module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 39
} | {
"line": 58,
"column": 0
} | [
{
"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": 102,
"column": 43
} | {
"line": 102,
"column": 79
} | {
"line": 102,
"column": 80
} | [
{
"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... | [
"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.map (algebraM... | LinearMap.BilinForm.baseChange_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 478,
"column": 10
} | {
"line": 478,
"column": 71
} | {
"line": 478,
"column": 72
} | [
{
"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).... | [
"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).fixedSubmodu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | {
"line": 301,
"column": 0
} | [
{
"pp": "case 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",
"ppTerm": "?hl",
"assigned": true,
"usedConstan... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | {
"line": 301,
"column": 0
} | [
{
"pp": "case 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",
"ppTerm": "?hr",
"assigned": true,
"usedConstan... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 23
} | {
"line": 134,
"column": 24
} | [
{
"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",
"ppTerm": "?m.57",
"assigned": true,
"usedConst... | [
"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⊢ reverse ((contractLeft d) (reverse (b * (ι Q) a))) = d a • b - (contractRight b) d * (ι Q) a"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 23
} | {
"line": 165,
"column": 24
} | [
{
"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)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CliffordAl... | [
"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⊢ reverse ((contractLeft d) (reverse ((ι Q) x))) = (algebraMap R (CliffordAlgebra Q)) (d x)"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 23
} | {
"line": 175,
"column": 24
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CliffordAlgebra.cont... | [
"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⊢ reverse ((contractLeft d) (reverse ((algebraMap R (CliffordAlgebra Q)) r))) = 0"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 28
} | {
"line": 179,
"column": 29
} | [
{
"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",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 28
} | {
"line": 183,
"column": 29
} | [
{
"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",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 56
} | {
"line": 631,
"column": 57
} | [
{
"pp": "R : 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 (transvection f... | [
"R : 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 (transvection f v)) = ↑1 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 23
} | {
"line": 197,
"column": 24
} | [
{
"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",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CliffordAlgebra.contr... | [
"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"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 197,
"column": 24
} | {
"line": 197,
"column": 41
} | {
"line": 197,
"column": 42
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Cl... | [
"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 (reverse ((contractLeft d) (reverse x))))) = 0"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 23
} | {
"line": 210,
"column": 24
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"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"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 24
} | {
"line": 210,
"column": 41
} | {
"line": 210,
"column": 42
} | [
{
"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",
"ppTerm": "?m.49",
"assig... | [
"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"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 42
} | {
"line": 210,
"column": 59
} | {
"line": 210,
"column": 60
} | [
{
"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",
"ppTe... | [
"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) d')))"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 60
} | {
"line": 210,
"column": 77
} | {
"line": 210,
"column": 78
} | [
{
"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... | [
"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 (reverse ((contractLeft d') (... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 13
} | {
"line": 275,
"column": 14
} | [
{
"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",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 104,
"column": 67
} | {
"line": 107,
"column": 85
} | {
"line": 109,
"column": 0
} | [
{
"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
} | {
"line": 90,
"column": 35
} | [
{
"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... | [
"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.range ⇑(ι Q))\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 132,
"column": 47
} | {
"line": 134,
"column": 59
} | {
"line": 136,
"column": 0
} | [
{
"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.CliffordAlgebra.SpinGroup | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 34
} | {
"line": 121,
"column": 35
} | [
{
"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 :... | [
"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 : M), involut... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 45
} | {
"line": 56,
"column": 46
} | [
{
"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",
"ppTerm": "?m.38",... | [
"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⊢ ιMulti_family R m (⇑b) s * ιMulti_family R n (⇑b) t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 45
} | {
"line": 61,
"column": 46
} | [
{
"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... | [
"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⊢ ιMulti_family R m (⇑b) s * ιMulti_family R n (⇑b) t =\n Equiv.Perm.sign (permOfDis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 34,
"column": 36
} | {
"line": 34,
"column": 62
} | {
"line": 34,
"column": 63
} | [
{
"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",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"ExteriorAlg... | [
"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 ∈ (ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 39,
"column": 19
} | {
"line": 39,
"column": 45
} | {
"line": 39,
"column": 46
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"ExteriorAlg... | [
"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 ∈ (ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 43,
"column": 65
} | {
"line": 43,
"column": 76
} | {
"line": 43,
"column": 77
} | [
{
"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",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"ModN._pro... | [
"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⊢ n • ↑φ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 33
} | {
"line": 216,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\n⊢ ((lift R A) fun {i} ↦ maps) ∘ₐ ι R A i = maps",
... | [] | ext a
simp [lift_apply, ι, ← ι_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 33
} | {
"line": 216,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\n⊢ ((lift R A) fun {i} ↦ maps) ∘ₐ ι R A i = maps",
... | [] | ext a
simp [lift_apply, ι, ← ι_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.CharP | {
"line": 27,
"column": 73
} | {
"line": 27,
"column": 98
} | {
"line": 28,
"column": 4
} | [
{
"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",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Lin... | [
"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⊢ (∀ (i : n), ↑k = 0) ↔ p ∣ k"
] | diagonal_eq_diagonal_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 96,
"column": 21
} | {
"line": 104,
"column": 25
} | {
"line": 105,
"column": 6
} | [
{
"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.Goursat | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 73
} | {
"line": 114,
"column": 6
} | [
{
"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... | [
"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' := (LinearMap.fst R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Goursat | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 73
} | {
"line": 118,
"column": 6
} | [
{
"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... | [
"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' := (LinearMap.fst R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.BaseChange | {
"line": 56,
"column": 25
} | {
"line": 56,
"column": 52
} | {
"line": 56,
"column": 53
} | [
{
"pp": "ι : 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_4 := Fra... | [
"ι : 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_4 := FractionRing S\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 63
} | {
"line": 69,
"column": 2
} | [
{
"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
} | {
"line": 73,
"column": 0
} | [
{
"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",
"ppTerm": "?mpr",
"assigned"... | [] | 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.FreeModule.Int | {
"line": 157,
"column": 24
} | {
"line": 157,
"column": 35
} | {
"line": 157,
"column": 36
} | [
{
"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",
"ppTerm": "?m.82",
"assigned": false,
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"S... | [
"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⊢ ∃ y, ↑y = A i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 49
} | {
"line": 102,
"column": 50
} | [
{
"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",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 20
} | {
"line": 172,
"column": 21
} | [
{
"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\nsnf : ↑(bN i) = a i • bM (f i)\nhi : a i = 0\n⊢ bN i = 0",
"ppTerm": "?m.67",
"assigned": false,
"usedConstan... | [
"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\nsnf : ↑(bN i) = a i • bM (f i)\nhi : a i = 0\n⊢ bN i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 174,
"column": 2
} | {
"line": 175,
"column": 34
} | {
"line": 175,
"column": 35
} | [
{
"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... | [
"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 : Fin n), ¬a x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 16
} | [
{
"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",
"ppTerm": "?refine_2",
"assigned": false,
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 19
} | {
"line": 92,
"column": 20
} | [
{
"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",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.... | [
"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 → ∑ x, M j x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 47,
"column": 42
} | {
"line": 47,
"column": 77
} | {
"line": 47,
"column": 78
} | [
{
"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)",
"ppTerm": "?zero",
"ass... | [
"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 // ker 0 = ⊥ } = ∏ i, (q ^ n - q ^ ↑i)"
] | Finsupp.linearCombination_fin_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 156,
"column": 10
} | {
"line": 156,
"column": 28
} | {
"line": 156,
"column": 29
} | [
{
"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\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → ... | [
"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\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function.Inj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 56
} | {
"line": 130,
"column": 57
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nha : ↑A 1 0 = 0\n⊢ ↑A 0 0 * ↑A 1 1 = m",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nha : ↑A 1 0 = 0\n⊢ ↑A 0 0 * ↑A 1 1 = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 99,
"column": 30
} | {
"line": 99,
"column": 55
} | {
"line": 99,
"column": 56
} | [
{
"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✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Rat",
"semigroupDvd",
"_private.Mathlib.Lin... | [
"m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\nx✝ : ℕ\n⊢ (∀ (i : n) (j : m), (A j i).den ∣ x✝) ↔ ∀ (i : m) (j : n), (A i j).den ∣ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 34
} | {
"line": 130,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).den = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"instOfNatNat",
"Nat.cast",
"Nat",
"Matrix.den",
"... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (diagonal fun x ↦ ↑a).den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 34
} | {
"line": 134,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).num = ↑a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"Int",
"Nat.cast",
"Matrix.num",
"instNatCastInt",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (diagonal fun x ↦ ↑a).num = diagonal fun x ↦ ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 34
} | {
"line": 148,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).den = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"instOfNatNat",
"Nat",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (diagonal fun x ↦ ↑a).den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 34
} | {
"line": 152,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).num = ↑a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"Matrix.num",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (diagonal fun x ↦ ↑a).num = diagonal fun x ↦ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 37
} | {
"line": 165,
"column": 38
} | [
{
"pp": "n : 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",
"ppTerm":... | [
"n : 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
} | {
"line": 177,
"column": 60
} | [
{
"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",
"ppTerm": "?step",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 62
} | {
"line": 178,
"column": 2
} | [
{
"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",
"ppTerm": "?step",
"assigned": true,
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"... | [] | 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
} | {
"line": 178,
"column": 2
} | [
{
"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",
"ppTerm": "?step",
"assigned": true,
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"... | [] | 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
} | {
"line": 178,
"column": 2
} | [
{
"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",
"ppTerm": "?step",
"assigned": true,
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"... | [] | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 65,
"column": 7
} | {
"line": 65,
"column": 18
} | {
"line": 65,
"column": 19
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ Aᴴ * Aᴴᴴ = ↑(Fintype.card n) • 1",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"NonA... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ Aᴴ * A = ↑(Fintype.card n) • 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 66,
"column": 7
} | {
"line": 66,
"column": 18
} | {
"line": 66,
"column": 19
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ Aᴴᴴ * Aᴴ = ↑(Fintype.card n) • 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"NonA... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ A * Aᴴ = ↑(Fintype.card n) • 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 70,
"column": 16
} | {
"line": 70,
"column": 27
} | {
"line": 70,
"column": 28
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : Aᴴ.IsHadamard\n⊢ A.IsHadamard",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : Aᴴ.IsHadamard\n⊢ A.IsHadamard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 82,
"column": 15
} | {
"line": 82,
"column": 26
} | {
"line": 82,
"column": 27
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ne₁ e₂ : m ≃ n\nh : (A.submatrix ⇑e₁ ⇑e₂).IsHadamard\n⊢ A.IsHadamard",
"ppTerm": "?m.24",
"assigned": false... | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ne₁ e₂ : m ≃ n\nh : (A.submatrix ⇑e₁ ⇑e₂).IsHadamard\n⊢ A.IsHadamard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 134,
"column": 8
} | {
"line": 136,
"column": 15
} | {
"line": 136,
"column": 16
} | [
{
"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... | [
"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) i j ↔ None... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 83,
"column": 16
} | {
"line": 83,
"column": 43
} | {
"line": 83,
"column": 44
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ne₁ e₂ : m ≃ n\nh : A.IsHadamard\n⊢ (A.submatrix ⇑e₁ ⇑e₂).IsHadamard",
"ppTerm": "?m.27",
"assigned": false... | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ne₁ e₂ : m ≃ n\nh : A.IsHadamard\n⊢ (A.submatrix ⇑e₁ ⇑e₂).IsHadamard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 43
} | {
"line": 121,
"column": 44
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhcol : ∀ (j : n), ∑ i, A i j = s\nj : n\n⊢ (1 ᵥ* A) j = (s • 1) j",
"ppTerm": "?m.50",
"assigned": tr... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhcol : ∀ (j : n), ∑ i, A i j = s\nj : n\n⊢ ∑ x, A x j = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 15
} | {
"line": 171,
"column": 16
} | [
{
"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... | [
"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\ni j : n\np :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 25
} | {
"line": 133,
"column": 26
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhcol : ∀ (j : n), ∑ i, A i j = s\nhvcol : 1 ᵥ* A = s • 1\nhconjcol : Aᴴ *ᵥ 1 = star s • 1\nhleft : 1 ᵥ* (A * ... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhcol : ∀ (j : n), ∑ i, A i j = s\nhvcol : 1 ᵥ* A = s • 1\nhconjcol : Aᴴ *ᵥ 1 = star s • 1\nhleft : 1 ᵥ* (A * Aᴴ) ⬝ᵥ 1 = ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 14
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhrow : ∀ (i : n), ∑ j, A i j = s\nhcol : ∀ (j : n), ∑ i, Aᴴ i j = star s\n⊢ ↑(Fintype.card n) = star s * s",
... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nA : Matrix n n R\ns : R\nhA : A.IsHadamard\nhcard : IsRegular ↑(Fintype.card n)\nhrow : ∀ (i : n), ∑ j, A i j = s\nhcol : ∀ (j : n), ∑ i, Aᴴ i j = star s\n⊢ ↑(Fintype.card n) = star s * s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 164,
"column": 16
} | {
"line": 164,
"column": 27
} | {
"line": 164,
"column": 28
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommSemiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : Aᵀ.IsHadamard\n⊢ A.IsHadamard",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommSemiring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : Aᵀ.IsHadamard\n⊢ A.IsHadamard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 42
} | {
"line": 200,
"column": 43
} | [
{
"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",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.to... | [
"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 < A b c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 47
} | {
"line": 173,
"column": 48
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Ring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ (-A).IsHadamard",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroC... | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Ring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.IsHadamard\n⊢ (∀ (i j : n), star (A i j) * A i j = 1 ∧ A i j * star (A i j) = 1) ∧\n A * Aᴴ = ↑(Fintype.card n) • 1 ∧ Aᴴ * A = ↑(Fintype.card n) • 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 178,
"column": 16
} | {
"line": 178,
"column": 27
} | {
"line": 178,
"column": 28
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Ring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : (-A).IsHadamard\n⊢ A.IsHadamard",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Ring R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : (-A).IsHadamard\n⊢ A.IsHadamard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 40
} | {
"line": 207,
"column": 41
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"D... | [
"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 ≤ A j i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 42
} | {
"line": 227,
"column": 43
} | [
{
"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",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Semi... | [
"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 ≤ A j i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 68
} | {
"line": 252,
"column": 69
} | [
{
"pp": "case step\nm : ℤ\nC : FixedDetMatrix (Fin 2) ℤ m → Prop\nA✝ : FixedDetMatrix (Fin 2) ℤ m\nhm : m ≠ 0\nh0 : ∀ (A : FixedDetMatrix (Fin 2) ℤ m), ↑A 1 0 = 0 → 0 < ↑A 0 0 → 0 ≤ ↑A 0 1 → |↑A 0 1| < |↑A 1 1| → C A\nhS : ∀ (B : FixedDetMatrix (Fin 2) ℤ m), C B → C (S • B)\nhT : ∀ (B : FixedDetMatrix (Fin 2) ℤ... | [
"case step\nm : ℤ\nC : FixedDetMatrix (Fin 2) ℤ m → Prop\nA✝ : FixedDetMatrix (Fin 2) ℤ m\nhm : m ≠ 0\nh0 : ∀ (A : FixedDetMatrix (Fin 2) ℤ m), ↑A 1 0 = 0 → 0 < ↑A 0 0 → 0 ≤ ↑A 0 1 → |↑A 0 1| < |↑A 1 1| → C A\nhS : ∀ (B : FixedDetMatrix (Fin 2) ℤ m), C B → C (S • B)\nhT : ∀ (B : FixedDetMatrix (Fin 2) ℤ m), C B → C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 37,
"column": 2
} | {
"line": 38,
"column": 67
} | {
"line": 39,
"column": 2
} | [
{
"pp": "case refine_1\nn : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\nσ : Perm n\nx✝ : σ ∈ univ\nhσ : σ ≠ 1\n⊢ ∏ i, diagonal d (σ i) i = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Equiv.... | [
"case refine_2\nn : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\n⊢ ∏ i, diagonal d (1 i) i = ∏ i, d i"
] | · match not_forall.mp (mt Equiv.ext hσ) with
| ⟨x, hx⟩ => exact Finset.prod_eq_zero (mem_univ x) (if_neg hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 39
} | {
"line": 248,
"column": 40
} | [
{
"pp": "n : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n ℤ\nhA : A.IsHadamard\nhcard : 2 < Fintype.card n\nhpm : ∀ (i j : n), A i j = 1 ∨ A i j = -1\nr s t : n\nhrs : r ≠ s\nhrt : r ≠ t\nhst : s ≠ t\ni k : n\nhik : i ≠ k\n⊢ ∑ j, A i j * A k j = 0",
"ppTerm": "?m.105",
"assigned":... | [
"n : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n ℤ\nhA : A.IsHadamard\nhcard : 2 < Fintype.card n\nhpm : ∀ (i j : n), A i j = 1 ∨ A i j = -1\nr s t : n\nhrs : r ≠ s\nhrt : r ≠ t\nhst : s ≠ t\ni k : n\nhik : i ≠ k\n⊢ ∑ j, A i j * A k j = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | {
"line": 45,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.perm... | [] | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | {
"line": 45,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.perm... | [] | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | {
"line": 45,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.perm... | [] | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 71
} | {
"line": 60,
"column": 72
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (... | [
"R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (j : κ i), M ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 98,
"column": 69
} | {
"line": 101,
"column": 43
} | {
"line": 103,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx : ⨂[R] (i : ι), A i\n⊢ (mul ((tprod R) 1)) x = x",
... | [] | by
induction x using PiTensorProduct.induction_on with
| smul_tprod => simp
| add _ _ h1 h2 => simp [map_add, h1, h2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.PiTensorProduct.Generators | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 52
} | {
"line": 126,
"column": 53
} | [
{
"pp": "case succ.e_f.h\nR : Type u_1\nN : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\nn : ℕ\nhn :\n ∀ {ι : Type u_2} [Finite ι] {M : ι → Type u_3} {γ : ι → Type u_5} [inst : (i : ι) → AddCommMonoid (M i)]\n [inst_1 : (i : ι) → Module R (M i)] {g : ⦃i : ι⦄ → γ i → M i}... | [
"case succ.e_f.h\nR : Type u_1\nN : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\nn : ℕ\nhn :\n ∀ {ι : Type u_2} [Finite ι] {M : ι → Type u_3} {γ : ι → Type u_5} [inst : (i : ι) → AddCommMonoid (M i)]\n [inst_1 : (i : ι) → Module R (M i)] {g : ⦃i : ι⦄ → γ i → M i},\n (∀ (i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 36
} | {
"line": 136,
"column": 37
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx✝ y✝ z✝ : ⨂[R] (i : ι), A i\nx y z : (i : ι) → A i\n⊢ (... | [
"ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx✝ y✝ z✝ : ⨂[R] (i : ι), A i\nx y z : (i : ι) → A i\n⊢ (tprod R) (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PiTensorProduct.Generators | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 15
} | {
"line": 136,
"column": 16
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁵ : Finite ι\nM : ι → Type u_3\nN : Type u_4\nγ : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : ⦃i : ι⦄ → γ i → M i\nhg : ∀ (i : ι), Submodule.span R (S... | [
"R : Type u_1\nι : Type u_2\ninst✝⁵ : Finite ι\nM : ι → Type u_3\nN : Type u_4\nγ : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : ⦃i : ι⦄ → γ i → M i\nhg : ∀ (i : ι), Submodule.span R (Set.range g) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PiTensorProduct.Generators | {
"line": 137,
"column": 58
} | {
"line": 137,
"column": 69
} | {
"line": 137,
"column": 70
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁵ : Finite ι\nM : ι → Type u_3\nN : Type u_4\nγ : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : ⦃i : ι⦄ → γ i → M i\nhg : ∀ (i : ι), Submodule.span R (S... | [
"R : Type u_1\nι : Type u_2\ninst✝⁵ : Finite ι\nM : ι → Type u_3\nN : Type u_4\nγ : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : ⦃i : ι⦄ → γ i → M i\nhg : ∀ (i : ι), Submodule.span R (Set.range g) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 13
} | {
"line": 45,
"column": 14
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Subsingleton V\nthis : IsEmpty { v // v ≠ 0 }\n⊢ IsEmpty (ℙ k V)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u_1\nV : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Subsingleton V\nthis : IsEmpty { v // v ≠ 0 }\n⊢ IsEmpty (ℙ k V)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 33
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\na✝ : Nontrivial V\nh : Finite k\nx✝ : Finite V ∨ Infinite V\n⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Fin... | [] | cases finite_or_infinite V with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 17
} | {
"line": 115,
"column": 18
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : Field k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Finite k\nn : ℕ\nh : Module.finrank k V = n\nthis✝ :\n ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_1 : AddCommGroup V] [inst_2 : Module k V] [Finite k] {n : ℕ},\n Module.finrank k V = n → ... | [
"k : Type u_1\nV : Type u_2\ninst✝³ : Field k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Finite k\nn : ℕ\nh : Module.finrank k V = n\nthis✝ :\n ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_1 : AddCommGroup V] [inst_2 : Module k V] [Finite k] {n : ℕ},\n Module.finrank k V = n → Finite V → N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 143,
"column": 10
} | {
"line": 143,
"column": 13
} | {
"line": 144,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : ℙ K V\n⊢ x ∈ ⊤ → x ∈ span Set.univ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"CompleteL... | [
"K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : ℙ K V\n_hx : x ∈ ⊤\n⊢ x ∈ span Set.univ"
] | _hx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 58
} | {
"line": 69,
"column": 59
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\ni : ι\n⊢ (Projectivization.rep ∘ f) i ∈ (Projectivization.submodule ∘ f) i",
"ppTerm": "?refine_2.refine_1",
... | [
"case refine_2.refine_1\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\ni : ι\n⊢ (f i).rep ∈ K ∙ (f i).rep"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.PSL.PSL2 | {
"line": 50,
"column": 26
} | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 38
} | [
{
"pp": "F : Type u_2\ninst✝ : Field F\nM : SL(2, F)\nx✝ : M ∈ ⊤\ni j : Fin 2\nhij : i ≠ j\na : F\n⊢ SpecialLinearGroup.transvection hij a ∈ lineStab (F ∙ Pi.single 0 1) ⊔ lineStab (F ∙ Pi.single 1 1)",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"F : Type u_2\ninst✝ : Field F\nM : SL(2, F)\nx✝ : M ∈ ⊤\ni j : Fin 2\nhij : i ≠ j\na : F\n⊢ SpecialLinearGroup.transvection hij a ∈ lineStab (F ∙ Pi.single 0 1) ⊔ lineStab (F ∙ Pi.single 1 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.PSL.PSL2 | {
"line": 132,
"column": 58
} | {
"line": 132,
"column": 69
} | {
"line": 132,
"column": 70
} | [
{
"pp": "F : Type u_2\ninst✝ : Field F\nthis : Finite F\nx : Fˣ\nhF : 4 ≤ orderOf x + 1\nhx : Function.Surjective fun x_1 ↦ x ^ x_1\nh : ↑x ^ 2 = 1\n⊢ ↑(x ^ 2) = ↑1",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne... | [
"F : Type u_2\ninst✝ : Field F\nthis : Finite F\nx : Fˣ\nhF : 4 ≤ orderOf x + 1\nhx : Function.Surjective fun x_1 ↦ x ^ x_1\nh : ↑x ^ 2 = 1\n⊢ x = 1 ∨ x = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 65
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis✝ : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : ... | [
"case pos\nK : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis✝ : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : FiniteDimens... | let s := (linearIndepOn_pair D D').extend (Set.subset_univ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.QuadraticForm.AlgClosed | {
"line": 38,
"column": 6
} | {
"line": 38,
"column": 17
} | {
"line": 38,
"column": 18
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Fintype ι\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : DecidableEq K\nw : ι → K\ni : ι\nh : ¬w i = 0\n⊢ w i ≠ 0",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Field.toSemifield",
"Semifield.toDivision... | [
"ι : Type u_1\ninst✝³ : Fintype ι\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : DecidableEq K\nw : ι → K\ni : ι\nh : ¬w i = 0\n⊢ ¬w i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 36
} | {
"line": 81,
"column": 37
} | [
{
"pp": "case refine_2\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ng : SpecialLinearGroup ι F\nL : Submodule F (ι → F)\nA : SpecialLinearGroup ι F\nhA : ∀ (w : ι → F), (g⁻¹ * A * g) • w - w ∈ L\nw : ι → F\n⊢ g⁻¹ • (A • w - w) ∈ L",
"ppTerm": "?refine_2",
"as... | [
"case refine_2\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ng : SpecialLinearGroup ι F\nL : Submodule F (ι → F)\nA : SpecialLinearGroup ι F\nhA : ∀ (w : ι → F), (g⁻¹ * A * g) • w - w ∈ L\nw : ι → F\n⊢ g⁻¹ • A • w - g⁻¹ • w ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 47
} | {
"line": 105,
"column": 48
} | [
{
"pp": "case refine_1\nR : Type u_3\nV : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\nv : V\nA : V →ₗ[R] V\nhAv : A v ∈ R ∙ v\nc : R\nhc : c • v = A v\na : R\nhw : a • v ∈ R ∙ v\n⊢ a • v ∈ Submodule.comap A (R ∙ v)",
"ppTerm": "?refine_1",
"assigned": true,
"used... | [
"case refine_1\nR : Type u_3\nV : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\nv : V\nA : V →ₗ[R] V\nhAv : A v ∈ R ∙ v\nc : R\nhc : c • v = A v\na : R\nhw : a • v ∈ R ∙ v\n⊢ a • A v ∈ R ∙ v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 54
} | {
"line": 92,
"column": 55
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Module.Free R V\nd1 : Module.finrank R V = 1\nu✝ v : SpecialLinearGroup R V\na✝ : Nontrivial R\nx✝ : V\nhx : ¬x✝ = 0\nu : SpecialLinearGroup R V\nx : V\nc : R := (LinearEquiv.smul_id_of_finr... | [
"case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Module.Free R V\nd1 : Module.finrank R V = 1\nu✝ v : SpecialLinearGroup R V\na✝ : Nontrivial R\nx✝ : V\nhx : ¬x✝ = 0\nu : SpecialLinearGroup R V\nx : V\nc : R := (LinearEquiv.smul_id_of_finrank_eq_one d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 12
} | {
"line": 228,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\n⊢ LinearEquiv.det u.toLinearEquiv = 1 → u ∈ Set.range ⇑toGeneralLinearGroup",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LinearEqu... | [
"case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\nhu : LinearEquiv.det u.toLinearEquiv = 1\n⊢ u ∈ Set.range ⇑toGeneralLinearGroup"
] | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 12
} | {
"line": 228,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\n⊢ LinearEquiv.det u.toLinearEquiv = 1 → u ∈ Set.range ⇑toGeneralLinearGroup",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LinearEqu... | [
"case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\nhu : LinearEquiv.det u.toLinearEquiv = 1\n⊢ u ∈ Set.range ⇑toGeneralLinearGroup"
] | intro hu | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer | {
"line": 142,
"column": 29
} | {
"line": 142,
"column": 40
} | {
"line": 142,
"column": 41
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → F\nhv : v ≠ 0\nx✝¹ x✝ : ↥(lineStab (F ∙ v))\nA : SpecialLinearGroup ι F\nhA : A ∈ lineStab (F ∙ v)\nB : SpecialLinearGroup ι F\nhB : B ∈ lineStab (F ∙ v)\n⊢ ⟨A, hA⟩ * ⟨B, hB⟩ = ⟨B, hB⟩ * ⟨A, hA⟩",
"ppTe... | [
"F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → F\nhv : v ≠ 0\nx✝¹ x✝ : ↥(lineStab (F ∙ v))\nA : SpecialLinearGroup ι F\nhA : A ∈ lineStab (F ∙ v)\nB : SpecialLinearGroup ι F\nhB : B ∈ lineStab (F ∙ v)\n⊢ A * B = B * A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 15
} | {
"line": 109,
"column": 16
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx✝ : M\nx : ι × ι\nhx : x.1 ∈ (bm.repr x✝).support ∧ x.2 ∈ (b... | [
"case h\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx✝ : M\nx : ι × ι\nhx : x.1 ∈ (bm.repr x✝).support ∧ x.2 ∈ (bm.repr x✝).s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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