module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Polynomial.Laurent | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 32
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℤ\n⊢ AddMonoidAlgebra.single n r = C r * T n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidAlgebra.instAddMonoid",
"HMul.hMul",
... | [] | simp [C, T, single_mul_single] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np : Submonoid R\nhp : p ≤ R⁰\nN : Type uM\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nf : M →ₗ[R] N\ninst✝ : IsLocalizedModule p f\n⊢ Module.rank R N = Module.rank R M",
"ppTerm": "?m.63",
"assigned"... | [
"R : Type uR\nM : Type uM\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np : Submonoid R\nhp : p ≤ R⁰\nN : Type uM\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nf : M →ₗ[R] N\ninst✝ : IsLocalizedModule p f\n⊢ Module.rank R N = Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 40
} | {
"line": 239,
"column": 41
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Semiring R\nM : R[T;T⁻¹] → Prop\np : R[T;T⁻¹]\nh_C : ∀ (a : R), M (C a)\nh_add : ∀ {p q : R[T;T⁻¹]}, M p → M q → M (p + q)\nh_C_mul_T : ∀ (n : ℕ) (a : R), M (C a * T ↑n) → M (C a * T (↑n + 1))\nh_C_mul_T_Z : ∀ (n : ℕ) (a : R), M (C a * T (-↑n)) → M (C a * T (-↑n - 1... | [
"case refine_1\nR : Type u_1\ninst✝ : Semiring R\nM : R[T;T⁻¹] → Prop\np : R[T;T⁻¹]\nh_C : ∀ (a : R), M (C a)\nh_add : ∀ {p q : R[T;T⁻¹]}, M p → M q → M (p + q)\nh_C_mul_T : ∀ (n : ℕ) (a : R), M (C a * T ↑n) → M (C a * T (↑n + 1))\nh_C_mul_T_Z : ∀ (n : ℕ) (a : R), M (C a * T (-↑n)) → M (C a * T (-↑n - 1))\nn : ℤ\na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 709,
"column": 2
} | {
"line": 709,
"column": 68
} | {
"line": 709,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nI p : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : IsLocalization.AtPrime S p\nhle : ¬I ≤ p\n⊢ ¬↑I ⊆ (↑p.primeCompl)ᶜ",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nI p : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : IsLocalization.AtPrime S p\nhle : ¬I ≤ p\n⊢ ∃ x ∈ I, x ∉ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 14
} | [
{
"pp": "R : Type uR\nS : Type uS\nM : Type uM\nN : Type uN\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : AddCommGroup N\ninst✝¹⁵ : Module R M\ninst✝¹⁴ : Module R N\ninst✝¹³ : Algebra R S\ninst✝¹² : Module S N\ninst✝¹¹ : IsScalarTower R S N\np : Submonoid R\ninst✝¹⁰ : IsLocali... | [
"R : Type uR\nS : Type uS\nM : Type uM\nN : Type uN\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : AddCommGroup N\ninst✝¹⁵ : Module R M\ninst✝¹⁴ : Module R N\ninst✝¹³ : Algebra R S\ninst✝¹² : Module S N\ninst✝¹¹ : IsScalarTower R S N\np : Submonoid R\ninst✝¹⁰ : IsLocalization p S\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 95
} | {
"line": 165,
"column": 2
} | [
{
"pp": "case inr\nR : Type uR\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nT : Type uT\ninst✝⁷ : CommRing T\ninst✝⁶ : NoZeroDivisors T\ninst✝⁵ : Algebra R T\ninst✝⁴ : FaithfulSMul R T\nP : Type uP\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : Module T P\ninst✝ : I... | [
"case inr\nR : Type uR\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nT : Type uT\ninst✝⁷ : CommRing T\ninst✝⁶ : NoZeroDivisors T\ninst✝⁵ : Algebra R T\ninst✝⁴ : FaithfulSMul R T\nP : Type uP\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : Module T P\ninst✝ : IsScalarTower... | replace inj : Function.Injective (algebraMap R FT) := (IsFractionRing.injective T _).comp inj | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nT : Type uT\ninst✝⁷ : CommRing T\ninst✝⁶ : NoZeroDivisors T\ninst✝⁵ : Algebra R T\ninst✝⁴ : FaithfulSMul R T\nP : Type uM\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : Module T P\ninst✝ : IsScalarTow... | [
"R : Type uR\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nT : Type uT\ninst✝⁷ : CommRing T\ninst✝⁶ : NoZeroDivisors T\ninst✝⁵ : Algebra R T\ninst✝⁴ : FaithfulSMul R T\nP : Type uM\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : Module T P\ninst✝ : IsScalarTower R T P\ng ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 15
} | {
"line": 203,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nthis : LinearIndependent R fun i ↦ r * ↑s ^ ↑i\n⊢ ↑n ≤ Module.rank R R",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nthis : LinearIndependent R fun i ↦ r * ↑s ^ ↑i\n⊢ ↑n ≤ Module.rank R R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 25
} | {
"line": 208,
"column": 26
} | [
{
"pp": "case cons\nn : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod * (List.map toMatrix L).prod = 1\n⊢ (List.map (toMatrix ∘ TransvectionS... | [
"case cons\nn : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod * (List.map toMatrix L).prod = 1\n⊢ (List.map (toMatrix ∘ TransvectionStruct.inv) L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 37
} | {
"line": 207,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nthis : ∀ (g : ℕ → R) (x : ℕ), ∑ i ∈ Finset.range n, g i • (r * ↑s ^ (i + x)) = 0 → ∀ i < n, g i = 0\ng : Fin n → R\nhg : ∑ i, g i • (r * ↑s ^ ↑i) = 0\ni : Fin n\n⊢ g i = 0",
"ppTe... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nthis : ∀ (g : ℕ → R) (x : ℕ), ∑ i ∈ Finset.range n, g i • (r * ↑s ^ (i + x)) = 0 → ∀ i < n, g i = 0\ng : Fin n → R\nhg : ∑ i, g i • (r * ↑s ^ ↑i) = 0\ni : Fin n\n⊢ g i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 115,
"column": 91
} | {
"line": 118,
"column": 16
} | {
"line": 120,
"column": 0
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ A.cramer b = Pi.single i A.det",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [] | by
rw [← transpose_transpose A, det_transpose]
convert! cramer_transpose_row_self Aᵀ i
exact funext h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 369,
"column": 12
} | {
"line": 369,
"column": 71
} | {
"line": 369,
"column": 72
} | [
{
"pp": "case mul_T.zero\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\n⊢ Q (toLaurent f * T (-↑0))",
"ppTerm": "?mul_T.zero",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"La... | [
"case mul_T.zero\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\n⊢ Q (toLaurent f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 370,
"column": 52
} | {
"line": 370,
"column": 81
} | {
"line": 370,
"column": 82
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n⊢ n < (listTransvecCol M).length",
"ppTerm": "?m.38",
"assigned": true,
"usedConstan... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n⊢ n < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 11
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case self\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\n⊢ ((List.drop r (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i",
"ppTerm": "?self",
"assigned": true,
"usedConstants": [
"Unit.unit",
"NonAssocSemir... | [] | | self => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 13
} | {
"line": 380,
"column": 14
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\n⊢ ((listTransvecCol M).prod * M) (inr ()) i = M (inr ()) i",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\n⊢ ((listTransvecCol M).prod * M) (inr ()) i = M (inr ()) i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 27
} | {
"line": 391,
"column": 28
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nH : ∀ k ≤ r, ((List.drop k (listTransvecCol M)).prod * M) (inl i) (inr ()) = if k ≤ ↑i then 0 else M (inl i) (inr ())\n⊢ ((listTransvecCol M).prod * M) (inl i) (inr ()) = 0",
... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nH : ∀ k ≤ r, ((List.drop k (listTransvecCol M)).prod * M) (inl i) (inr ()) = if k ≤ ↑i then 0 else M (inl i) (inr ())\n⊢ ((listTransvecCol M).prod * M) (inl i) (inr ()) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 395,
"column": 52
} | {
"line": 395,
"column": 81
} | {
"line": 395,
"column": 82
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\n⊢ n < (listTransvecCol M).length",
... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\n⊢ n < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 506,
"column": 21
} | {
"line": 508,
"column": 28
} | {
"line": 510,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nn : ℕ\n⊢ IsLocalization.mk' R[T;T⁻¹] p ⟨X ^ n, ⋯⟩ * T ↑n = toLaurent p",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"Eq.mpr",
"Int.instAddCommMonoid",
"AddMonoidAlgebra.semiring",
... | [] | by
rw [← toLaurent_X_pow, ← algebraMap_eq_toLaurent, IsLocalization.mk'_spec,
algebraMap_eq_toLaurent] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 532,
"column": 4
} | {
"line": 532,
"column": 52
} | {
"line": 532,
"column": 53
} | [
{
"pp": "R : Type u_1\nS✝ : Type u_2\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℕ\n⊢ IsUnit ((eval₂RingHom f ↑x) ↑⟨(fun x ↦ X ^ x) n, ⋯⟩)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Polynomial.eval₂... | [
"R : Type u_1\nS✝ : Type u_2\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℕ\n⊢ IsUnit (↑x ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 54
} | {
"line": 278,
"column": 2
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nr : α\nA : Matrix n n α\n⊢ (r • A).adjugate = r ^ (Fintype.card n - 1) • A.adjugate",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Matrix.smu... | [
"n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nr : α\nA : Matrix n n α\n⊢ (of fun i ↦ (r ^ (Fintype.card n - 1) • Aᵀ.cramer) (Pi.single i 1)) =\n r ^ (Fintype.card n - 1) • of fun i ↦ Aᵀ.cramer (Pi.single i 1)"
] | rw [adjugate, adjugate, transpose_smul, cramer_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 41
} | {
"line": 290,
"column": 42
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nM : Matrix n n α\nv : n → α\nh : M *ᵥ v = 0\ni : n\nhi : v i ∈ nonZeroDivisors α\n⊢ M.det * v i = 0",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nM : Matrix n n α\nv : n → α\nh : M *ᵥ v = 0\ni : n\nhi : v i ∈ nonZeroDivisors α\n⊢ M.det * v i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 420,
"column": 10
} | {
"line": 420,
"column": 55
} | {
"line": 420,
"column": 56
} | [
{
"pp": "case neg.inr.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTr... | [
"case neg.inr.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCol M)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 421,
"column": 10
} | {
"line": 421,
"column": 35
} | {
"line": 421,
"column": 36
} | [
{
"pp": "case neg.inr.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTr... | [
"case neg.inr.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCol M)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 32
} | {
"line": 279,
"column": 33
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (x * A).trace = (x * B).trace\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (x * A).trace = (x * B).trace\ni : m\nj : n\n⊢ A i j = B i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 32
} | {
"line": 287,
"column": 33
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (A * x).trace = (B * x).trace\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (A * x).trace = (B * x).trace\ni : m\nj : n\n⊢ A i j = B i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 29
} | {
"line": 426,
"column": 30
} | [
{
"pp": "case self.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk : ℕ\n⊢ ¬r ≤ ↑i",
"ppTerm": "?self.hnc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",... | [
"case self.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk : ℕ\n⊢ ↑i < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 11
} | {
"line": 423,
"column": 4
} | [
{
"pp": "case self\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk : ℕ\n⊢ ((List.drop r (listTransvecCol M)).prod * M) (inl i) (inr ()) = if r ≤ ↑i then 0 else M (inl i) (inr ())",
"ppTerm": "?self",
"assigned": true,
"... | [] | | self => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 26
} | {
"line": 435,
"column": 4
} | [
{
"pp": "case succ\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nIH : k ≤ r → (M * (List.take k (listTransvecRow M)).prod) i (inr ()) = M i (inr ())\nhk : k + 1 ≤ r\n⊢ (M * (List.take (k + 1) (listTransvecRow M)).prod) i (inr ()) = M i (inr ())",
... | [
"case succ\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nIH : k ≤ r → (M * (List.take k (listTransvecRow M)).prod) i (inr ()) = M i (inr ())\nhk : k + 1 ≤ r\nhkr : k < r\n⊢ (M * (List.take (k + 1) (listTransvecRow M)).prod) i (inr ()) = M i (inr ())"
... | have hkr : k < r := hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 71
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case inl\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nh_card : Fintype.card n = 0\nthis : IsEmpty n\n⊢ A.adjugate.det = A.det ^ (Fintype.card n - 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [h_card, Nat.zero_sub, pow_zero, adjugate_subsingleton, det_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 13
} | {
"line": 451,
"column": 14
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nA : (listTransvecRow M).length = r\n⊢ (M * (List.take r (listTransvecRow M)).prod) i (inr ()) = M i (inr ())",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nA : (listTransvecRow M).length = r\n⊢ (M * (List.take r (listTransvecRow M)).prod) i (inr ()) = M i (inr ())"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Kronecker | {
"line": 236,
"column": 25
} | {
"line": 236,
"column": 37
} | {
"line": 236,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nα : Type u_3\nβ : Type u_5\nγ : Type u_7\nm : Type u_10\nn : Type u_11\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : AddCommMonoid α\ninst✝⁶ : AddCommMonoid β\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : Module R α\ninst✝³ : Module R γ\... | [
"R : Type u_1\nS : Type u_2\nα : Type u_3\nβ : Type u_5\nγ : Type u_7\nm : Type u_10\nn : Type u_11\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : AddCommMonoid α\ninst✝⁶ : AddCommMonoid β\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : Module R α\ninst✝³ : Module R γ\ninst✝² : Mo... | Matrix.diag, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 44
} | {
"line": 509,
"column": 45
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow ((listTransvecCol M).prod * M)\n⊢ ((listTransvecCol M).prod * M) (inr ()) (inr ()) ≠ 0",
"ppTerm": "?m.61",
"assigned": true,
... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow ((listTransvecCol M).prod * M)\n⊢ ¬M (inr ()) (inr ()) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 44
} | {
"line": 520,
"column": 45
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M * (listTransvecRow M).prod)\n⊢ (M * (listTransvecRow M).prod) (inr ()) (inr ()) ≠ 0",
"ppTerm": "?m.73",
"assigned": true,
... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M * (listTransvecRow M).prod)\n⊢ ¬M (inr ()) (inr ()) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 285,
"column": 18
} | {
"line": 285,
"column": 70
} | {
"line": 285,
"column": 71
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\nx✝¹ x✝ : Matrix n m α\nh : (fun x ↦ A * x) x✝¹ = (fun x ↦ A * x) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"use... | [
"m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\nx✝¹ x✝ : Matrix n m α\nh : (fun x ↦ A * x) x✝¹ = (fun x ↦ A * x) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 289,
"column": 20
} | {
"line": 289,
"column": 73
} | {
"line": 289,
"column": 74
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\na x : Matrix m n α\nhax : (fun x ↦ x * A) a = (fun x ↦ x * A) x\n⊢ a = x",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\na x : Matrix m n α\nhax : (fun x ↦ x * A) a = (fun x ↦ x * A) x\n⊢ a = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 56
} | {
"line": 307,
"column": 57
} | [
{
"pp": "l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\nh : A * B = 1\nx✝¹ x✝ : Matrix l m α\ng : (fun x ↦ x * A) x✝¹ = (fun x ↦ x * A) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.17",
"assigne... | [
"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\nh : A * B = 1\nx✝¹ x✝ : Matrix l m α\ng : (fun x ↦ x * A) x✝¹ = (fun x ↦ x * A) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 311,
"column": 18
} | {
"line": 311,
"column": 74
} | {
"line": 311,
"column": 75
} | [
{
"pp": "l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\nh : A * B = 1\nx✝¹ x✝ : Matrix m l α\ng : (fun x ↦ B * x) x✝¹ = (fun x ↦ B * x) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.17",
"assigne... | [
"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\nh : A * B = 1\nx✝¹ x✝ : Matrix m l α\ng : (fun x ↦ B * x) x✝¹ = (fun x ↦ B * x) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 559,
"column": 35
} | {
"line": 559,
"column": 46
} | {
"line": 559,
"column": 47
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nH : M.IsTwoBlockDiagonal\n⊢ ((List.map toMatrix []).prod * M * (List.map toMatrix []).prod).IsTwoBlockDiagonal",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.IsTwoBl... | [
"𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nH : M.IsTwoBlockDiagonal\n⊢ M.IsTwoBlockDiagonal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 56
} | {
"line": 577,
"column": 4
} | [
{
"pp": "case pos.inl\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) = 0\nH :\n (¬∀ (i : Fin r) (j : Unit), of (fun i j ↦ M (inl i) (inr j)) i j = 0 i j) ∨\n ¬∀ (i : Unit) (j : Fin r), of (fun i j ↦ M (inr i) (inl j)) i j = 0 i j\ni : Fin r\nh ... | [
"case pos.inl\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) = 0\nH :\n (¬∀ (i : Fin r) (j : Unit), of (fun i j ↦ M (inl i) (inr j)) i j = 0 i j) ∨\n ¬∀ (i : Unit) (j : Fin r), of (fun i j ↦ M (inr i) (inl j)) i j = 0 i j\ni : Fin r\nh : M (inl i) ... | let M' := transvection (inr Unit.unit) (inl i) 1 * M | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 584,
"column": 4
} | {
"line": 584,
"column": 51
} | {
"line": 585,
"column": 4
} | [
{
"pp": "case pos.inr\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) = 0\nH :\n (¬∀ (i : Fin r) (j : Unit), of (fun i j ↦ M (inl i) (inr j)) i j = 0 i j) ∨\n ¬∀ (i : Unit) (j : Fin r), of (fun i j ↦ M (inr i) (inl j)) i j = 0 i j\ni : Fin r\nh ... | [
"case pos.inr\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) = 0\nH :\n (¬∀ (i : Fin r) (j : Unit), of (fun i j ↦ M (inl i) (inr j)) i j = 0 i j) ∨\n ¬∀ (i : Unit) (j : Fin r), of (fun i j ↦ M (inr i) (inl j)) i j = 0 i j\ni : Fin r\nh : M (inr ())... | let M' := M * transvection (inl i) (inr unit) 1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 414,
"column": 85
} | {
"line": 414,
"column": 98
} | {
"line": 415,
"column": 6
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni : n\nthis✝ : Nonempty n\nn' : ℕ\nhn' : Fintype.card n = n'.succ\nx✝ : Trunc (n ≃ Fin n'.succ)\ne : n ≃ Fin n'.succ\nA' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A\nthis : A.det = ∑ ... | [
"n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni : n\nthis✝ : Nonempty n\nn' : ℕ\nhn' : Fintype.card n = n'.succ\nx✝ : Trunc (n ≃ Fin n'.succ)\ne : n ≃ Fin n'.succ\nA' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A\nthis : A.det = ∑ i_1, A (e.sy... | ← e.sum_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 102,
"column": 60
} | {
"line": 102,
"column": 71
} | {
"line": 102,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhM : M.BlockTriangular b\nx✝ : (M + N).BlockTriangular b\n⊢ N.BlockTriangular b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhM : M.BlockTriangular b\nx✝ : (M + N).BlockTriangular b\n⊢ N.BlockTriangular b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 105,
"column": 60
} | {
"line": 105,
"column": 71
} | {
"line": 105,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhN : N.BlockTriangular b\nx✝ : (M + N).BlockTriangular b\n⊢ M.BlockTriangular b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhN : N.BlockTriangular b\nx✝ : (M + N).BlockTriangular b\n⊢ M.BlockTriangular b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 108,
"column": 60
} | {
"line": 108,
"column": 71
} | {
"line": 108,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhM : M.BlockTriangular b\nx✝ : (M - N).BlockTriangular b\n⊢ N.BlockTriangular b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhM : M.BlockTriangular b\nx✝ : (M - N).BlockTriangular b\n⊢ N.BlockTriangular b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 111,
"column": 60
} | {
"line": 111,
"column": 71
} | {
"line": 111,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhN : N.BlockTriangular b\nx✝ : (M - N).BlockTriangular b\n⊢ M.BlockTriangular b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM N : Matrix m m R\nb : m → α\ninst✝¹ : LT α\ninst✝ : AddGroup R\nhN : N.BlockTriangular b\nx✝ : (M - N).BlockTriangular b\n⊢ M.BlockTriangular b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 47
} | {
"line": 635,
"column": 48
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit A ↔ IsUnit B\n⊢ A⁻¹ + B⁻¹ = A⁻¹ * (A + B) * B⁻¹",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.add",
"HMul.hMul",
"con... | [
"n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit A ↔ IsUnit B\n⊢ A⁻¹ʳ + B⁻¹ʳ = A⁻¹ʳ * (A + B) * B⁻¹ʳ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 47
} | {
"line": 640,
"column": 48
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit A ↔ IsUnit B\n⊢ A⁻¹ - B⁻¹ = A⁻¹ * (B - A) * B⁻¹",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGr... | [
"n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit A ↔ IsUnit B\n⊢ A⁻¹ʳ - B⁻¹ʳ = A⁻¹ʳ * (B - A) * B⁻¹ʳ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 734,
"column": 75
} | {
"line": 743,
"column": 14
} | {
"line": 745,
"column": 0
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nP : Matrix n n 𝕜 → Prop\nM : Matrix n n 𝕜\nhMdet : M.det ≠ 0\nhdiag : ∀ (D : n → 𝕜), (diagonal D).det ≠ 0 → P (diagonal D)\nhtransvec : ∀ (t : TransvectionStruct n 𝕜), P t.toMatrix\nhmul : ∀ (A B : Matrix n n... | [] | by
let Q : Matrix n n 𝕜 → Prop := fun N => det N ≠ 0 ∧ P N
have : Q M := by
apply diagonal_transvection_induction Q M
· grind
· intro t
exact ⟨by simp, htransvec t⟩
· intro A B QA QB
exact ⟨by simp [QA.1, QB.1], hmul A B QA.1 QB.1 QA.2 QB.2⟩
exact this.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 40
} | {
"line": 283,
"column": 41
} | [
{
"pp": "m : Type u_3\nR : Type v\ninst✝³ : CommRing R\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"ppTerm": "?m.40",
"assigned": true,
"used... | [
"m : Type u_3\nR : Type v\ninst✝³ : CommRing R\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 46
} | {
"line": 144,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R[X]\nr : R\n⊢ eval ((scalar n) r) (matPolyEquiv M) = M.map (eval r)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R[X]\nr : R\n⊢ eval ((scalar n) r) (matPolyEquiv M) = M.map (eval r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 31
} | {
"line": 149,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R[X]\nr : R\ni j : n\n⊢ eval ((scalar n) r) (matPolyEquiv M) i j = eval r (M i j)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | [
"R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R[X]\nr : R\ni j : n\n⊢ M.map (eval r) i j = eval r (M i j)"
] | matPolyEquiv_eval_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 47
} | {
"line": 142,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nm : Type u_3\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nM : Matrix m m R\nt : R\ni j : m\n⊢ (evalRingHom t).mapMatrix ((scalar m) X - C.mapMatrix M) i j = ((scalar m) t - M) i j",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Polynomial.C... | [] | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 15
} | {
"line": 366,
"column": 16
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p ... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i ↦ ¬p i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 22
} | {
"line": 367,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p ... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nh_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i ↦ ¬p i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 43
} | {
"line": 382,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\nA : Type u_8\nM N : Matrix m m R\nb : m → α\ninst✝⁶ : CommRing R\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝... | [
"α : Type u_1\nβ : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : α → Type u_6\nn' : α → Type u_7\nR : Type v\nA : Type u_8\nM N : Matrix m m R\nb : m → α\ninst✝⁶ : CommRing R\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrder α\ninst✝ : Invertibl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 15
} | {
"line": 396,
"column": 16
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nq : m → Prop := fun i ↦ ¬b i < k\nh_sum : M⁻¹.toBlock q p... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nq : m → Prop := fun i ↦ ¬b i < k\nh_sum : M⁻¹.toBlock q p * M.toBlock... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 397,
"column": 65
} | {
"line": 397,
"column": 85
} | {
"line": 397,
"column": 86
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nq : m → Prop := fun i ↦ ¬b i < k\nh_sum : M⁻¹.toBlock q p... | [
"α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nq : m → Prop := fun i ↦ ¬b i < k\nh_sum : M⁻¹.toBlock q p * M.toBlock... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.LinearCombination.Lemmas | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 27
} | {
"line": 169,
"column": 28
} | [
{
"pp": "α : Type u_1\na a' b b' : α\ninst✝² : AddCommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelAddMonoid α\np : a ≤ b\nH : a' + b < b' + b\n⊢ a' < b'",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\na a' b b' : α\ninst✝² : AddCommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelAddMonoid α\np : a ≤ b\nH : a' + b < b' + b\n⊢ a' < b'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Basic | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 13
} | {
"line": 288,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : (Matrix n n R)ˣ\nN : Matrix n n R\n⊢ ((↑M)⁻¹ * N * ↑M).charpoly = N.charpoly",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : (Matrix n n R)ˣ\nN : Matrix n n R\n⊢ ((↑M)⁻¹ * N * ↑M).charpoly = N.charpoly"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsUnit x\nH : IsNilpotent x\n⊢ False",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsUnit x\nH : IsNilpotent x\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 95,
"column": 32
} | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsNilpotent x\n⊢ ¬¬IsNilpotent x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"NPow.toPow",
"Semiring.toMonoid",
"IsNilpotent",
"Ring.toSemiring",
"Eq",
... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsNilpotent x\n⊢ IsNilpotent x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 67,
"column": 4
} | {
"line": 69,
"column": 51
} | {
"line": 69,
"column": 52
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix l m α\nB : Matrix l n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible D\n... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix l m α\nB : Matrix l n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible D\n⊢ fromBlocks... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 54,
"column": 26
} | {
"line": 54,
"column": 52
} | {
"line": 54,
"column": 53
} | [
{
"pp": "R : Type u_1\nr : R\ninst✝ : Semiring R\nk : ℕ\n⊢ C r ^ k = 0 ↔ r ^ k = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"id",
"Polynomial",
"NPow.toPow",
"_private.Mathlib.RingTheory... | [
"R : Type u_1\nr : R\ninst✝ : Semiring R\nk : ℕ\n⊢ C (r ^ k) = 0 ↔ r ^ k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 79,
"column": 40
} | {
"line": 79,
"column": 83
} | {
"line": 79,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : R[X]\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nx✝ : i ∈ P.support\n⊢ IsNilpotent ((fun n a ↦ (monomial n) a) i (P.coeff i))",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"CommSemiring.to... | [
"R : Type u_1\ninst✝ : CommRing R\nP : R[X]\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nx✝ : i ∈ P.support\n⊢ IsNilpotent (P.coeff i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 98
} | {
"line": 117,
"column": 8
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A B 0 D)\... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A B 0 D)\nthis :\n f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 98
} | {
"line": 123,
"column": 8
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A B 0 D)\... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A B 0 D)\nthis :\n f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 48
} | {
"line": 84,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\ni k : ℕ\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n⊢ r ^ k = 0",
"ppTerm": "?m.103",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\ni k : ℕ\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n⊢ r ^ k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 23
} | {
"line": 86,
"column": 24
} | [
{
"pp": "case refine_2.zero\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\nhp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\n⊢ IsNilpotent ((p + C r).coeff 0)",
"ppTerm": "?refine_2.zero",
"assigned": tru... | [
"case refine_2.zero\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\nhp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\n⊢ IsNilpotent r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 98
} | {
"line": 135,
"column": 8
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A 0 C D)\... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A 0 C D)\nthis :\n (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 16
} | [
{
"pp": "case refine_2.succ\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\nhp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : ℕ\n⊢ IsNilpotent p",
"ppTerm": "?refine_2.succ",
"assigned": false,
"us... | [
"case refine_2.succ\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nr : R\nhp₀ : p.coeff 0 = 0\nx✝ : r ≠ 0\nhp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : ℕ\n⊢ IsNilpotent p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 98
} | {
"line": 142,
"column": 8
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A 0 C D)\... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A 0 C D)\nthis :\n (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 15
} | {
"line": 92,
"column": 16
} | [
{
"pp": "case refine_3.succ\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nx✝ : p ≠ 0\nhnp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : ℕ\n⊢ IsNilpotent ((p * X).coeff (i + 1))",
"ppTerm": "?refine_3.succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_3.succ\nR : Type u_1\ninst✝ : CommRing R\nP p : R[X]\nx✝ : p ≠ 0\nhnp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : ℕ\n⊢ IsNilpotent (p.coeff i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 43
} | {
"line": 98,
"column": 44
} | [
{
"pp": "case refine_1.inl\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent ((reflect N P).coeff i)\ni : ℕ\nhi : i ≤ N\n⊢ IsNilpotent (P.coeff i)",
"ppTerm": "?refine_1.inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"case refine_1.inl\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent ((reflect N P).coeff i)\ni : ℕ\nhi : i ≤ N\n⊢ IsNilpotent (P.coeff i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 30
} | {
"line": 100,
"column": 31
} | [
{
"pp": "case refine_2.inl\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nhi : i ≤ N\n⊢ IsNilpotent ((reflect N P).coeff i)",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.re... | [
"case refine_2.inl\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nhi : i ≤ N\n⊢ IsNilpotent (P.coeff (N - i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 40
} | {
"line": 101,
"column": 41
} | [
{
"pp": "case refine_2.inr\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nhi : N < i\n⊢ IsNilpotent ((reflect N P).coeff i)",
"ppTerm": "?refine_2.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.re... | [
"case refine_2.inr\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nN : ℕ\nhN : P.natDegree ≤ N\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\ni : ℕ\nhi : N < i\n⊢ IsNilpotent (P.coeff i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 47
} | {
"line": 98,
"column": 6
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤\n⊢ ⨅ s, (↑↑s).encard ≤ ⨅ s, ⨅ (_ : span R s = p), s.encard",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Subm... | [
"case a\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤\ns : Set M\nh : span R s = p\n⊢ ⨅ s, (↑↑s).encard ≤ s.encard"
] | refine le_iInf fun s ↦ le_iInf fun h ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 67,
"column": 78
} | {
"line": 83,
"column": 37
} | {
"line": 85,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\n⊢ (M.charpoly - ∏ i, (X - C (M i i))).degree < ↑(Fintype.card n - 1)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"Finset.mem_univ",... | [] | by
rw [charpoly, det_apply', ← insert_erase (mem_univ (Equiv.refl n)),
sum_insert (notMem_erase (Equiv.refl n) univ), add_comm]
simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl]
rw [← mem_degreeLT]
apply Submodule.sum_me... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.SpanRank | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 22
} | {
"line": 116,
"column": 23
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : ⨅ s, #↑↑s < ℵ₀\ns : { s // span R s = p }\nhs : (fun s ↦ #↑↑s) s = ⨅ s, #↑↑s\n⊢ (↑s).Finite",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"case mp\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : ⨅ s, #↑↑s < ℵ₀\ns : { s // span R s = p }\nhs : (fun s ↦ #↑↑s) s = ⨅ s, #↑↑s\n⊢ (↑s).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 35
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ ↑(toNat p.spanRank) = p.spanRank ↔ p.spanRank < ℵ₀",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Cardinal.cast_toNat_eq_iff_lt_aleph0",
"Submodule.spanR... | [] | exact cast_toNat_eq_iff_lt_aleph0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Finiteness.Subalgebra | {
"line": 58,
"column": 58
} | {
"line": 58,
"column": 80
} | {
"line": 58,
"column": 81
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nh : M.FG\nn✝ n : ℕ\nih : (M ^ n).FG\n⊢ (M ^ n.succ).FG",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"HMul.hMul",
"IsS... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nh : M.FG\nn✝ n : ℕ\nih : (M ^ n).FG\n⊢ (M ^ n * M).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 66,
"column": 2
} | {
"line": 68,
"column": 27
} | {
"line": 69,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Fintype ι\nM : Type u_2\ninst✝² : AddCommGroup M\nR : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Module R M\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\nx y : Module.End R M\ne : (fromEnd R b) x = (fromEnd R b) y\nm : M\n⊢ x m = y m",
"ppTerm": "?m.43",
"assigned": tr... | [
"ι : Type u_1\ninst✝³ : Fintype ι\nM : Type u_2\ninst✝² : AddCommGroup M\nR : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Module R M\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\nx y : Module.End R M\ne : (fromEnd R b) x = (fromEnd R b) y\nm : ι → R\n⊢ x ((Fintype.linearCombination R b) m) = y ((Fintype.linearCom... | obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.linearCombination R b) := by
rw [(Fintype.range_linearCombination R b).trans hb]
exact Submodule.mem_top | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Module.SpanRank | {
"line": 198,
"column": 32
} | {
"line": 198,
"column": 43
} | {
"line": 198,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : s.encard = ↑p.spanFinrank\nhs₂ : span R s = p\ns_f : s.Finite\n⊢ span R ↑s_f.toFinset = p",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : s.encard = ↑p.spanFinrank\nhs₂ : span R s = p\ns_f : s.Finite\n⊢ span R s = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 198,
"column": 29
} | {
"line": 198,
"column": 47
} | {
"line": 198,
"column": 47
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : s.encard = ↑p.spanFinrank\nhs₂ : span R s = p\ns_f : s.Finite\n⊢ span R ↑s_f.toFinset = p",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
... | [] | by simpa using hs₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.SpanRank | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 61
} | {
"line": 199,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : s.encard = ↑p.spanFinrank\nhs₂ : span R s = p\ns_f : s.Finite\n⊢ s_f.toFinset.card = p.spanFinrank",
"ppTerm": "?m.61",
"assigned": false,
"usedConstant... | [
"R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : s.encard = ↑p.spanFinrank\nhs₂ : span R s = p\ns_f : s.Finite\n⊢ s_f.toFinset.card = p.spanFinrank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 42
} | {
"line": 256,
"column": 43
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nhp : p.FG\n⊢ p.generators.Finite",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"Partia... | [
"R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nhp : p.FG\n⊢ #↑p.generators < ℵ₀"
] | ← Cardinal.lt_aleph0_iff_set_finite, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 51
} | {
"line": 133,
"column": 52
} | [
{
"pp": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nb : ι → M\ninst✝ : DecidableEq ι\nr : R\n⊢ Represents b ((Algebra.algebraMap R (Matrix ι ι R)) r) ((Algebra.algebraMap R (Module.End R M)) r)",
"ppTerm": "?m.25",
"as... | [
"ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nb : ι → M\ninst✝ : DecidableEq ι\nr : R\n⊢ Represents b (r • 1) (r • 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 291,
"column": 49
} | {
"line": 291,
"column": 84
} | {
"line": 291,
"column": 85
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.spanFinrank = 1\nfg : p.FG\n⊢ ∃ a, p.generators = {a}",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.spanFinrank = 1\nfg : p.FG\n⊢ ∃ a, p.generators = {a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 13
} | {
"line": 317,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\np : Submodule R M\n⊢ (map f p).spanRank ≤ p.spanRank",
"ppTerm"... | [
"R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\np : Submodule R M\n⊢ (map f p).spanRank ≤ p.spanRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 332,
"column": 34
} | {
"line": 332,
"column": 86
} | {
"line": 332,
"column": 87
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nL : Type v\ninst✝² : AddCommMonoid L\ninst✝¹ : Module S L\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] L\nhf : Function.Injective ⇑f\np : Submodule R M\ns : Set M\... | [
"R : Type u_1\nS : Type u_2\nM : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nL : Type v\ninst✝² : AddCommMonoid L\ninst✝¹ : Module S L\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] L\nhf : Function.Injective ⇑f\np : Submodule R M\ns : Set M\nhs : Cardin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 13
} | {
"line": 337,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\nhf : Function.Injective ⇑f\np : Submodule R M\n⊢ (map f p).spanRank... | [
"R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\nhf : Function.Injective ⇑f\np : Submodule R M\n⊢ (map f p).spanRank = p.spanRan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | {
"line": 347,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\n⊢ f.range.spanRank ≤ ⊤.spanRank",
"ppTerm": "?m.50",
"assig... | [
"R : Type u_1\nS : Type u_2\nM N : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module S N\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] N\n⊢ f.range.spanRank ≤ ⊤.spanRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 13
} | {
"line": 351,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ ⊤.spanRank = p.spanRank",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ ⊤.spanRank = p.spanRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 220,
"column": 15
} | {
"line": 220,
"column": 26
} | {
"line": 220,
"column": 27
} | [
{
"pp": "M : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nI : Ideal R\nh✝ : Nontrivial R\ns : Finset M\nhs_card : s.card = ⊤.spanFinrank\nhs_span : Submodule.span R ↑s = ⊤\nthis : Submodule.span R (Set.range Subtype.val) = ⊤\nA : ↥(isRepre... | [
"M : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nI : Ideal R\nh✝ : Nontrivial R\ns : Finset M\nhs_card : s.card = ⊤.spanFinrank\nhs_span : Submodule.span R ↑s = ⊤\nthis : Submodule.span R (Set.range Subtype.val) = ⊤\nA : ↥(isRepresentation R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 13
} | {
"line": 407,
"column": 14
} | [
{
"pp": "R S : Type u\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nI : Ideal R\n⊢ Submodule.spanRank (map f I) ≤ Submodule.spanRank I",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R S : Type u\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nI : Ideal R\n⊢ Submodule.spanRank (map f I) ≤ Submodule.spanRank I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRank | {
"line": 412,
"column": 2
} | {
"line": 412,
"column": 13
} | {
"line": 412,
"column": 14
} | [
{
"pp": "R S : Type u\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R ≃+* S\nI : Ideal R\n⊢ Submodule.spanRank (map f I) = Submodule.spanRank I",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R S : Type u\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R ≃+* S\nI : Ideal R\n⊢ Submodule.spanRank (map f I) = Submodule.spanRank I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 75
} | {
"line": 179,
"column": 8
} | [
{
"pp": "case succ.h₀.succ\nR : Type u\ninst✝ : CommRing R\nn : ℕ\nIH : ∀ (M : Matrix (Fin (n + 1)) (Fin (n + 1)) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace\nM : Matrix (Fin (n + 1 + 1)) (Fin (n + 1 + 1)) R\ni : Fin (n + 1).succ\na✝ : i ∈ univ\nhi : i ≠ 0\nj : Fin (n + 1)\n⊢ j.succ ≠ Fin.castSucc ... | [
"case succ.h₀.succ.h\nR : Type u\ninst✝ : CommRing R\nn : ℕ\nIH : ∀ (M : Matrix (Fin (n + 1)) (Fin (n + 1)) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace\nM : Matrix (Fin (n + 1 + 1)) (Fin (n + 1 + 1)) R\ni : Fin (n + 1).succ\na✝ : i ∈ univ\nhi : i ≠ 0\nj : Fin (n + 1)\n⊢ Fin.castSucc 0 < i"
] | · exact (bne_iff_ne (a := Fin.succ j) (b := Fin.castSucc 0)).mp rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 179,
"column": 34
} | {
"line": 179,
"column": 78
} | {
"line": 180,
"column": 4
} | [
{
"pp": "case succ.h₀.succ.h\nR : Type u\ninst✝ : CommRing R\nn : ℕ\nIH : ∀ (M : Matrix (Fin (n + 1)) (Fin (n + 1)) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace\nM : Matrix (Fin (n + 1 + 1)) (Fin (n + 1 + 1)) R\ni : Fin (n + 1).succ\na✝ : i ∈ univ\nhi : i ≠ 0\nj : Fin (n + 1)\n⊢ 0 < i",
"ppTerm"... | [] | exact lt_of_le_of_ne (Fin.zero_le _) hi.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 66,
"column": 31
} | {
"line": 66,
"column": 76
} | {
"line": 66,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Ring T\nf : R →+* S\ng : S →+* T\nx : T\np : R[X]\nhp : p.Monic\nhx : eval₂ (g.comp f) x p = 0\n⊢ eval₂ g x (Polynomial.map f p) = 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"E... | [
"R : Type u_1\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Ring T\nf : R →+* S\ng : S →+* T\nx : T\np : R[X]\nhp : p.Monic\nhx : eval₂ (g.comp f) x p = 0\n⊢ eval x (Polynomial.map (g.comp f) p) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 48
} | {
"line": 211,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nr : R\nM : Matrix n n R\n⊢ (1 + r • M).det = 1 + M.trace * r + eval r (1 + X • M.map ⇑C).det.divX.divX * r ^ 2",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nr : R\nM : Matrix n n R\n⊢ (1 + r • M).det = 1 + M.trace * r + eval r (1 + X • M.map ⇑C).det.divX.divX * r ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 72
} | {
"line": 222,
"column": 8
} | [
{
"pp": "case pos.«_@».Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff.4094078725._hygCtx._hyg.105.«2»\nR : Type u\ninst✝³ : CommRing R\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix n n R\ninst✝ : Nontrivial R\nhn : Fintype.card n = 2\nthis : Nonempty n\n⊢ M.charpoly.coeff 2 = (X ^ 2 - C M.tra... | [
"case pos.«_@».Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff.4094078725._hygCtx._hyg.105.«2»\nR : Type u\ninst✝³ : CommRing R\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix n n R\ninst✝ : Nontrivial R\nhn : Fintype.card n = 2\nthis : Nonempty n\n⊢ M.charpoly.coeff 2 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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