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.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 48
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nf g : α → ℝ\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s μ\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s μ\nhgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ\nhs : Me... | have h_meas_nonpos_f : MeasurableSet {x | f x ≤ 0} :=
hf.measurableSet_le stronglyMeasurable_const | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 378,
"column": 29
} | {
"line": 387,
"column": 81
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ klDiv μ ν = 0 ↔ μ = ν",
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSeminormedCommRing",
"MeasureTheory.lintegral_eq_zero_iff",
"False",
"Real.p... | by
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ klDiv_self _⟩
have h_ne : klDiv μ ν ≠ ⊤ := by simp [h]
rw [klDiv_ne_top_iff] at h_ne
rw [klDiv_eq_lintegral_klFun, if_pos h_ne.1, lintegral_eq_zero_iff (by fun_prop)] at h
refine (Measure.rnDeriv_eq_one_iff_eq h_ne.1).mp ?_
filter_upwards [h] with x hx
simp only [Pi.zero... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 220,
"column": 37
} | {
"line": 220,
"column": 64
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : ¬m ≤ m₀\n⊢ Integrable 0 μ",
"usedConstants": [
"PseudoMetricSpace.toUniformSpace",
"MeasureTheory.integrable_z... | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 222,
"column": 50
} | {
"line": 222,
"column": 77
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ Integrable 0 μ",
"usedConstants": [
"PseudoMetricSpace.toUniformSpace",
... | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 38
} | [
{
"pp": "α : Type u_1\nm m₀ : MeasurableSpace α\nμ : Measure α\nR : Type u_5\ninst✝³ : NormedRing R\ninst✝² : NormedSpace ℝ R\ninst✝¹ : CompleteSpace R\nn : ℕ\ninst✝ : n.AtLeastTwo\nf : α → R\n⊢ μ[OfNat.ofNat n * f | m] =ᶠ[ae μ] OfNat.ofNat n * μ[f | m]",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 64
} | [
{
"pp": "case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : AffineSpace V P\ninst✝⁴ : Ring k\ninst✝³ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Finite ι'\ninst✝ : DecidableEq ι'\np : ι' → P\nA : Matrix ι ι' k\nhA : b.toMatrix p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 35
} | [
{
"pp": "R : Type u_4\nV : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\n⊢ e.fixedReduce = refl R (V ⧸ (↑e).fixedSubmodule) ↔ ∀ (v : V), e v - v ∈ (↑e).fixedSubmodule",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.Quotient.addCommMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 29
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 106,
"column": 18
} | {
"line": 106,
"column": 29
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι) (r : R), i ≠ j → f * (b.coord i).transvection (r • b j) = (b.coord i).transvection (r • b j) * f\ni j : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 15
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\ni j : ι\nhij : ¬j = i\n⊢ (b.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 29
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\nh_allEq : ∀ (i j : ι), (b.coord i) (f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 84,
"column": 42
} | {
"line": 84,
"column": 53
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nh_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))\nhcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 80,
"column": 36
} | {
"line": 80,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 41
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nh : ∀ (i_1 : ι), (b.repr (s •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 17
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nthis : t = 0 ∨ (b.r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 84,
"column": 2
} | {
"line": 86,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S... | suffices f.toLinearMap.comp (linearMapRightBaseChangeHom S M ε) =
(finitePow ι ibc).equiv.toLinearMap.comp e'.toLinearMap by
simp [h', this, ← LinearEquiv.trans_assoc e'.symm e'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.Center | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 43
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 60
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 65
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 64
} | [
{
"pp": "case refine_2\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 19
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nthis : ∀ e ∈ dilatransvections R V, e.symm ∈ dilatransvections R V\n⊢ e.symm ∈ dilatransvections R V → e ∈ dilatransvections R V",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 37
} | [
{
"pp": "case h.h\nV : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhe : Module.rank K ↥u.range ≤ 1\nhu : u + LinearMap.id = ↑e\nhr : Subsingleton ↥u.range\nx : V\n⊢ u x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 479,
"column": 10
} | {
"line": 479,
"column": 71
} | [
{
"pp": "V : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top : (↑e).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 56
} | [
{
"pp": "case a\nR : Type u_3\nV : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : IsDomain R\nf : Dual R V\nv : V\nK : Type u_3 := FractionRing R\nthis✝ : Field K := inferInstance\nthis : (algebraMap R K) (LinearMap.det (transv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 39
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nv : V\n⊢ (algebraMap A (CliffordAlgebra (QuadraticForm.baseChange A Q))) ((QuadraticForm.baseChange A Q) (... | rw [QuadraticForm.baseChange_tmul, one_mul, ← Algebra.algebraMap_eq_smul_one,
← IsScalarTower.algebraMap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 101,
"column": 43
} | {
"line": 101,
"column": 79
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nthis✝ : Invertible 2 := (Invertible.map (algebraMap R A) 2).copy 2 ⋯\nthis : Invertible 2 := (Invertible.m... | LinearMap.BilinForm.baseChange_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 160,
"column": 8
} | {
"line": 160,
"column": 20
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₂ : Type uM₂\nN₁ : Type uN₁\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : Algebra R A\ninst✝³ : Module R N₁\ninst✝² : Module A N₁\ninst✝¹ : IsScalarTower R A N₁\ninst✝ : Module R M₂\nQ₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) ... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | [
{
"pp": "case a.hl\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\n⊢ ((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R =\n ((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R",
"usedConstants": [
"AlgHom.toLinearMap",
... | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | [
{
"pp": "case a.hr\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\n⊢ ((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R =\n ((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R",
"usedConstants": [
"AlgHom.toLinearMap",
... | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 23
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n⊢ (contractRight (b * (ι Q) a)) d = d a • b - (contractRight b) d * (ι Q) a",
"usedConstants": [
"CliffordAlgebra.contractRight... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 23
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : M\n⊢ (contractRight ((ι Q) x)) d = (algebraMap R (CliffordAlgebra Q)) (d x)",
"usedConstants": [
"CliffordAlgebra.contractRight_eq",
"Eq.mpr",
... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 23
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nr : R\n⊢ (contractRight ((algebraMap R (CliffordAlgebra Q)) r)) d = 0",
"usedConstants": [
"CliffordAlgebra.contractRight_eq",
"Eq.mpr",
"Algebra... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 28
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractLeft d) 1 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 28
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractRight 1) d = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 23
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ (contractRight ((contractRight x) d)) d = 0",
"usedConstants": [
"CliffordAlgebra.contractRight_eq",
"Eq.mpr",
"Algebra.... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 196,
"column": 24
} | {
"line": 196,
"column": 41
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d) (reverse ((contractRight x) d))) = 0",
"usedConstants": [
"CliffordAlgebra.contractRight_eq",
"Eq.mp... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 15
} | [
{
"pp": "case mk.algebraMap\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : EvenHom Q A\nx val✝ : CliffordAlgebra Q\nproperty✝ : val✝ ∈ even Q\nr : R\npf✝⁴ : SMulCommClass R R A\npf✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 23
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ (contractRight ((contractRight x) d)) d' = -(contractRight ((contractRight x) d')) d",
"usedConstants": [
"CliffordAlgebra.contra... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 209,
"column": 24
} | {
"line": 209,
"column": 41
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse ((contractRight x) d))) = -(contractRight ((contractRight x) d')) d",
"usedConstants": [
"Cli... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 209,
"column": 42
} | {
"line": 209,
"column": 59
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -(contractRight ((contractRight x) d')) d",
"used... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 209,
"column": 60
} | {
"line": 209,
"column": 77
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -reverse ((contractLeft d) (reverse ((contractRight x... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nB : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\n⊢ (changeForm h) 1 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 103,
"column": 67
} | {
"line": 106,
"column": 85
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁷ : CommSemiring ι\ninst✝⁶ : Module ι (Additive ℤˣ)\ninst✝⁵ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝² : (i : ι) → AddCommGroup (ℬ i)\ninst✝¹ : (i : ι) → Module R (𝒜 i)\ninst✝ : (i : ι) → Mo... | by
refine TensorProduct.directSum R R 𝒜 ℬ ≪≫ₗ ?_ ≪≫ₗ (TensorProduct.directSum R R ℬ 𝒜).symm
exact LinearEquiv.ofLinear (gradedCommAux _ _ _) (gradedCommAux _ _ _)
(gradedCommAux_comp_gradedCommAux _ _ _) (gradedCommAux_comp_gradedCommAux _ _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\ny z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.ra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx y z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy : ∀ (b :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 46
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 46
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : ¬Disjoint ↑s ↑t\n⊢ b.ExteriorAlgebra ↑s * b.ExteriorAlgebra ↑t = 0",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : Disjoint ↑s ↑t\n⊢ b.ExteriorAlgebra ↑s * b.ExteriorAlgebra ↑t = Equiv.Perm.sign (permOfDisj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 34,
"column": 36
} | {
"line": 34,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ ⋀[R]^1 M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"Ring.toNonAssocRing",
"ExteriorAlgebra",
"co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 39,
"column": 19
} | {
"line": 39,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ ⋀[R]^1 M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"Ring.toNonAssocRing",
"ExteriorAlgebra",
"co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 132,
"column": 47
} | {
"line": 134,
"column": 59
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring ι\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CommRing R\ninst✝⁵ : Ring A\ninst✝⁴ : Ring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝¹ : GradedAlgebra 𝒜\ninst✝ : GradedAlgebra ℬ\n... | by
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 43,
"column": 65
} | {
"line": 43,
"column": 76
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : AddMonoid M\nφ : { φ // ∀ (g : G), n • φ g = 0 }\ng : G\n⊢ ((LinearMap.lsmul ℤ G) ↑n) g ∈ (↑φ).ker",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"ModN._proof_2",
"AddMonoidHom.instAddMonoidHomCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 76
} | [
{
"pp": "G : Type u_1\nH✝ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : NeZero n\nι : Type u_4\nb : Basis ι ℤ G\nψ : G →+ G := zsmulAddGroupHom ↑n\nnG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range\nH : Type u_1 := G ⧸ nG\nφ : G →ₗ[ℤ] H := nG.mkQ\nmod : (ι →₀ ℤ) →ₗ[ℤ] ι →₀ ZMod n := map... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 43
} | [
{
"pp": "case refine_1\nG : Type u_1\nH✝ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : NeZero n\nι : Type u_4\nb : Basis ι ℤ G\nψ : G →+ G := zsmulAddGroupHom ↑n\nnG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range\nH : Type u_1 := G ⧸ nG\nφ : G →ₗ[ℤ] H := nG.mkQ\nmod : (ι →₀ ℤ) →ₗ[ℤ] ι →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Goursat | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 9
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\n⊢ L.goursatFst.prod L.goursatSnd ≤ L",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.goursatSnd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Goursat | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (Linea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Goursat | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (Linea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 96,
"column": 21
} | {
"line": 104,
"column": 25
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Fintype ι\ninst✝ : Module R M\nN : Submodule R M\nbM : Basis ι R M\nbN : Basis (Fin n) R ↥N\nf : Fin n ↪ ι\na : Fin n → R\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nN' : Submodule R (ι → R) := S... | by
simp only [hj.choose_spec, ↓reduceIte]
rw [mul_comm]
conv_rhs =>
rw [← hj.choose_spec, (h (f hj.choose)).choose_spec]
simp only [EmbeddingLike.apply_eq_iff_eq, exists_eq, ↓reduceDIte, Classical.choose_eq]
congr!
· exa... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.BaseChange | {
"line": 56,
"column": 25
} | {
"line": 56,
"column": 52
} | [
{
"pp": "case h\nι : Type u_1\nι' : Type u_2\ninst✝⁵ : Finite ι'\nR : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : IsDomain S\nv✝ : ι → ι' → R\nh : LinearIndependent R v✝\nthis : IsDomain R\nK : Type u_3 := FractionRing R\nL : Type u_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.CharP | {
"line": 27,
"column": 73
} | {
"line": 27,
"column": 98
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝³ : AddMonoidWithOne R\ninst✝² : DecidableEq n\ninst✝¹ : Nonempty n\np : ℕ\ninst✝ : CharP R p\nk : ℕ\n⊢ ((diagonal fun x ↦ ↑k) = diagonal fun x ↦ 0) ↔ p ∣ k",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.LinearAlgebra.Matrix.CharP.0.Matrix.charP._simp_... | diagonal_eq_diagonal_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 157,
"column": 24
} | {
"line": 157,
"column": 35
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Fintype ι\ninst✝¹ : Infinite R\ninst✝ : Module R M\nN : Submodule R M\nsnf : SmithNormalForm N ι n\nh : ¬n = Fintype.card ι\n⊢ n ≤ Fintype.card ι",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.AbsoluteValue | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : Nontrivial R\ninst✝⁴ : CommRing S\ninst✝³ : LinearOrder S\ninst✝² : IsStrictOrderedRing S\nn : Type u_3\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nι : Type u_4\ns : Finset ι\nc : ι → R\nA : ι → Matrix n n R\nabv : AbsoluteValue R S\nx : S\nhx :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 20
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\ni : Fin n\nhi : a i = 0\nsnf : ↑(bN i) = a i • bM (f i)\n⊢ bN i = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 174,
"column": 2
} | {
"line": 175,
"column": 34
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nha : ∀ (i : Fin n), a i ≠ 0\nh : n = Fintype.card ι\n⊢ ¬∏ x, (Submodule.toAddSubgroup... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 15
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝ : Finite ι\nN : Submodule ℤ (ι → ℤ)\nn : ℕ\nthis : Fintype ι\nbN : Module.Basis (Fin n) ℤ ↥N\nx✝ : Nonempty (↥N ≃ₗ[ℤ] ι → ℤ)\ne : ↥N ≃ₗ[ℤ] ι → ℤ\nhc : Fintype.card (Fin n) = Fintype.card ι\n⊢ n = Fintype.card ι",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin (n + 1)) R\ni₀ j₀ : Fin (n + 1)\nhv : ∀ (i : Fin (n + 1)), i ≠ i₀ → ∑ j, M i j = 0\n⊢ ∀ (j : Fin (n + 1)), j ≠ i₀ → ∑ i, Mᵀ i j = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 63
} | [
{
"pp": "case mp\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝² : CommRing R\nA : Matrix m n R\nι : Type w\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : ι → m\ng : ι → n\nhA : (A.submatrix (f ∘ ⇑(Fintype.equivFin ι).symm) (g ∘ ⇑(Fintype.equivFin ι).symm)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).... | rwa [← submatrix_submatrix, det_submatrix_equiv_self] at hA | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 63
} | [
{
"pp": "case mpr\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nk : ℕ\nf : Fin k → m\ng : Fin k → n\nhA : (A.submatrix (f ∘ ⇑Equiv.ulift) (g ∘ ⇑Equiv.ulift)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).det ∈ Set.range SignType.cast",
"usedConstants": [
"SignTy... | rwa [← submatrix_submatrix, det_submatrix_equiv_self] at hA | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | [
{
"pp": "m : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nhA : ∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), (A.submatrix f g).det ∈ Set.range SignType.cast\ni : m\nj : n\n⊢ A i j ∈ Set.range SignType.cast",
"usedConstants": [
"SignType.cast",
"Eq.mpr",
"NegZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 49
} | [
{
"pp": "m : Type u_1\nm' : Type u_2\nn : Type u_3\nn' : Type u_4\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nem : m ≃ m'\nen : n ≃ n'\nhA : ((reindex em en) A).IsTotallyUnimodular\n⊢ A.IsTotallyUnimodular",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 156,
"column": 10
} | {
"line": 156,
"column": 28
} | [
{
"pp": "case h\nm : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function.Injective g → ((A.fromRows B).submatr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 47,
"column": 42
} | {
"line": 47,
"column": 77
} | [
{
"pp": "case zero\nK : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nhk : 0 ≤ n\nthis : Unique { s // ⊤ = ⊥ }\n⊢ card { s // (Finsupp.linearCombination K s).ker = ⊥ } = ∏ i, (q ^ n - q ^ ↑i)",
"usedConstants": [
"E... | Finsupp.linearCombination_fin_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 99,
"column": 30
} | {
"line": 99,
"column": 55
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\nx✝ : ℕ\n⊢ Aᵀ.den ∣ x✝ ↔ A.den ∣ x✝",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Rat",
"semigroupDvd",
"_private.Mathlib.LinearAlgebra.Matrix.Integer.0.Matrix.den_transp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 34
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).den = 1",
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"instOfNatNat",
"Nat.cast",
"Nat",
"Matrix.den",
"OfNat.ofNat",
"Eq",
"Rat.semirin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 34
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).num = ↑a",
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"Int",
"Nat.cast",
"Matrix.num",
"instNatCastInt",
"Eq",
"Rat.semiring",
"Rat.ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 34
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).den = 1",
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"instOfNatNat",
"Nat",
"Matrix.den",
"OfNat.ofNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 34
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).num = ↑a",
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"Matrix.num",
"instIntCastInt",
"Eq",
"Rat.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 134,
"column": 8
} | {
"line": 136,
"column": 15
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\nm : ℕ\nih : ∀ (i j : n), 0 < (A ^ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 56
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nha : ↑A 1 0 = 0\n⊢ ↑A 0 0 * ↑A 1 1 = m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 49
} | [
{
"pp": "case «0».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)| ≤ |m|",
"usedConstants": [
"Nat.le_refl",
"abs",
"Matrix",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 49
} | [
{
"pp": "case «0».«1»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)| ≤ |m|",
"usedConstants": [
"Nat.le_refl",
"abs",
"Matrix",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 49
} | [
{
"pp": "case «1».«1»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)| ≤ |m|",
"usedConstants": [
"Nat.le_refl",
"abs",
"Matrix",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 15
} | [
{
"pp": "case mp\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis✝ : Quiver n := A.toQuiver\nh_irr : A.IsIrreducible\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 37
} | [
{
"pp": "case a.h\nn : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommRing R\nm k : ℤ\nH : Finset ℤ := Finset.Icc (-|k|) |k|\nH4 : Type := Fin 2 → Fin 2 → ↥H\nM N : ↑(reps k)\nh : (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) M = (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) N\ni j : Fin 2\n⊢ ↑↑M i j = ↑↑N i j",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 59
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 62
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"Membership.mem",
"Eq.mp",
"instOf... | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 62
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"Membership.mem",
"Eq.mp",
"instOf... | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 62
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"Membership.mem",
"Eq.mp",
"instOf... | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 42
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : LinearOrder R\nA : Matrix n n R\ni j : n\nthis : Quiver n := A.toQuiver\nb c : n\nq : Path i b\ne : b ⟶ c\nih : Path b i\n⊢ 0 < Aᵀ c b",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 40
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : LinearOrder R\nA : Matrix n n R\nhA : A.IsIrreducible\ni j : n\n⊢ 0 ≤ Aᵀ i j",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 35
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : Semiring R\nn : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\nI J : Ideal R\neq : matrix n I = matrix n J\nx : R\nthis : (∀ (x_1 x_2 : n), x ∈ I) ↔ ∀ (x_1 x_2 : n), x ∈ J\n⊢ x ∈ I ↔ x ∈ J",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 42
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : LinearOrder R\nA : Matrix n n R\nhA_nonneg : ∀ (i j : n), 0 ≤ A i j\nh : Aᵀ.IsIrreducible\ni j : n\n⊢ 0 ≤ Aᵀ i j",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 144,
"column": 21
} | {
"line": 144,
"column": 32
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon R\ni j : n\nx y : R\nh : (matrix n c) (Matrix.single i j x) (Matrix.single i j y)\n⊢ c x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 20
} | [
{
"pp": "case inl\nR : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon R\ni j : n\nx y : R\nh : c x y\ni' j' : n\nhi : i ≠ i'\n⊢ c (Matrix.single i j x i' j') (Matrix.single i j y i' j')",
"usedConstants": [
"Eq.mpr",
"False",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 20
} | [
{
"pp": "case inr.inl\nR : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon R\ni j : n\nx y : R\nh : c x y\nj' : n\nhj : j ≠ j'\n⊢ c (Matrix.single i j x i j') (Matrix.single i j y i j')",
"usedConstants": [
"Eq.mpr",
"False",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | [
{
"pp": "case inr.inr\nR : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon R\ni j : n\nx y : R\nh : c x y\n⊢ c (Matrix.single i j x i j) (Matrix.single i j y i j)",
"usedConstants": [
"Eq.mpr",
"RingCon.instFunLikeForallProp",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nI J : RingCon R\neq : matrix n I = matrix n J\nr s : R\nthis :\n (matrix n I) (Matrix.of fun x x_1 ↦ r) (Matrix.of fun x x_1 ↦ s) =\n (matrix n J) (Matrix.of fun x x_1 ↦ r) (Matrix.of fun x x_1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 182,
"column": 29
} | {
"line": 182,
"column": 40
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon (Matrix n n R)\nw✝ x✝ y✝ z✝ : R\nh₁ : ∀ (i j : n), c (single i j w✝) (single i j x✝)\nh₂ : ∀ (i j : n), c (single i j y✝) (single i j z✝)\ni j : n\n⊢ c (single i j (w✝ * y✝)) (single... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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