module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 339,
"column": 55
} | {
"line": 339,
"column": 83
} | {
"line": 339,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : Linea... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : LinearIndependent... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 431,
"column": 8
} | {
"line": 431,
"column": 19
} | {
"line": 431,
"column": 20
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | [
"case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.root i = ∑ j ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 13
} | {
"line": 340,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : Linea... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : LinearIndependent... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 356,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\nin... | [] | refine Or.inl fun k₁ k₂ ↦ ?_
have hk := B.pairing_mul_eq_pairing_mul_swap k₁ k₂
rcases this with h₀ | h₀ <;> rcases key k₁ with h₁ | h₁ <;> rcases key k₂ with h₂ | h₂ <;>
simp only [h₁, h₂, h₀, ← mul_assoc, mul_comm, mul_eq_mul_right_iff] at hk <;>
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 356,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\nin... | [] | refine Or.inl fun k₁ k₂ ↦ ?_
have hk := B.pairing_mul_eq_pairing_mul_swap k₁ k₂
rcases this with h₀ | h₀ <;> rcases key k₁ with h₁ | h₁ <;> rcases key k₂ with h₂ | h₂ <;>
simp only [h₁, h₂, h₀, ← mul_assoc, mul_comm, mul_eq_mul_right_iff] at hk <;>
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 409,
"column": 2
} | {
"line": 411,
"column": 26
} | {
"line": 412,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni✝ j✝ : ι\nh✝ : Li... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni✝ j✝ : ι\nh✝ : LinearIndepend... | have h₂ : P.reflection i (P.root <| P.chainBotIdx i j) ∈ range P.root := by
rw [← root_reflectionPerm]
exact mem_range_self _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 374,
"column": 2
} | {
"line": 375,
"column": 39
} | {
"line": 375,
"column": 40
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\ninst✝¹ : P.IsRed... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\ninst✝¹ : P.IsReduced\ninst✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Star | {
"line": 29,
"column": 36
} | {
"line": 29,
"column": 66
} | {
"line": 29,
"column": 67
} | [
{
"pp": "n : ℕ\nhn : Even n\nx : ℤ\nhx : x ∈ nonneg ℤ\n⊢ x = x.natAbs • 1 ^ n",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"abs"... | [
"n : ℕ\nhn : Even n\nx : ℤ\nhx : x ∈ nonneg ℤ\n⊢ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Star | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 23
} | {
"line": 35,
"column": 24
} | [
{
"pp": "⊢ closure (range fun x ↦ x * x) = nonneg ℤ",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ closure (range fun x ↦ x * x) = nonneg ℤ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 177,
"column": 27
} | {
"line": 177,
"column": 43
} | {
"line": 177,
"column": 44
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nhli : ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nhli : LinearIndepO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 178,
"column": 27
} | {
"line": 178,
"column": 43
} | {
"line": 178,
"column": 44
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nhli : ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nhli : LinearIndepO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 557,
"column": 67
} | {
"line": 557,
"column": 78
} | {
"line": 557,
"column": 79
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallographic\ninst✝... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 65
} | {
"line": 65,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K₁ K₂ : α\nε0 : 0 < ε\na₁ a₂ b₁ b₂ : β\nM : α := max 1 (max K₁ K₂)\nK0 : 0 < M\nεK : 0 < ε / 2 / M\nh₁ : abv (a₁ - b₁) < ε / 2 / M\nh₂ : abv... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K₁ K₂ : α\nε0 : 0 < ε\na₁ a₂ b₁ b₂ : β\nM : α := max 1 (max K₁ K₂)\nK0 : 0 < M\nεK : 0 < ε / 2 / M\nh₁ : abv (a₁ - b₁) < ε / 2 / M\nh₂ : abv (a₂ - b₂) <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 15
} | {
"line": 117,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.... | [
"case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.succ))\nhR :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 44
} | {
"line": 212,
"column": 45
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 125,
"column": 31
} | {
"line": 125,
"column": 53
} | {
"line": 125,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ abv ((fun x_1 ↦ x) j - (fun x_1 ↦ x) 0) < ε",
"ppTerm": "?m.23",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 32
} | {
"line": 368,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\nε : α\nε0 : ε > 0\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑f j) < ε / 2 ∧ abv (↑g j) < ε / 2\nj : ℕ\ni... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\nε : α\nε0 : ε > 0\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑f j) < ε / 2 ∧ abv (↑g j) < ε / 2\nj : ℕ\nij : j ≥ x✝\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 35
} | {
"line": 389,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\n⊢ (f - g).LimZero",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\n⊢ (f + -g).LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 13
} | {
"line": 392,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhfg : (f - g).LimZero\n⊢ (g - f).LimZero",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"use... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhfg : (f - g).LimZero\n⊢ (g - f).LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 395,
"column": 31
} | {
"line": 395,
"column": 57
} | {
"line": 395,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ abv (↑0 j) < ε",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 408,
"column": 21
} | {
"line": 408,
"column": 32
} | {
"line": 408,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ : CauSeq β abv\nf : (x✝ - y✝).LimZero\nε : α\nhε : ε > 0\n⊢ ∃ i, ∀ j ≥ i, abv (↑(y✝ - x✝) j) < ε",
"ppTerm": "?m.44",
"assigned"... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ : CauSeq β abv\nf : (x✝ - y✝).LimZero\nε : α\nhε : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), i ≤ j → abv (↑y✝ j - ↑x✝ j) < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 409,
"column": 20
} | {
"line": 409,
"column": 31
} | {
"line": 409,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ z✝ : CauSeq β abv\nfg : (x✝ - y✝).LimZero\ngh : (y✝ - z✝).LimZero\n⊢ (x✝ - z✝).LimZero",
"ppTerm": "?m.64",
"assigned": false,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ z✝ : CauSeq β abv\nfg : (x✝ - y✝).LimZero\ngh : (y✝ - z✝).LimZero\n⊢ (x✝ - z✝).LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 418,
"column": 28
} | {
"line": 418,
"column": 61
} | {
"line": 418,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 - g1 ≈ f2 - g2",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants"... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 + -g1 ≈ f2 + -g2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 428,
"column": 15
} | {
"line": 428,
"column": 26
} | {
"line": 428,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : f.LimZero\n⊢ g.LimZero",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : f.LimZero\n⊢ g.LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 428,
"column": 70
} | {
"line": 428,
"column": 81
} | {
"line": 428,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : g.LimZero\n⊢ f.LimZero",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : g.LimZero\n⊢ f.LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 220,
"column": 35
} | {
"line": 220,
"column": 46
} | {
"line": 220,
"column": 47
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 469,
"column": 30
} | {
"line": 469,
"column": 41
} | {
"line": 469,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\n⊢ ¬f.LimZero",
"ppTerm": "?m.80",
"as... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\n⊢ ¬f.LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 617,
"column": 4
} | {
"line": 617,
"column": 15
} | {
"line": 617,
"column": 16
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : CharZero R\ninst✝² : Finite ι\ninst✝¹ : IsDomain R\ninst✝ : P.IsCrystallo... | [
"case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : CharZero R\ninst✝² : Finite ι\ninst✝¹ : IsDomain R\ninst✝ : P.IsCrystallographic\ni :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 470,
"column": 30
} | {
"line": 470,
"column": 41
} | {
"line": 470,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\nhf' : ¬f.LimZero\n⊢ ¬g.LimZero",
"ppTerm"... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\nhf' : ¬f.LimZero\n⊢ ¬g.LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 494,
"column": 2
} | {
"line": 494,
"column": 44
} | {
"line": 495,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Ring β\nabv : β → α\ninst✝² : IsAbsoluteValue abv\nG : Type u_3\ninst✝¹ : SMul G β\ninst✝ : IsScalarTower G β β\nf1 f2 : CauSeq β abv\nc : G\nhf : f1 ≈ f2\n⊢ c • f1 ≈ c • f2",
"ppTerm": "?... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Ring β\nabv : β → α\ninst✝² : IsAbsoluteValue abv\nG : Type u_3\ninst✝¹ : SMul G β\ninst✝ : IsScalarTower G β β\nf1 f2 : CauSeq β abv\nc : G\nhf : f1 ≈ f2\n⊢ c • f1 ≈ c • f2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 500,
"column": 17
} | {
"line": 500,
"column": 45
} | {
"line": 500,
"column": 46
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 : CauSeq β abv\nhf : f1 ≈ f2\nn : ℕ\nih : f1 ^ n ≈ f2 ^ n\n⊢ f1 ^ (n + 1) ≈ f2 ^ (n + 1)",
"ppTerm": "?succ",
"assign... | [
"case succ\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 : CauSeq β abv\nhf : f1 ≈ f2\nn : ℕ\nih : f1 ^ n ≈ f2 ^ n\n⊢ f1 * f1 ^ n ≈ f2 * f2 ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 512,
"column": 68
} | {
"line": 512,
"column": 79
} | {
"line": 512,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : Field α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Ring β\ninst✝¹ : IsDomain β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nh : const abv 1 ≈ const abv 0\nthis : ∀ ε > 0, ∃ i, ∀ (k : ℕ), i ≤ k → abv (1 - 0) < ε\nh2 : 0 < abv 1\ni : ℕ\nhi : ∀ (k ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : Field α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Ring β\ninst✝¹ : IsDomain β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nh : const abv 1 ≈ const abv 0\nthis : ∀ ε > 0, ∃ i, ∀ (k : ℕ), i ≤ k → abv (1 - 0) < ε\nh2 : 0 < abv 1\ni : ℕ\nhi : ∀ (k : ℕ), i ≤ k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 549,
"column": 21
} | {
"line": 549,
"column": 93
} | {
"line": 549,
"column": 94
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 553,
"column": 21
} | {
"line": 553,
"column": 93
} | {
"line": 553,
"column": 94
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 556,
"column": 47
} | {
"line": 556,
"column": 60
} | {
"line": 556,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nhx : x ≠ 0\n⊢ ¬(const abv x).LimZero",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nhx : x ≠ 0\n⊢ ¬x = 0"
] | const_limZero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 619,
"column": 4
} | {
"line": 625,
"column": 50
} | {
"line": 627,
"column": 0
} | [
{
"pp": "case refine_3\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : CharZero R\ninst✝² : Finite ι\ninst✝¹ : IsDomain R\ninst✝ : P.IsCrystallo... | [] | by_cases hm : m < n
· have : m = (⟨m, hm⟩ : Fin n).castSucc := rfl
rw [this, Fin.sum_Iic_castSucc]
simp only [Fin.snoc_castSucc, h₄]
· replace hm : m = n := by lia
replace hm : Finset.Iic m = Finset.univ := by ext; simp [hm, Fin.le_def, Fin.is_le]
simp [hm, Fin.sum_univ_castSucc, ← h₃, ←... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 619,
"column": 4
} | {
"line": 625,
"column": 50
} | {
"line": 627,
"column": 0
} | [
{
"pp": "case refine_3\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : CharZero R\ninst✝² : Finite ι\ninst✝¹ : IsDomain R\ninst✝ : P.IsCrystallo... | [] | by_cases hm : m < n
· have : m = (⟨m, hm⟩ : Fin n).castSucc := rfl
rw [this, Fin.sum_Iic_castSucc]
simp only [Fin.snoc_castSucc, h₄]
· replace hm : m = n := by lia
replace hm : Finset.Iic m = Finset.univ := by ext; simp [hm, Fin.le_def, Fin.is_le]
simp [hm, Fin.sum_univ_castSucc, ← h₃, ←... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.Completion | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 69
} | {
"line": 142,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx y : β\nh : ofRat x = ofRat y\n⊢ x = y",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"α : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx y : β\nh : ofRat x = ofRat y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Completion | {
"line": 200,
"column": 8
} | {
"line": 200,
"column": 73
} | {
"line": 201,
"column": 8
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf g : CauSeq β abv\nfg : f ≈ g\nthis : f.LimZero ↔ g.LimZero\nhf : ¬f.LimZero\nhg : ¬g.LimZero\n⊢ mk (f.inv ... | [
"case neg\nα : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf g : CauSeq β abv\nfg : f ≈ g\nthis : f.LimZero ↔ g.LimZero\nhf : ¬f.LimZero\nhg : ¬g.LimZero\nIf : mk (f.inv hf) * mk ... | have If : mk (inv f hf) * mk f = 1 := mk_eq.2 (inv_mul_cancel hf) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 248,
"column": 38
} | {
"line": 248,
"column": 49
} | {
"line": 248,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 662,
"column": 4
} | {
"line": 662,
"column": 15
} | {
"line": 662,
"column": 16
} | [
{
"pp": "case ind.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallo... | [
"case ind.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallographic\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 246,
"column": 6
} | {
"line": 249,
"column": 40
} | {
"line": 250,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [] | refine ⟨a ri, ?_, this⟩
by_contra contra
replace contra : a ri = 0 := by simpa using contra
simp [contra, P.ne_zero j] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 246,
"column": 6
} | {
"line": 249,
"column": 40
} | {
"line": 250,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [] | refine ⟨a ri, ?_, this⟩
by_contra contra
replace contra : a ri = 0 := by simpa using contra
simp [contra, P.ne_zero j] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 251,
"column": 54
} | {
"line": 251,
"column": 70
} | {
"line": 251,
"column": 71
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 252,
"column": 32
} | {
"line": 252,
"column": 52
} | {
"line": 252,
"column": 53
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 15
} | {
"line": 340,
"column": 16
} | [
{
"pp": "case h.h.h\nx : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝² ≤ mk y✝¹ → mk y✝¹ ≤ mk y✝ → mk y✝² ≤ mk y✝",
"ppTerm": "?h.h.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"Rat",
"Rat.linearOrder",
"id",
"CauSeq.instLE... | [
"case h.h.h\nx : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝² ≤ y✝¹ → y✝¹ ≤ y✝ → y✝² ≤ y✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 15
} | {
"line": 332,
"column": 16
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ < mk y✝ ↔ mk y✝¹ ≤ mk y✝ ∧ ¬mk y✝ ≤ mk y✝¹",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CauSeq.instLTAbs",
"Real.instLE",
"Real",
"abs",
"congrArg",
"Rat",
"Rat.linearOrde... | [
"case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹ < y✝ ↔ y✝¹ ≤ y✝ ∧ ¬y✝ ≤ y✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 23
} | {
"line": 344,
"column": 24
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ ≤ mk y✝ → mk y✝ ≤ mk y✝¹ → mk y✝¹ = mk y✝",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"Rat.linearOrde... | [
"case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹ ≤ y✝ → y✝ ≤ y✝¹ → y✝¹ ≈ y✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 810,
"column": 2
} | {
"line": 810,
"column": 36
} | {
"line": 810,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊔ b ≈ a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊔ b ≈ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 813,
"column": 2
} | {
"line": 813,
"column": 36
} | {
"line": 813,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : a ≤ b\n⊢ a ⊓ b ≈ a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : a ≤ b\n⊢ a ⊓ b ≈ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 46
} | {
"line": 378,
"column": 47
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ 0 < mk y✝¹ → 0 < mk y✝ → 0 < mk y✝¹ * mk y✝",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"CauSeq.Pos",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
... | [
"case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹.Pos → y✝.Pos → (y✝¹ * y✝).Pos"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 40,
"column": 74
} | {
"line": 42,
"column": 76
} | {
"line": 44,
"column": 0
} | [
{
"pp": "n : Type u_2\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Fintype n\nv w : n → R\nh : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u\n⊢ v = w",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"dotProduct",
"congrArg",
... | [] | by
funext x
classical rw [← dotProduct_single_one v x, ← dotProduct_single_one w x, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 48
} | {
"line": 133,
"column": 49
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix m p R\n⊢ A * Aᴴ * B = 0 ↔ Aᴴ * B = 0",
"ppTerm":... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix m p R\n⊢ A * Aᴴ * B = 0 ↔ Aᴴ * B = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 48
} | {
"line": 142,
"column": 49
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix p n R\n⊢ B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0",
"ppTerm... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix p n R\n⊢ B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 71
} | {
"line": 147,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ (Aᴴ * A) *ᵥ v = 0 ↔ A *ᵥ v = 0",
"ppTerm": "?m.31",
"ass... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ (Aᴴ * A) *ᵥ v = 0 ↔ A *ᵥ v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 48
} | {
"line": 151,
"column": 49
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0",
"ppTerm": "?m.35",
"as... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 71
} | {
"line": 156,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ v ᵥ* (Aᴴ * A) = 0 ↔ v ᵥ* Aᴴ = 0",
"ppTerm": "?m.35",
"as... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ v ᵥ* (Aᴴ * A) = 0 ↔ v ᵥ* Aᴴ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 48
} | {
"line": 160,
"column": 49
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ v ᵥ* (A * Aᴴ) = 0 ↔ v ᵥ* A = 0",
"ppTerm": "?m.31",
"ass... | [
"m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ v ᵥ* (A * Aᴴ) = 0 ↔ v ᵥ* A = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 13
} | {
"line": 178,
"column": 14
} | [
{
"pp": "n : Type u_2\nR : Type u_4\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\n⊢ 0 < v ⬝ᵥ star v ↔ v ≠ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Pi.instStarFora... | [
"n : Type u_2\nR : Type u_4\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\n⊢ 0 < v ⬝ᵥ star v ↔ ¬v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 15
} | {
"line": 265,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ ... | [
"case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 15
} | {
"line": 487,
"column": 16
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ ≤ mk y✝ ∨ mk y✝ ≤ mk y✝¹",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"congrArg",
"Rat",
"Rat.linearOrder",
"id",
"CauSeq.instLEAbs",
... | [
"case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹ ≤ y✝ ∨ y✝ ≤ y✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 515,
"column": 74
} | {
"line": 515,
"column": 90
} | {
"line": 515,
"column": 90
} | [
{
"pp": "x : ℝ\nq : ℚ\n⊢ { cauchy := ↑q.num } / ↑q.den = ↑q.num / ↑q.den",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Semiring.toNatCast",
"Int.cast",
"Eq.mpr",
"Real",
"Rat.num",
"instHDiv",
"abs",
"congrArg",
"Real.instDivInvMonoi... | [
"x : ℝ\nq : ℚ\n⊢ ↑q.num / ↑q.den = ↑q.num / ↑q.den"
] | ofCauchy_intCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Basic | {
"line": 570,
"column": 58
} | {
"line": 571,
"column": 48
} | {
"line": 572,
"column": 6
} | [
{
"pp": "b : ℕ\nhb : ∀ {a : ℝ}, 0 < a → a * ↑b + 1 ≤ (a + 1) ^ b\na : ℝ\nha' : 0 < a\n⊢ a * ↑(b + 1) + 1 = (0 + 1) ^ b * a + (a * ↑b + 1)",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"MulOne.toOne",
"Real",
"HMul.hMul... | [] | by
simp [mul_add, add_assoc, add_left_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | {
"line": 134,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ 1 ⊙ A = diagonal d ↔ A.diag = d",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ 1 ⊙ A = diagonal d ↔ A.diag = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 13
} | {
"line": 137,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ A ⊙ 1 = diagonal d ↔ A.diag = d",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ A ⊙ 1 = diagonal d ↔ A.diag = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ 1 ⊙ A = 0 ↔ A.diag = 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ 1 ⊙ A = 0 ↔ A.diag = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 13
} | {
"line": 143,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ A ⊙ 1 = 0 ↔ A.diag = 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ A ⊙ 1 = 0 ↔ A.diag = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 89,
"column": 25
} | {
"line": 89,
"column": 37
} | {
"line": 89,
"column": 38
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_1\ninst✝³ : AddCommMonoid R\ninst✝² : Mul R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA B : Matrix m n R\n⊢ A.vec ⬝ᵥ B.vec = ∑ i, (Aᵀ * B).diag i",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.diag",
"HMul.h... | [
"m : Type u_2\nn : Type u_3\nR : Type u_1\ninst✝³ : AddCommMonoid R\ninst✝² : Mul R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA B : Matrix m n R\n⊢ A.vec ⬝ᵥ B.vec = ∑ x, (Aᵀ * B) x x"
] | Matrix.diag, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 41
} | {
"line": 318,
"column": 42
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\ns : Set... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 319,
"column": 32
} | {
"line": 319,
"column": 43
} | {
"line": 319,
"column": 44
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\ns : Set... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 47
} | {
"line": 169,
"column": 0
} | [
{
"pp": "case hB\nl : Type u_2\nm : Type u_3\nn : Type u_1\nR : Type u_4\ninst✝³ : Semiring R\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix l m R\nB : Matrix m n R\nx : R\ni j : n\n⊢ Commute x (1 i j)",
"ppTerm": "?hB",
"assigned": true,
"usedConstants": [
"NonAsso... | [] | 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.Hermitian | {
"line": 104,
"column": 15
} | {
"line": 104,
"column": 26
} | {
"line": 104,
"column": 27
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\ne : m ≃ n\nh : (A.submatrix ⇑e ⇑e).IsHermitian\n⊢ A.IsHermitian",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\ne : m ≃ n\nh : (A.submatrix ⇑e ⇑e).IsHermitian\n⊢ A.IsHermitian"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 14
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsHermitian\n⊢ A.IsHermitian",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsHermitian\n⊢ A.IsHermitian"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 47
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case hA\nm : Type u_1\nn : Type u_2\np : Type u_4\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq m\nA : Matrix m n R\nB : Matrix n p R\nx : R\ni j : m\n⊢ Commute (1 i j) x",
"ppTerm": "?hA",
"assigned": true,
"usedConstants": [
"NonAsso... | [] | 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.Hermitian | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 28
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝² : AddMonoid α\ninst✝¹ : StarAddMonoid α\ninst✝ : DecidableEq n\nM : n → Matrix m m α\n⊢ (blockDiagonal M).IsHermitian ↔ ∀ (i : n), (M i).IsHermitian",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.block... | [
"α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝² : AddMonoid α\ninst✝¹ : StarAddMonoid α\ninst✝ : DecidableEq n\nM : n → Matrix m m α\n⊢ (fun k ↦ (M k)ᴴ) = M ↔ ∀ (i : n), (M i)ᴴ = M i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | {
"line": 54,
"column": 14
} | [
{
"pp": "case h\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nha : IsUnit A.det\nh : n ≤ m\n⊢ IsUnit (A ^ n).det",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",... | [
"case h\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nha : IsUnit A.det\nh : n ≤ m\n⊢ IsUnit (A.det ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | {
"line": 100,
"column": 14
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\n⊢ A ^ (-1) = A⁻¹",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\n⊢ A ^ (-1) = A⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 15
} | {
"line": 110,
"column": 16
} | [
{
"pp": "case ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit (A ^ ofNat n).det",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"Com... | [
"case ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit (A.det ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | {
"line": 111,
"column": 16
} | [
{
"pp": "case negSucc\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit (A ^ -[n+1]).det",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
... | [
"case negSucc\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit A.det"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 54
} | {
"line": 62,
"column": 55
} | [
{
"pp": "R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis : ∀ (i : n), HasEigenvalue ((toLin b b)... | [
"R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis : ∀ (i : n), HasEigenvalue ((toLin b b) (diagonal d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 16
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis : ∀ (i : n), HasEigenvalue ((... | [] | · rintro ⟨i, rfl⟩
exact this i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 60
} | {
"line": 108,
"column": 61
} | [
{
"pp": "R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nd : n → R\nμ : R\nb : Basis n R M\ninst✝ : IsDomain R\nx : M\nhx : x ∈ maxGenEigenspace ((toLin b b) (diagonal d)) μ\nk : ℕ\nhk : ∀ (j : n), (diagonal... | [
"R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nd : n → R\nμ : R\nb : Basis n R M\ninst✝ : IsDomain R\nx : M\nhx : x ∈ maxGenEigenspace ((toLin b b) (diagonal d)) μ\nk : ℕ\nhk : ∀ (j : n), (diagonal ((d - μ • 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 17
} | {
"line": 197,
"column": 18
} | [
{
"pp": "case inr.ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nhB : B⁻¹ = 0\na✝ : ℕ\n⊢ Commute A (B ^ ofNat a✝)",
"ppTerm": "?inr.ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
... | [
"case inr.ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nhB : B⁻¹ = 0\na✝ : ℕ\n⊢ Commute A (B ^ a✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 73,
"column": 22
} | {
"line": 73,
"column": 33
} | {
"line": 73,
"column": 34
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nd : n → R\nx✝ : (diagonal d).PosSemidef\ni : n\nleft✝ : (diagonal d).IsHermitian\nhP : ∀ (x : n →₀ R), 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d... | [
"n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nd : n → R\nx✝ : (diagonal d).PosSemidef\ni : n\nleft✝ : (diagonal d).IsHermitian\nhP : ∀ (x : n →₀ R), 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d i j * xj\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 61
} | {
"line": 83,
"column": 62
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosSemidef\ne : m → n\nx : m →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M.submatrix e e i j * xj",
"ppTerm": "?m.30",
"assigned": true,
"usedConstant... | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosSemidef\ne : m → n\nx : m →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M (e i) (e j) * xj"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 48
} | {
"line": 89,
"column": 49
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosSemidef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ 0 ≤ x.sum fun x' v' ↦ x.sum fun x v ↦ star v * Mᵀ x x' * v'",
"ppTerm": "?m.60",
"assigned": true,... | [
"n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosSemidef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ 0 ≤ x.sum fun x' v' ↦ x.sum fun x v ↦ v' * M x' x * star v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 100,
"column": 7
} | {
"line": 100,
"column": 18
} | {
"line": 100,
"column": 19
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nx✝ : Mᴴ.PosSemidef\n⊢ M.PosSemidef",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nx✝ : Mᴴ.PosSemidef\n⊢ M.PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 34
} | {
"line": 105,
"column": 35
} | [
{
"pp": "m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : m →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A + B) i j * xj",
"ppTerm": "?m.45",
"assigned": true,
"... | [
"m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : m →₀ R\n⊢ 0 ≤\n (x.sum fun a b ↦ x.sum fun a_1 ↦ HMul.hMul (star b * A a a_1)) +\n x.sum fun a b ↦ x.sum fun a_1 ↦ HMul.hMul (star ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 65
} | {
"line": 111,
"column": 66
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁹ : Ring R\ninst✝⁸ : PartialOrder R\ninst✝⁷ : StarRing R\nα : Type u_5\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : StarRing α\ninst✝³ : StarOrderedRing α\ninst✝² : Algebra α R\ninst✝¹ : StarModule α R\ninst✝ : PosSMulMono α R\nx : Matrix n n R\nhx : x.Po... | [
"n : Type u_2\nR : Type u_3\ninst✝⁹ : Ring R\ninst✝⁸ : PartialOrder R\ninst✝⁷ : StarRing R\nα : Type u_5\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : StarRing α\ninst✝³ : StarOrderedRing α\ninst✝² : Algebra α R\ninst✝¹ : StarModule α R\ninst✝ : PosSMulMono α R\nx : Matrix n n R\nhx : x.PosSemidef\na ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 49
} | {
"line": 117,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"ppTerm": "?m.43",
"assigned": true,
"usedConstan... | [] | 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.PosDef | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 49
} | {
"line": 117,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"ppTerm": "?m.43",
"assigned": true,
"usedConstan... | [] | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 49
} | {
"line": 117,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"ppTerm": "?m.43",
"assigned": true,
"usedConstan... | [] | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ A i i",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ A i i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 146,
"column": 15
} | {
"line": 146,
"column": 26
} | {
"line": 146,
"column": 27
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\ne : m ≃ n\nh : (M.submatrix ⇑e ⇑e).PosSemidef\n⊢ M.PosSemidef",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\ne : m ≃ n\nh : (M.submatrix ⇑e ⇑e).PosSemidef\n⊢ M.PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 61
} | {
"line": 176,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosDef\ne : m → n\nhe : Function.Injective e\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M.submatrix e e i j * xj",
"ppTerm": "?m.33",
... | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosDef\ne : m → n\nhe : Function.Injective e\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M (e i) (e j) * xj"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 64
} | {
"line": 183,
"column": 6
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosDef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ x ≠ 0 → 0 < x.sum fun x' v' ↦ x.sum fun x v ↦ star v * Mᵀ x x' * v'",
"ppTerm": "?m.60",
"assigned": t... | [
"n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosDef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ ¬x = 0 → 0 < x.sum fun x' v' ↦ x.sum fun x v ↦ v' * M x' x * star v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 187,
"column": 7
} | {
"line": 187,
"column": 18
} | {
"line": 187,
"column": 19
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nx✝ : Mᵀ.PosDef\n⊢ M.PosDef",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nx✝ : Mᵀ.PosDef\n⊢ M.PosDef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 196,
"column": 26
} | {
"line": 196,
"column": 55
} | {
"line": 196,
"column": 56
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : DecidableEq n\ninst✝ : NoZeroDivisors R\nd : n → R\nh : ∀ (i : n), 0 < d i\nx : n →₀ R\nhx : x ≠ 0\n⊢ ?m.73",
"ppTerm": "?m.78",
"assigned": false,
"usedConstants"... | [
"n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : DecidableEq n\ninst✝ : NoZeroDivisors R\nd : n → R\nh : ∀ (i : n), 0 < d i\nx : n →₀ R\nhx : x ≠ 0\n⊢ ?m.73"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 28
} | {
"line": 205,
"column": 29
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i",
"ppTerm": "?m.26",
"assigned": false,
"usedC... | [
"n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 33
} | {
"line": 249,
"column": 34
} | [
{
"pp": "m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosDef\nhB : B.PosSemidef\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A + B) i j * xj",
"ppTerm": "?m.46",
"assigned": tru... | [
"m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosDef\nhB : B.PosSemidef\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 <\n (x.sum fun a b ↦ x.sum fun a_1 ↦ HMul.hMul (star b * A a a_1)) +\n x.sum fun a b ↦ x.sum fun a_1 ↦ HMul.hMu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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