninst✝ : Invertible x\nh : ∃ i ∈ Ideal.map (algebraMap R ↥R[x]) I, ∃ j ∈ Ideal.span {⟨x, ⋯⟩}, i + j = 1\ny : ↥R[x]\nhy : y ∈ Ideal.map (algebraMap R ↥R[x]) I\nz : ↥R[x]\nhz : z ∈ I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 40
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.degree < ↑d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.natDegree < d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 40
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.natDegree < d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.degree < ↑d",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 130,
"column": 39
} | {
"line": 130,
"column": 50
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 77
} | [
{
"pp": "A : Type u_4\nB : Type u_5\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\npb : PowerBasis A B\nb : B\n⊢ ∃ a, pb.gen ∣ b - (algebraMap A B) a",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"Submodule.Quotient.instZeroQuotient",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 36,
"column": 17
} | {
"line": 36,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 1 < n\na : K\n⊢ X.natDegree < (X ^ n - C a).natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.natDegree_X",
"Polynomial.natDegree_X_pow_sub_C",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
... | by rwa [natDegree_X_pow_sub_C, natDegree_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 80,
"column": 16
} | {
"line": 80,
"column": 70
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\ne : n = 0\n⊢ Irreducible (C (1 - a))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 59
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nm : ℕ\nhm' : m ≠ 1\nb : K\nk : ℕ\nhn : m * k ≠ 0\nq : K[X]\nH : Irreducible ((X ^ k - C b) * q)\nhq : (X ^ k) ^ m - C b ^ m = (X ^ k - C b) * q\n⊢ q.degree = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 139,
"column": 21
} | {
"line": 139,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), Irreducible x → ¬x * x ∣ 0\n⊢ 0 = 0 ∧ ∀ (x : R), ¬Irreducible x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Irreducible",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"And",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 17
} | [
{
"pp": "case refine_2.inl\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), ¬Irreducible x\n⊢ ∀ (x : R), Irreducible x → ¬x * x ∣ 0",
"usedConstants": [
"Eq.mpr",
"False",
"Dvd.dvd",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\n⊢ (a.sum fun n a ↦ eval₂ (C.comp (Quotient.mk I)) X ((monomial n) a)) = 0",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Polynomial.C",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Ide... | refine Finset.sum_eq_zero fun n _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Squarefree.Basic | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nr : R\nhr : r ≠ 0\n⊢ Squarefree r ↔ ∀ (x : R), Irreducible x → ¬x * x ∣ r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 19
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\nn : ℕ\nx✝ : n ∈ a.support\nm : ℕ\nh : m = 0\n⊢ (if m = 0 then (Quotient.mk I) (a.coeff n) else 0) = coeff 0 m",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Ideal.Quo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 41
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\n⊢ ∑ x ∈ a.support, eval₂ (C.comp (Ideal.Quotient.mk I)) X ((monomial x) (coeff x a)) = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Ideal.Quotient.commSemiring",
"Nat.... | refine Finset.sum_eq_zero fun n _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 67
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\n⊢ eval₂ (C.comp (Ideal.Quotient.mk I)) X ((monomial n) (coeff n a)) = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Ideal.Quotient.commSemirin... | simp only [eval₂_monomial, Function.comp_apply, RingHom.coe_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 213,
"column": 8
} | {
"line": 213,
"column": 30
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\nthis : coeff n a ∈ I\n⊢ C ((Ideal.Quotient.mk I) (coeff n a)) = 0",
"usedConstants": [
"RingHom.instRingHomClass",
"Semiring.toModule",
"congr... | ← @Ideal.mk_ker R _ I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 12
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\n⊢ ¬IsUnit a → emultiplicity a x ≤ 1",
"usedConstants": [
"CommM... | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 12
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\n⊢ ¬IsUnit a → emultiplicity a x ≤ 1",
"usedConstants": [
"CommM... | intro hu | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.FieldTheory.Separable | {
"line": 171,
"column": 2
} | {
"line": 175,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : R[X]\nhq : ¬IsUnit q\nhsep : p.Separable\n⊢ emultiplicity q p ≤ 1",
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Preorder.toLT",
"Dvd.dvd",
"instAddMonoidWithOneENat",
"HMul.hMul",
"Monoid.... | contrapose! hq
apply isUnit_of_self_mul_dvd_separable hsep
rw [← sq]
apply pow_dvd_of_le_emultiplicity
exact Order.add_one_le_of_lt hq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 171,
"column": 2
} | {
"line": 175,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : R[X]\nhq : ¬IsUnit q\nhsep : p.Separable\n⊢ emultiplicity q p ≤ 1",
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Preorder.toLT",
"Dvd.dvd",
"instAddMonoidWithOneENat",
"HMul.hMul",
"Monoid.... | contrapose! hq
apply isUnit_of_self_mul_dvd_separable hsep
rw [← sq]
apply pow_dvd_of_le_emultiplicity
exact Order.add_one_le_of_lt hq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : R\n⊢ (X - C x).Separable",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"RingHom"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 58
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nt : Multiset R\nhs : (Multiset.map (fun a ↦ X - C a) (a ::ₘ a ::ₘ t)).prod.Separable\n⊢ (X - C a) * (X - C a) ∣ (Multiset.map (fun a ↦ X - C a) (a ::ₘ a ::ₘ t)).prod",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Dvd.d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 13
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nn : ℕ\na b c : R\nhn : ↑n = 0\nhb✝ : IsUnit b\nf : R[X] := C a * X ^ n + C b * X + C c\ne : R\nhb : e * b = 1\nhderiv : derivative f = C b\n⊢ -derivative f * f + (f + C e) * derivative f = 1",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"P... | hderiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 383,
"column": 15
} | {
"line": 383,
"column": 29
} | [
{
"pp": "case h.inr.succ\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nhp : Nat.Prime p\nf : F[X]\nhf : Irreducible f\nh1 : ¬f.Separable\nN : ℕ\nih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f\nhn : f.natDegree = N + 1\nn : ℕ\ng : F[X]\nhg4 ... | expand_expand, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Perfect | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 17
} | [
{
"pp": "M : Type u_1\np q : ℕ\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\ninst✝ : PerfectRing M q\n⊢ Bijective fun x ↦ x ^ 1",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Monoid.toPow",
"funext",
"HPow.hPow",
"CommMonoid.toMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 30
} | [
{
"pp": "case inr\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn₁ n₂ : ℕ\nthis :\n ∀ {F : Type u} [inst : Field F] (p : ℕ) [HF : CharP F p] {f : F[X]},\n Irreducible f →\n 0 < p →\n ∀ (n₁ n₂ : ℕ),\n n₁ ≤ n₂ →\n ∀ (g₁ : F[X]),\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 76
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 451,
"column": 45
} | {
"line": 451,
"column": 61
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : p.Separable\nhsplit : (map (algebraMap F K) p).Splits\n⊢ Fintype.card ↥(p.aroots K).toFinset = p.natDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"Finse... | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Separable | {
"line": 490,
"column": 36
} | {
"line": 490,
"column": 47
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nx : F\nh : F[X]\nh_sep : h.Separable\nh_root : eval x h = 0\nh_splits : (map i h).Splits\nh_roots : ∀ y ∈ (map i h).roots, y = i x\nh_ne_zero : h ≠ 0\nthis : (map i h).roots = {i x}\n⊢ map i h = map i (C h.leadingCoeff * (X - C x))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AnnihilatingPolynomial | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 57
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : annIdealGenerator 𝕜 a = 0\np : 𝕜[X]\np_monic : p.Monic\nhp : (aeval a) p = 0\n⊢ p ∈ ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 751,
"column": 8
} | {
"line": 751,
"column": 50
} | [
{
"pp": "A₁ : Type u_1\nB₁ : Type u_2\nA₂ : Type u_3\nB₂ : Type u_4\ninst✝⁵ : Field A₁\ninst✝⁴ : Ring B₁\ninst✝³ : Field A₂\ninst✝² : Ring B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx✝ : B₁\nh : IsSeparable... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 400,
"column": 77
} | {
"line": 400,
"column": 97
} | [
{
"pp": "case a\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : Fintype ι\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf : ι → M →ₗ[R] M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\nB : Basis (Free.ChooseBasisIndex R... | Equiv.prodComm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.AdjointAction.Basic | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁴ : Field K\ninst✝³ : PerfectField K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\na : Module.End K V\nha : a.IsSemisimple\nthis : (Polynomial.aeval ((Algebra.lmul K (Module.End K V)) a)) (minpoly K a) = 0\n⊢ (Polynomial.aeval (LinearMap.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 462,
"column": 6
} | {
"line": 462,
"column": 17
} | [
{
"pp": "K : Type u_5\nV : Type u_6\nW : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\nF : Type u_8\ninst✝¹ : EquivLike F (End K V) (End K W)\ninst✝ : AlgEquivClass F K (End K V) (End K W)\nf : F\nx : End K V\nw✝ : V ≃ₗ[K] W\nh : ↑f = Lin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 56
} | [
{
"pp": "K : Type u_5\nm : Type u_6\nn : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nF : Type u_8\ninst✝¹ : EquivLike F (Matrix m m K) (Matrix n n K)\ninst✝ : AlgEquivClass F K (Matrix m m K) (Matrix n n K)\nf : F\nx : Matrix m m K\n⊢ (f x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : M ≃ₗ[R] M'\nv : Basis ι R M\nv' : Basis ι R M'\n⊢ ?B * (toMatrix v v') ↑f = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 578,
"column": 6
} | {
"line": 578,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), f x = f' (e x)\n⊢ Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f')",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MonoidHom.instFu... | ← mul_one (LinearMap.det f'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 620,
"column": 62
} | {
"line": 620,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ne : Basis ι R M\ninst✝ : Nontrivial R\nh : e.det = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 19
} | [
{
"pp": "case intro\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_5\ninst✝ : Finite ι\ne : Basis ι R M\nf : M [⋀^ι]→ₗ[R] R\nh : f ⇑e = 0\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\n⊢ f = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 677,
"column": 51
} | {
"line": 680,
"column": 18
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\ne : Basis ι A M\nf : M →ₗ[A] M\nv : ι → M\n⊢ e.det (⇑f ∘ v) = LinearMap.det f * e.det v",
"usedConstants": [
"AlternatingMap",
"Eq.mpr"... | by
rw [det_apply, det_apply, ← f.det_toMatrix e, ← Matrix.det_mul,
e.toMatrix_eq_toMatrix_constr (f ∘ v), e.toMatrix_eq_toMatrix_constr v, ← toMatrix_comp,
e.constr_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Determinant | {
"line": 700,
"column": 30
} | {
"line": 700,
"column": 40
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\nb b' b'' : Basis ι A M\nthis : ⇑b'' = ⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b'\n| LinearMap.det ↑(b'.equiv b'' (Equiv.refl ι)) * b.det ⇑b'",
"usedCons... | det_basis, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 49
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (Set.range v)\ni k : ι\nhik : k ≠ i\n⊢ e.det v * ((Basis.mk hli hsp... | rw [mk_coord_apply_ne hik, mul_zero, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Determinant | {
"line": 823,
"column": 6
} | {
"line": 823,
"column": 35
} | [
{
"pp": "case a.inl.inl\nR : Type u_6\nV : Type u_7\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\nW : Submodule R V\ninst✝² : Free R ↥W\ninst✝¹ : Module.Finite R ↥W\ninst✝ : Free R (V ⧸ W)\ne : V →ₗ[R] V\nhe : W ≤ comap e W\nm : Type u_7 := Free.ChooseBasisIndex... | apply sumQuot_repr_inl_of_mem | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AdjoinRoot | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 15
} | [
{
"pp": "case h.h₁.a\nR : Type u_1\nT : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Semiring T\np : R[X]\nf g : AdjoinRoot p →+* T\nhAlg : f.comp (of p) = g.comp (of p)\nhRoot : f (root p) = g (root p)\nx : R\n⊢ ((f.comp (Ideal.Quotient.mk (span {p}))).comp C) x = ((g.comp (Ideal.Quotient.mk (span {p}))).comp C) x",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 389,
"column": 24
} | {
"line": 389,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g✝ : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\np : R[X]\nq : S[X]\nr : T[X]\nhf : q ∣ Polynomial.map f p\nhg : r ∣ Polynomial.map g q\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 397,
"column": 6
} | {
"line": 397,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ p ∣ Polynomial.map (↑f.symm) q",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 405,
"column": 6
} | {
"line": 405,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ Associated (Polynomial.map (↑f.symm) q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 422,
"column": 24
} | {
"line": 422,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing R\nf✝ g✝ : R[X]\ninst✝⁵ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f✝ = 0\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S →ₐ[R] T\ng : T →ₐ[R] U... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 434,
"column": 6
} | {
"line": 434,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing R\nf✝ g : R[X]\ninst✝⁵ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S ≃ₐ[R] T\np : S[X]\nq :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 445,
"column": 6
} | {
"line": 446,
"column": 13
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing R\nf✝ g : R[X]\ninst✝⁵ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S ≃ₐ[R] T\np : S[X]\nq :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 48
} | [
{
"pp": "case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := rang... | rintro ⟨x₁, y₁⟩ - h₁ ⟨x₂, y₂⟩ - h₂ h₃ h₄ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.AdjoinRoot | {
"line": 758,
"column": 59
} | {
"line": 758,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nx✝ : ↥R[x]\ny₁ : S\ny₂ : y₁ ∈ R[x]\nh : ⟨y₁, y₂⟩ ∈ Subtype.val ⁻¹' {x}\n⊢ (toAdjoin R x).toRingHom ((mk (minpoly R x)) X) = ⟨y₁, y₂⟩",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 852,
"column": 6
} | {
"line": 852,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf : R[X]\nx : AdjoinRoot f\n⊢ (quotMapOfEquivQuotMapCMapMk I f).symm\n ((Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I))) x) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) x",
"usedConstants": [
"Ideal.quotEquiv... | quotMapOfEquivQuotMapCMapMk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.AdjointAction.Derivation | {
"line": 73,
"column": 35
} | {
"line": 74,
"column": 47
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : LieAlgebra.center R L = ⊥\n⊢ Function.Injective ⇑(ad R L)",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Eq.mpr",
"LieDerivation",
"LieSubmodule.instBot",
"Li... | by
rw [← LieHom.ker_eq_bot, ad_ker_eq_center, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Dynamics.Newton | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 15
} | [
{
"pp": "case succ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : ℕ\nih : IsNilpotent (P.newtonMap^[n] x - x)\n⊢ IsNilpotent ((aeval (P.newtonMap^[n] x)) P)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ IsSemisimple 0",
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"Eq.mpr",
"Submodule",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ IsSemisimple LinearMap.id",
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"LinearMap.id",
"Eq.mpr",
"Submodule",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nM₂ : Type u_3\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\ng : End R M₂\ne : M ≃ₗ[R] M₂\nhe : ↑e ∘ₗ f = g ∘ₗ ↑e\nx : M\n⊢ (e ≪≫ₗ AEval'.of g) (f • x) = X • (e ≪≫ₗ AEval'.of g) x",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 156,
"column": 28
} | {
"line": 156,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 57
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ x ∈ Submodule.comap f p",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ x ∈ Submodule.comap f p",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 58,
"column": 6
} | {
"line": 59,
"column": 13
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : a * P ^ k + b * P' = 1\n⊢ (aeval f) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 40
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : (aeval f) b * (aeval f) P' = 1\n⊢ ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 309,
"column": 26
} | {
"line": 309,
"column": 41
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Module K M\nf g : End K M\ninst✝¹ : FiniteDimensional K M\ninst✝ : PerfectField K\ncomm : Commute f g\nhf : f.IsSemisimple\nhg : g.IsSemisimple\na : End K M\nha : a ∈ K[f, g]\nR : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 310,
"column": 19
} | {
"line": 310,
"column": 63
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Module K M\nf g : End K M\ninst✝¹ : FiniteDimensional K M\ninst✝ : PerfectField K\ncomm : Commute f g\nhf : f.IsSemisimple\nhg : g.IsSemisimple\na : End K M\nha : a ∈ K[f, g]\nR : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 15
} | [
{
"pp": "case refine_2\nK : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk✝ : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k✝\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\nsep' : IsUnit ((aeval ff) P')\ns : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 311,
"column": 4
} | {
"line": 312,
"column": 11
} | [
{
"pp": "case refine_3\nM : Type u_2\ninst✝⁴ : AddCommGroup M\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Module K M\nf g : End K M\ninst✝¹ : FiniteDimensional K M\ninst✝ : PerfectField K\ncomm : Commute f g\nhf : f.IsSemisimple\nhg : g.IsSemisimple\na : End K M\nha : a ∈ K[f, g]\nR : Type u_3 := K[X] ⧸ Ideal.spa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 15
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module ℚ M\ninst✝ : Module ℚ N\nfM : End R M\nfN : End R N\ng : M →ₗ[R] N\nh : fN ∘ₗ g = g ∘ₗ fM\nm : M\nk l : ℕ\nkl : ℕ := max k l\nhfM : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 247,
"column": 48
} | {
"line": 247,
"column": 59
} | [
{
"pp": "R : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : NonUnitalNonAssocRing B\ninst✝³ : Module R B\ninst✝² : SMulCommClass R B B\ninst✝¹ : IsScalarTower R B B\ninst✝ : Module ℚ B\nD : B →ₗ[R] B\nh_der : ∀ (x y : B), D (x * y) = x * D y + D x * y\nh_nil : IsNilpotent D\nx y : B\nDL : End R (B ⊗[R] B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Semisimple.Defs | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 53
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieAlgebra R L\ninst✝ : Subsingleton L\n⊢ radical R L = ⊥",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 155,
"column": 53
} | {
"line": 155,
"column": 64
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nhB : B.Nondegenerate\nx : M\nhx : x ∈ B.orthogonal ⊤\n⊢ ∀ (x_1 : M), (B x_1) x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 417,
"column": 4
} | {
"line": 417,
"column": 45
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ derivedSeriesOfIdeal R L k₁ I",
... | exact derivedSeriesOfIdeal_antitone I h₁₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Solvable | {
"line": 410,
"column": 2
} | {
"line": 418,
"column": 47
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ IsLieAbelian ↥(derivedSeriesOfIdeal R L k I) ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"usedConstants": [
"LieAlgebra.toModule",
"Iff.mpr",... | rw [abelian_iff_derived_succ_eq_bot]
let s := { k | derivedSeriesOfIdeal R L k I = ⊥ }
change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
have hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by
intro k₁ k₂ h₁₂ h₁
suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this
change derivedSeriesOfIdeal... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Solvable | {
"line": 410,
"column": 2
} | {
"line": 418,
"column": 47
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ IsLieAbelian ↥(derivedSeriesOfIdeal R L k I) ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"usedConstants": [
"LieAlgebra.toModule",
"Iff.mpr",... | rw [abelian_iff_derived_succ_eq_bot]
let s := { k | derivedSeriesOfIdeal R L k I = ⊥ }
change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
have hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by
intro k₁ k₂ h₁₂ h₁
suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this
change derivedSeriesOfIdeal... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 53
} | [
{
"pp": "case h.mp\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : ∀ φ ∈ (domRestrict B W).range, φ x = 0\ny : V\nhy : y ∈ W\n⊢ B.IsOrtho y x",
"usedConstants": [
"Submodule",
"Semiring.toModule",
... | exact hx (B.domRestrict W ⟨y, hy⟩) ⟨⟨y, hy⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.BooleanGenerators | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nC : Set α\nhC : ∀ x ∈ C, IsCompactElement x\nha : sSup C ≤ sSup S\ns : Finset α\nhs₁ : ↑s ⊆ S\nt : Finset α\nht : t ⊆ s\nhb : IsCompactElement (t.sup id)\nhbS : t.sup id ≤ sSup S\nhs₂ : t.sup ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 30
} | [
{
"pp": "case h.refine_1.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : (B.restrict p) ⟨z, hz⟩ = 0\nx y : M\nhx : x ∈ p\nhy : y ∈ q\n⊢ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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