module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 129,
"column": 46
} | {
"line": 129,
"column": 51
} | {
"line": 129,
"column": 51
} | [
{
"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\nf : M →+ ℚ\nhf : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 129,
"column": 46
} | {
"line": 129,
"column": 51
} | {
"line": 129,
"column": 51
} | [
{
"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\nf : M →+ ℚ\nhf : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 55
} | {
"line": 427,
"column": 4
} | [
{
"pp": "case inr\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 inr\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 ∈... | refine (Finset.sum_neg' (fun i _ ↦ neg.le i) ?_).ne | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inl.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inl.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inr.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inr.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inl.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 11
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case e'_4.e'_6\nι : 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\nf... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inl.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inr.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 9
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inr.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 63
} | {
"line": 147,
"column": 4
} | [
{
"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\nf : M →+ ℚ\nhf : ... | [
"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\nf : M →+ ℚ\nhf : ∀ (i : ι), f... | rw [range_comp, ← span_span_of_tower ℚ, span_image, h_span] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 161,
"column": 21
} | {
"line": 161,
"column": 26
} | {
"line": 161,
"column": 26
} | [
{
"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 : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 161,
"column": 21
} | {
"line": 161,
"column": 26
} | {
"line": 161,
"column": 26
} | [
{
"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 : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 161,
"column": 21
} | {
"line": 161,
"column": 26
} | {
"line": 161,
"column": 26
} | [
{
"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 : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inl.inl.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inl.inl.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inl.inr.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inl.inr.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inr.inl.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inr.inl.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inr.inr.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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 9
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case neg.inr.inr.inr.inr\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 : Is... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 174,
"column": 24
} | {
"line": 174,
"column": 29
} | {
"line": 174,
"column": 29
} | [
{
"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 : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 174,
"column": 24
} | {
"line": 174,
"column": 29
} | {
"line": 174,
"column": 29
} | [
{
"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 : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 174,
"column": 24
} | {
"line": 174,
"column": 29
} | {
"line": 174,
"column": 29
} | [
{
"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 : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 547,
"column": 21
} | {
"line": 547,
"column": 26
} | {
"line": 547,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝¹ : CharZero R\ninst✝ : P.IsCrystallographic\ni : ι\nhi : b.IsPos (-i)\nj : ι\nhj : j ∈ b.su... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 547,
"column": 21
} | {
"line": 547,
"column": 26
} | {
"line": 547,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝¹ : CharZero R\ninst✝ : P.IsCrystallographic\ni : ι\nhi : b.IsPos (-i)\nj : ι\nhj : j ∈ b.su... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 547,
"column": 21
} | {
"line": 547,
"column": 26
} | {
"line": 547,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝¹ : CharZero R\ninst✝ : P.IsCrystallographic\ni : ι\nhi : b.IsPos (-i)\nj : ι\nhj : j ∈ b.su... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 206,
"column": 46
} | {
"line": 206,
"column": 51
} | {
"line": 206,
"column": 51
} | [
{
"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... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 470,
"column": 4
} | {
"line": 470,
"column": 62
} | {
"line": 470,
"column": 62
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\ni j : ι... | [
"case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : P.... | chainBotCoeff_add_chainTopCoeff_eq_pairingIn_chainTopIdx h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 7
} | {
"line": 474,
"column": 0
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\ni j : ι... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 661,
"column": 6
} | {
"line": 661,
"column": 11
} | {
"line": 662,
"column": 4
} | [
{
"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✝... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Real.Basic | {
"line": 203,
"column": 24
} | {
"line": 203,
"column": 88
} | {
"line": 204,
"column": 2
} | [
{
"pp": "x a b c : ℝ\n⊢ a * (b + c) = a * b + a * c",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Real",
"Real.cauchy",
"HMul.hMul",
"Real.ext_cauchy",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
... | [] | by apply ext_cauchy; simp only [cauchy_add, cauchy_mul, mul_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 274,
"column": 56
} | {
"line": 274,
"column": 61
} | {
"line": 274,
"column": 61
} | [
{
"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... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 274,
"column": 56
} | {
"line": 274,
"column": 61
} | {
"line": 274,
"column": 61
} | [
{
"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... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 274,
"column": 56
} | {
"line": 274,
"column": 61
} | {
"line": 274,
"column": 61
} | [
{
"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... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 184,
"column": 35
} | {
"line": 184,
"column": 40
} | {
"line": 184,
"column": 40
} | [
{
"pp": "case inl\nα : Type u_1\nm : Type u_2\nn : Type u_3\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroClass α\nia : m\nja : n\nib : m\njb : n\na b : α\nh✝ : ¬ia = ib\n⊢ ((of fun i' j' ↦ if ia = i' ∧ ja = j' then a else 0) ⊙ of fun i' j' ↦ if ib = i' ∧ jb = j' then b else 0) = 0",
"ppTe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 184,
"column": 35
} | {
"line": 184,
"column": 40
} | {
"line": 184,
"column": 40
} | [
{
"pp": "case inr\nα : Type u_1\nm : Type u_2\nn : Type u_3\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroClass α\nia : m\nja : n\nib : m\njb : n\na b : α\nh✝ : ¬ja = jb\n⊢ ((of fun i' j' ↦ if ia = i' ∧ ja = j' then a else 0) ⊙ of fun i' j' ↦ if ib = i' ∧ jb = j' then b else 0) = 0",
"ppTe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 20
} | {
"line": 87,
"column": 2
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ n ≠ 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"cast",
... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 178,
"column": 14
} | {
"line": 191,
"column": 64
} | {
"line": 193,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA X Y : M\nhx : IsUnit X.det\nhy : IsUnit Y.det\nh : SemiconjBy A X Y\nn : ℕ\n⊢ SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | by
have hx' : IsUnit (X ^ n.succ).det := by
rw [det_pow]
exact hx.pow n.succ
have hy' : IsUnit (Y ^ n.succ).det := by
rw [det_pow]
exact hy.pow n.succ
rw [zpow_negSucc, zpow_negSucc, nonsing_inv_apply _ hx', nonsing_inv_apply _ hy', SemiconjBy]
refine (isRegular_of_isLeftRegular_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 264,
"column": 13
} | {
"line": 264,
"column": 23
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case empty\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\nhs : ∅.Nonempty\nhA : ∀ i ∈ ∅, (A i).PosDef\n⊢ (∑ i ∈ ∅, A i).PosDef",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
... | [] | simp at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 264,
"column": 13
} | {
"line": 264,
"column": 23
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case empty\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\nhs : ∅.Nonempty\nhA : ∀ i ∈ ∅, (A i).PosDef\n⊢ (∑ i ∈ ∅, A i).PosDef",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
... | [] | simp at hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 264,
"column": 13
} | {
"line": 264,
"column": 23
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case empty\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\nhs : ∅.Nonempty\nhA : ∀ i ∈ ∅, (A i).PosDef\n⊢ (∑ i ∈ ∅, A i).PosDef",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
... | [] | simp at hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ZMatrix | {
"line": 49,
"column": 57
} | {
"line": 49,
"column": 62
} | {
"line": 50,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\nμ ρ : R\nhρ : HasEigenvalu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 9
} | {
"line": 98,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\nS : Type u_5\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra S R\ninst✝⁷ : IsDomain R\ninst✝⁶ : NeZero 2\ninst✝⁵ : FaithfulSMul S R\ninst✝... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 144,
"column": 37
} | {
"line": 144,
"column": 42
} | {
"line": 145,
"column": 2
} | [
{
"pp": "ι : 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 : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 144,
"column": 37
} | {
"line": 144,
"column": 42
} | {
"line": 145,
"column": 2
} | [
{
"pp": "ι : 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 : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 144,
"column": 37
} | {
"line": 144,
"column": 42
} | {
"line": 145,
"column": 2
} | [
{
"pp": "ι : 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 : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 91
} | {
"line": 73,
"column": 92
} | [
{
"pp": "case succ\nS : Type u\ninst✝² : Semiring S\ninst✝¹ : Nontrivial S\ninst✝ : NoZeroDivisors S\nn : ℕ\nhn : (ascPochhammer S n).Monic\nthis : (X + 1).leadingCoeff = 1\n⊢ X.leadingCoeff * ((ascPochhammer S n).comp (X + 1)).leadingCoeff = 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [
"case succ\nS : Type u\ninst✝² : Semiring S\ninst✝¹ : Nontrivial S\ninst✝ : NoZeroDivisors S\nn : ℕ\nhn : (ascPochhammer S n).Monic\nthis : (X + 1).leadingCoeff = 1\n⊢ X.leadingCoeff * ((ascPochhammer S n).leadingCoeff * (X + 1).leadingCoeff ^ (ascPochhammer S n).natDegree) = 1"
] | leadingCoeff_comp (ne_zero_of_eq_one <| natDegree_X_add_C 1 : natDegree (X + 1) ≠ 0), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 215,
"column": 50
} | {
"line": 215,
"column": 70
} | {
"line": 215,
"column": 70
} | [
{
"pp": "S : Type u_2\ninst✝ : Semiring S\nn : ℕ\n⊢ ↑(ascFactorial 1 n) = ↑n !",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Nat.ascFactorial",
"id",
"AddMonoidWithOne.toNatCast",
... | [
"S : Type u_2\ninst✝ : Semiring S\nn : ℕ\n⊢ ↑n ! = ↑n !"
] | Nat.one_ascFactorial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 318,
"column": 98
} | {
"line": 319,
"column": 36
} | {
"line": 321,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn : ℕ\nh : n ≠ 0\n⊢ eval 0 (descPochhammer R n) = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"eq_false",
"congrArg",
"descPochhammer",
"AddGroupWithOne.toAddMonoidWithOne",
"instOfNatNat",
... | [] | by
simp [descPochhammer_eval_zero, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 7
} | {
"line": 213,
"column": 0
} | [
{
"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\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 7
} | {
"line": 228,
"column": 0
} | [
{
"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\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 78
} | {
"line": 246,
"column": 6
} | [
{
"pp": "case mem\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\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Fini... | [
"case mem\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\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst✝¹... | obtain ⟨l, hl, rfl⟩ : ∃ l : b.support, p l ∧ P.root l = x := by simp_all | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Basis | {
"line": 119,
"column": 48
} | {
"line": 119,
"column": 53
} | {
"line": 121,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\n⊢ b.symm.symm = b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Basis | {
"line": 119,
"column": 48
} | {
"line": 119,
"column": 53
} | {
"line": 121,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\n⊢ b.symm.symm = b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Basis | {
"line": 119,
"column": 48
} | {
"line": 119,
"column": 53
} | {
"line": 121,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\n⊢ b.symm.symm = b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 50
} | {
"line": 210,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv : ι → F\ni : ι\n⊢ Lagrange.basis {i} v i = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.prod_empty",
"MulOne.toOne",
"Polynomial.instOne",
"Monoid.toMulOneCl... | [] | rw [Lagrange.basis, erase_singleton, prod_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 50
} | {
"line": 210,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv : ι → F\ni : ι\n⊢ Lagrange.basis {i} v i = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.prod_empty",
"MulOne.toOne",
"Polynomial.instOne",
"Monoid.toMulOneCl... | [] | rw [Lagrange.basis, erase_singleton, prod_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 50
} | {
"line": 210,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv : ι → F\ni : ι\n⊢ Lagrange.basis {i} v i = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.prod_empty",
"MulOne.toOne",
"Polynomial.instOne",
"Monoid.toMulOneCl... | [] | rw [Lagrange.basis, erase_singleton, prod_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 315,
"column": 2
} | {
"line": 318,
"column": 38
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.B... | [
"case inr\nι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.Base\np : Sub... | · obtain ⟨j, hj, hj₀⟩ := b.exists_mem_support_pos_pairingIn_ne_zero i
refine ⟨fun i ↦ P.pairingIn ℤ i j, subset_span ⟨⟨j, hj⟩, rfl⟩, ?_⟩
rw [ne_eq, P.pairingIn_eq_zero_iff] at hj₀
simpa [f, ne_eq, Int.cast_eq_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 322,
"column": 6
} | {
"line": 322,
"column": 11
} | {
"line": 323,
"column": 4
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.Base\np : S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 49
} | {
"line": 232,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\nn : ℕ\ninst✝ : IsDomain R\nv : Fin n → R\n⊢ (∃ i j, v i = v j ∧ i ≠ j) → (vandermonde v).det = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"instDecidableEqFin... | [] | · simp only [Ne, forall_exists_index, and_imp]
refine fun i j h₁ h₂ => Matrix.det_zero_of_row_eq h₂ (funext fun k => ?_)
rw [vandermonde_apply, vandermonde_apply, h₁] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 25
} | {
"line": 91,
"column": 2
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 25
} | {
"line": 91,
"column": 2
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 25
} | {
"line": 91,
"column": 2
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.EngelSubalgebra | {
"line": 125,
"column": 2
} | {
"line": 137,
"column": 32
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsArtinian R L\nH : LieSubalgebra R L\nx : L\nh : engel R x ≤ H\nN : LieSubalgebra R L := ⋯\naux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H\naux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N\ndx : ↥N →ₗ[R] ↥N := ⋯\nk : ℕ\nhk : Codisjoin... | [
"case a\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsArtinian R L\nH : LieSubalgebra R L\nx : L\nh : engel R x ≤ H\nN : LieSubalgebra R L := ⋯\naux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H\naux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N\ndx : ↥N →ₗ[R] ↥N := ⋯\nk : ℕ\nhk : Codisjoint (dx ^ (k +... | · rw [← Submodule.map_le_iff_le_comap]
apply le_sup_of_le_left
rw [Submodule.map_le_iff_le_comap]
intro y hy
simp only [Submodule.mem_comap, mem_engel_iff, mem_toSubmodule]
use k + 1
clear hk; revert hy
generalize k + 1 = k
induction k generalizing y with
| zero =>
cases y; int... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Lagrange | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 9
} | {
"line": 527,
"column": 0
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_3\ninst✝ : Finite ι\nx y : ι → F\nhwd : ∀ (i j : ι), x i = x j → y i = y j\nthis : Fintype ι\nhinj : Set.InjOn (fun d ↦ d) ↑(image x univ)\nv : F → F := fun z ↦ if h : ∃ i, x i = z then y h.choose else 0\nv_def : v = fun z ↦ if h : ∃ i, x i = z then ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Lagrange | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 9
} | {
"line": 527,
"column": 0
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_3\ninst✝ : Finite ι\nx y : ι → F\nhwd : ∀ (i j : ι), x i = x j → y i = y j\nthis : Fintype ι\nhinj : Set.InjOn (fun d ↦ d) ↑(image x univ)\nv : F → F := fun z ↦ if h : ∃ i, x i = z then y h.choose else 0\nv_def : v = fun z ↦ if h : ∃ i, x i = z then ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 9
} | {
"line": 527,
"column": 0
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_3\ninst✝ : Finite ι\nx y : ι → F\nhwd : ∀ (i j : ι), x i = x j → y i = y j\nthis : Fintype ι\nhinj : Set.InjOn (fun d ↦ d) ↑(image x univ)\nv : F → F := fun z ↦ if h : ∃ i, x i = z then y h.choose else 0\nv_def : v = fun z ↦ if h : ∃ i, x i = z then ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 67
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 67
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 67
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 243,
"column": 2
} | {
"line": 244,
"column": 5
} | {
"line": 246,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ (map f) ((bind₁ g) φ) = (bind₁ fun i ↦ (map f) (g i)) ((map f) φ)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [... | [] | rw [hom_bind₁, map_comp_C, ← eval₂Hom_map_hom]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 243,
"column": 2
} | {
"line": 244,
"column": 5
} | {
"line": 246,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ (map f) ((bind₁ g) φ) = (bind₁ fun i ↦ (map f) (g i)) ((map f) φ)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [... | [] | rw [hom_bind₁, map_comp_C, ← eval₂Hom_map_hom]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 682,
"column": 28
} | {
"line": 682,
"column": 84
} | {
"line": 684,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nj : ι\nhj : j ∈ s\n⊢ C (r j) * Lagrange.basis s v j = C (nodalWeight s v j) * (nodal s v / (X - C (v j))) * C (r j)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"NonUnitalNonAs... | [] | by rw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Basis | {
"line": 195,
"column": 29
} | {
"line": 195,
"column": 36
} | {
"line": 195,
"column": 36
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny : L\nthis : ∀ (i : ι), ∀ x ∈ lieSpan R L (range b.f), ⁅b.e i, x⁆ ∈ b.cartan.toLieSubmodule ⊔ b.borelLower\nu v : L\nhx✝ : u ∈ lieSpan R L (range b.e... | [
"case add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny : L\nthis : ∀ (i : ι), ∀ x ∈ lieSpan R L (range b.f), ⁅b.e i, x⁆ ∈ b.cartan.toLieSubmodule ⊔ b.borelLower\nu v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈... | add_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 349,
"column": 17
} | {
"line": 349,
"column": 22
} | {
"line": 367,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : σ → MvPolynomial τ R\n⊢ ∀ {α β : Type u_6} (f : MvPolynomial (α → β) R) (x : MvPolynomial α R),\n (do\n let x_1 ← f\n x_1 <$> x) =\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 349,
"column": 17
} | {
"line": 349,
"column": 22
} | {
"line": 367,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : σ → MvPolynomial τ R\n⊢ ∀ {α β : Type u_6} (f : MvPolynomial (α → β) R) (x : MvPolynomial α R),\n (do\n let x_1 ← f\n x_1 <$> x) =\n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 349,
"column": 17
} | {
"line": 349,
"column": 22
} | {
"line": 367,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : σ → MvPolynomial τ R\n⊢ ∀ {α β : Type u_6} (f : MvPolynomial (α → β) R) (x : MvPolynomial α R),\n (do\n let x_1 ← f\n x_1 <$> x) =\n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 161,
"column": 53
} | {
"line": 161,
"column": 58
} | {
"line": 162,
"column": 4
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 161,
"column": 53
} | {
"line": 161,
"column": 58
} | {
"line": 162,
"column": 4
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 161,
"column": 53
} | {
"line": 161,
"column": 58
} | {
"line": 162,
"column": 4
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 7
} | {
"line": 243,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CharZero R\ninst✝⁴ : IsDomain R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : IsNoetherian R L\ninst✝ : Module.Free R L\nh : ∀ (x y : L), y ∈ derivedSeries R L 1 → ((killingForm R L) x) y = 0\n⊢ ∀ x ∈ ⊤, ∀ y ∈ ⁅⊤, ⊤⁆, ((killingForm R L)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 596,
"column": 36
} | {
"line": 596,
"column": 41
} | {
"line": 596,
"column": 41
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\ni : ι\nhi : 0 < i\nhx : x ∈ 𝒜 i\n⊢... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 596,
"column": 36
} | {
"line": 596,
"column": 41
} | {
"line": 596,
"column": 41
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\ni : ι\nhi : 0 < i\nhx : x ∈ 𝒜 i\n⊢... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 596,
"column": 36
} | {
"line": 596,
"column": 41
} | {
"line": 596,
"column": 41
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\ni : ι\nhi : 0 < i\nhx : x ∈ 𝒜 i\n⊢... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 59
} | {
"line": 433,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nφ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ coeff d (∑ i ∈ Finite.toFinset ⋯, (weightedHomogeneousComponent w i) φ) = coeff d φ",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Fin... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nφ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ (∑ x ∈ Finite.toFinset ⋯, if (weight w) d = x then coeff d φ else 0) = coeff d φ"
] | simp only [coeff_sum, coeff_weightedHomogeneousComponent] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 165,
"column": 44
} | {
"line": 165,
"column": 72
} | {
"line": 165,
"column": 72
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nι₁ : Type u_4\nι₂ : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Finite ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\nf : M₁ ... | [
"R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nι₁ : Type u_4\nι₂ : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Finite ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\nf : M₁ →ₗ[R] M₂\ni ... | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 397,
"column": 11
} | {
"line": 397,
"column": 33
} | {
"line": 397,
"column": 34
} | [
{
"pp": "case h\nR : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ ... | [
"case h\nR : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ Polynomial.e... | eval_eq_eval_mv_eval', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 442,
"column": 51
} | {
"line": 442,
"column": 56
} | {
"line": 443,
"column": 4
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nN : ℕ\nIH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0\nF : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhnR : ↑n ≤ #R\nhdeg : ((finSuccEquiv R N) F).natDegree < n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 442,
"column": 51
} | {
"line": 442,
"column": 56
} | {
"line": 443,
"column": 4
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nN : ℕ\nIH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0\nF : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhnR : ↑n ≤ #R\nhdeg : ((finSuccEquiv R N) F).natDegree < n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 442,
"column": 51
} | {
"line": 442,
"column": 56
} | {
"line": 443,
"column": 4
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nN : ℕ\nIH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0\nF : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhnR : ↑n ≤ #R\nhdeg : ((finSuccEquiv R N) F).natDegree < n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 556,
"column": 64
} | {
"line": 556,
"column": 92
} | {
"line": 556,
"column": 92
} | [
{
"pp": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : IsDomain R\nh : ↑... | [
"case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : IsDomain R\nh : ↑(finrank R M... | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 254,
"column": 28
} | {
"line": 254,
"column": 69
} | {
"line": 258,
"column": 4
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [] | rwa [← LieSubmodule.Quotient.mk_eq_zero'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 283,
"column": 24
} | {
"line": 283,
"column": 30
} | {
"line": 283,
"column": 31
} | [
{
"pp": "l : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix l l... | [
"l : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix l l R := V * C ... | hsymm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 9
} | {
"line": 79,
"column": 2
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\nh : ((A.submatrix r id).submatrix id c).cRank ≤ (A.submatrix r id).cRank\nf : (m → R) →ₗ[R] m₀ → R := LinearMap.funLeft R R r\n⊢ span R ((fun a ↦ f (of.symm Aᵀ a)) '' Set.u... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 94
} | {
"line": 306,
"column": 2
} | [
{
"pp": "l : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix l l... | [
"case refine_1\nl : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix ... | refine ⟨X.submatrix s s, IsSymm.submatrix ?_ s, (isUnit_submatrix_equiv s.symm s.symm).1 ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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