module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Separable | {
"line": 763,
"column": 6
} | {
"line": 763,
"column": 43
} | {
"line": 763,
"column": 44
} | [
{
"pp": "A₁ : Type u_1\nB₁ : Type u_2\nA₂ : Type u_3\nB₂ : Type u_4\ninst✝⁵ : Field A₁\ninst✝⁴ : Ring B₁\ninst✝³ : Field A₂\ninst✝² : Ring B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx✝ : Algebra.IsSeparable... | [
"A₁ : Type u_1\nB₁ : Type u_2\nA₂ : Type u_3\nB₂ : Type u_4\ninst✝⁵ : Field A₁\ninst✝⁴ : Ring B₁\ninst✝³ : Field A₂\ninst✝² : Ring B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx✝ : Algebra.IsSeparable A₂ B₂\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 469,
"column": 6
} | {
"line": 469,
"column": 17
} | {
"line": 469,
"column": 18
} | [
{
"pp": "K : Type u_5\nV : Type u_6\nW : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\nF : Type u_8\ninst✝¹ : EquivLike F (End K V) (End K W)\ninst✝ : AlgEquivClass F K (End K V) (End K W)\nf : F\nx : End K V\nw✝ : V ≃ₗ[K] W\nh : ↑f = Lin... | [
"K : Type u_5\nV : Type u_6\nW : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\nF : Type u_8\ninst✝¹ : EquivLike F (End K V) (End K W)\ninst✝ : AlgEquivClass F K (End K V) (End K W)\nf : F\nx : End K V\nw✝ : V ≃ₗ[K] W\nh : ↑f = LinearEquiv.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 56
} | {
"line": 475,
"column": 4
} | [
{
"pp": "K : Type u_5\nm : Type u_6\nn : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nF : Type u_8\ninst✝¹ : EquivLike F (Matrix m m K) (Matrix n n K)\ninst✝ : AlgEquivClass F K (Matrix m m K) (Matrix n n K)\nf : F\nx : Matrix m m K\n⊢ (f x)... | [
"K : Type u_5\nm : Type u_6\nn : Type u_7\ninst✝⁶ : Field K\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nF : Type u_8\ninst✝¹ : EquivLike F (Matrix m m K) (Matrix n n K)\ninst✝ : AlgEquivClass F K (Matrix m m K) (Matrix n n K)\nf : F\nx : Matrix m m K\n⊢ (f x).det = x.det... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 13
} | {
"line": 501,
"column": 14
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : M ≃ₗ[R] M'\nv : Basis ι R M\nv' : Basis ι R M'\n⊢ ?B * (toMatrix v v') ↑f = ... | [
"case h\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : M ≃ₗ[R] M'\nv : Basis ι R M\nv' : Basis ι R M'\n⊢ ?B * (toMatrix v v') ↑f = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 35
} | {
"line": 586,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), f x = f' (e x)\n⊢ Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f')",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), f x = f' (e x)\n⊢ Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f' * 1)"
] | ← mul_one (LinearMap.det f'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 629,
"column": 62
} | {
"line": 629,
"column": 77
} | {
"line": 629,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ne : Basis ι R M\ninst✝ : Nontrivial R\nh : e.det = 0\n⊢ False",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ne : Basis ι R M\ninst✝ : Nontrivial R\nh : e.det = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 673,
"column": 4
} | {
"line": 673,
"column": 19
} | {
"line": 673,
"column": 20
} | [
{
"pp": "case intro\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_5\ninst✝ : Finite ι\ne : Basis ι R M\nf : M [⋀^ι]→ₗ[R] R\nh : f ⇑e = 0\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\n⊢ f = 0",
"ppTerm": "?intro",
"assigned": fa... | [
"case intro\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_5\ninst✝ : Finite ι\ne : Basis ι R M\nf : M [⋀^ι]→ₗ[R] R\nh : f ⇑e = 0\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 686,
"column": 51
} | {
"line": 689,
"column": 18
} | {
"line": 691,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\ne : Basis ι A M\nf : M →ₗ[A] M\nv : ι → M\n⊢ e.det (⇑f ∘ v) = LinearMap.det f * e.det v",
"ppTerm": "?m.55",
"assigned": true,
"usedConstan... | [] | by
rw [det_apply, det_apply, ← f.det_toMatrix e, ← Matrix.det_mul,
e.toMatrix_eq_toMatrix_constr (f ∘ v), e.toMatrix_eq_toMatrix_constr v, ← toMatrix_comp,
e.constr_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Determinant | {
"line": 709,
"column": 30
} | {
"line": 709,
"column": 40
} | {
"line": 709,
"column": 41
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\nb b' b'' : Basis ι A M\nthis : ⇑b'' = ⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b'\n| LinearMap.det ↑(b'.equiv b'' (Equiv.refl ι)) * b.det ⇑b'",
"ppTerm":... | [
"M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\nb b' b'' : Basis ι A M\nthis : ⇑b'' = ⇑↑(b'.equiv b'' (Equiv.refl ι)) ∘ ⇑b'\n| b'.det ⇑b'' * b.det ⇑b'"
] | det_basis, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 763,
"column": 4
} | {
"line": 763,
"column": 49
} | {
"line": 764,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (Set.range v)\ni k : ι\nhik : k ≠ i\n⊢ e.det v * ((Basis.mk hli hsp... | [
"case inr\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (Set.range v)\ni k : ι\nhik : k ≠ i\n⊢ ↑e.det (update v i (v k)) = 0"
] | rw [mk_coord_apply_ne hik, mul_zero, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Determinant | {
"line": 832,
"column": 6
} | {
"line": 832,
"column": 35
} | {
"line": 833,
"column": 4
} | [
{
"pp": "case inl.inl\nR : Type u_6\nV : Type u_7\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\nW : Submodule R V\ninst✝² : Free R ↥W\ninst✝¹ : Module.Finite R ↥W\ninst✝ : Free R (V ⧸ W)\ne : V →ₗ[R] V\nhe : W ≤ comap e W\nm : Type u_7 := Free.ChooseBasisIndex R... | [] | apply sumQuot_repr_inl_of_mem | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AdjoinRoot | {
"line": 394,
"column": 24
} | {
"line": 394,
"column": 56
} | {
"line": 394,
"column": 57
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g✝ : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\np : R[X]\nq : S[X]\nr : T[X]\nhf : q ∣ Polynomial.map f p\nhg : r ∣ Polynomial.map g q\n⊢... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g✝ : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\np : R[X]\nq : S[X]\nr : T[X]\nhf : q ∣ Polynomial.map f p\nhg : r ∣ Polynomial.map g q\n⊢ Polynomial.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 38
} | {
"line": 402,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ p ∣ Polynomial.map (↑f.symm) q",
"... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ p ∣ Polynomial.map (↑f.symm) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 32
} | {
"line": 113,
"column": 33
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ IsSemisimple 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"Eq.mpr",
"Submodule",
... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ ∀ (p : Submodule R M), ∃ q, IsCompl p q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 32
} | {
"line": 117,
"column": 33
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ IsSemisimple LinearMap.id",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"LinearMap.id",
"E... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ ∀ (p : Submodule R M), ∃ q, IsCompl p q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 34
} | {
"line": 127,
"column": 35
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nM₂ : Type u_3\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\ng : End R M₂\ne : M ≃ₗ[R] M₂\nhe : ↑e ∘ₗ f = g ∘ₗ ↑e\nx : M\n⊢ (e ≪≫ₗ AEval'.of g) (f • x) = X • (e ≪≫ₗ AEval'.of g) x",
"ppTerm"... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nM₂ : Type u_3\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\ng : End R M₂\ne : M ≃ₗ[R] M₂\nhe : ↑e ∘ₗ f = g ∘ₗ ↑e\nx : M\n⊢ e (f x) = g (e x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 38
} | {
"line": 410,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ Associated (Polynomial.map (↑f.symm) q... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ Associated (Polynomial.map (↑f.symm) q) p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 151,
"column": 28
} | {
"line": 151,
"column": 62
} | {
"line": 151,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodu... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodule.span R {x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 57
} | {
"line": 158,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodu... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhs : f.IsFinitelySemisimple\nthis : ∀ (p : Submodule R M) (hp₁ : p ∈ f.invtSubmodule), Module.Finite R ↥p → LinearMap.restrict f hp₁ = 0\nx : M\nk : ℕ\nhk : f ^ k = 0\np : Submodule R M := Submodule.span R {x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 13
} | {
"line": 166,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ x ∈ Submodule.comap f p",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ f x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ x ∈ Submodule.comap f p",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\np : Submodule R M\nh : p ≤ Submodule.comap (f - (algebraMap R (End R M)) μ) p\nx : M\nhx : x ∈ p\n⊢ f x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 763,
"column": 59
} | {
"line": 763,
"column": 70
} | {
"line": 763,
"column": 71
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nx✝ : ↥R[x]\ny₁ : S\ny₂ : y₁ ∈ R[x]\nh : ⟨y₁, y₂⟩ ∈ Subtype.val ⁻¹' {x}\n⊢ (toAdjoin R x).toRingHom ((mk (minpoly R x)) X) = ⟨y₁, y₂⟩",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nx✝ : ↥R[x]\ny₁ : S\ny₂ : y₁ ∈ R[x]\nh : ⟨y₁, y₂⟩ ∈ Subtype.val ⁻¹' {x}\n⊢ x = y₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 48
} | {
"line": 119,
"column": 8
} | [
{
"pp": "case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := rang... | [
"case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁✝ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄✝ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := range (N + 1)\... | rintro ⟨x₁, y₁⟩ - h₁ ⟨x₂, y₂⟩ - h₂ h₃ h₄ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.AdjoinRoot | {
"line": 857,
"column": 6
} | {
"line": 857,
"column": 34
} | {
"line": 857,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf : R[X]\nx : AdjoinRoot f\n⊢ (quotMapOfEquivQuotMapCMapMk I f).symm\n ((Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I))) x) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) x",
"ppTerm": "?m.56",
"assigned": true,
... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf : R[X]\nx : AdjoinRoot f\n⊢ (quotEquivOfEq ⋯).symm ((Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I))) x) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) x"
] | quotMapOfEquivQuotMapCMapMk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.AdjointAction.Derivation | {
"line": 73,
"column": 35
} | {
"line": 74,
"column": 47
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : LieAlgebra.center R L = ⊥\n⊢ Function.Injective ⇑(ad R L)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Eq.mpr",
"LieDerivat... | [] | by
rw [← LieHom.ker_eq_bot, ad_ker_eq_center, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Dynamics.Newton | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 15
} | {
"line": 68,
"column": 16
} | [
{
"pp": "case succ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : ℕ\nih : IsNilpotent (P.newtonMap^[n] x - x)\n⊢ IsNilpotent ((aeval (P.newtonMap^[n] x)) P)",
"ppTerm": "?succ",
"assigned": false,
"usedC... | [
"case succ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : ℕ\nih : IsNilpotent (P.newtonMap^[n] x - x)\n⊢ IsNilpotent ((aeval (P.newtonMap^[n] x)) P)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 1003,
"column": 2
} | {
"line": 1005,
"column": 8
} | {
"line": 1006,
"column": 2
} | [
{
"pp": "case refine_2\nR✝ : Type u_1\nS✝ : Type u_2\nT✝ : Type u_3\nU✝ : Type u_4\nK : Type u_5\nR : Type u_6\nS : Type u_7\nT : Type u_8\nU : Type u_9\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : CommRing U\ninst✝ : Algebra R U\np✝ p : S[... | [
"case refine_3\nR✝ : Type u_1\nS✝ : Type u_2\nT✝ : Type u_3\nU✝ : Type u_4\nK : Type u_5\nR : Type u_6\nS : Type u_7\nT : Type u_8\nU : Type u_9\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : CommRing U\ninst✝ : Algebra R U\np✝ p : S[X]\nq : (T ⊗... | · ext
· simp [Algebra.ofId_apply]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 58,
"column": 6
} | {
"line": 59,
"column": 13
} | {
"line": 59,
"column": 14
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : a * P ^ k + b * P' = 1\n⊢ (aeval f) ... | [
"K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : a * P ^ k + b * P' = 1\n⊢ (aeval f) b * (aeval f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 40
} | {
"line": 61,
"column": 41
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : (aeval f) b * (aeval f) P' = 1\n⊢ ↑(... | [
"K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\na b : K[X]\nh : (aeval f) b * (aeval f) P' = 1\n⊢ (aeval f) b * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 15
} | {
"line": 66,
"column": 16
} | [
{
"pp": "case refine_2\nK : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk✝ : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k✝\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\nsep' : IsUnit ((aeval ff) P')\ns : ... | [
"case refine_2\nK : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk✝ : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k✝\nff : ↥K[f] := ⟨f, ⋯⟩\nP' : K[X] := derivative P\nnil' : IsNilpotent ((aeval ff) P)\nsep' : IsUnit ((aeval ff) P')\ns : End K V\nmem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 15
} | {
"line": 232,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module ℚ M\ninst✝ : Module ℚ N\nfM : End R M\nfN : End R N\ng : M →ₗ[R] N\nh : fN ∘ₗ g = g ∘ₗ fM\nm : M\nk l : ℕ\nkl : ℕ := max k l\nhfM : ... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module ℚ M\ninst✝ : Module ℚ N\nfM : End R M\nfN : End R N\ng : M →ₗ[R] N\nh : fN ∘ₗ g = g ∘ₗ fM\nm : M\nk l : ℕ\nkl : ℕ := max k l\nhfM : fM ^ kl = 0\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Semisimple.Defs | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 53
} | {
"line": 83,
"column": 54
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieAlgebra R L\ninst✝ : Subsingleton L\n⊢ radical R L = ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Li... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieAlgebra R L\ninst✝ : Subsingleton L\n⊢ ⊤ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 417,
"column": 4
} | {
"line": 417,
"column": 45
} | {
"line": 418,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ derivedSeriesOfIdeal R L k₁ I",
... | [] | exact derivedSeriesOfIdeal_antitone I h₁₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Solvable | {
"line": 410,
"column": 2
} | {
"line": 418,
"column": 47
} | {
"line": 420,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ IsLieAbelian ↥(derivedSeriesOfIdeal R L k I) ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | rw [abelian_iff_derived_succ_eq_bot]
let s := { k | derivedSeriesOfIdeal R L k I = ⊥ }
change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
have hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by
intro k₁ k₂ h₁₂ h₁
suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this
change derivedSeriesOfIdeal... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Solvable | {
"line": 410,
"column": 2
} | {
"line": 418,
"column": 47
} | {
"line": 420,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ IsLieAbelian ↥(derivedSeriesOfIdeal R L k I) ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | rw [abelian_iff_derived_succ_eq_bot]
let s := { k | derivedSeriesOfIdeal R L k I = ⊥ }
change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
have hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by
intro k₁ k₂ h₁₂ h₁
suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this
change derivedSeriesOfIdeal... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanGenerators | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 40
} | {
"line": 84,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nC : Set α\nhC : ∀ x ∈ C, IsCompactElement x\nha : sSup C ≤ sSup S\ns : Finset α\nhs₁ : ↑s ⊆ S\nt : Finset α\nht : t ⊆ s\nhb : IsCompactElement (t.sup id)\nhbS : t.sup id ≤ sSup S\nhs₂ : t.sup ... | [
"α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nC : Set α\nhC : ∀ x ∈ C, IsCompactElement x\nha : sSup C ≤ sSup S\ns : Finset α\nhs₁ : ↑s ⊆ S\nt : Finset α\nht : t ⊆ s\nhb : IsCompactElement (t.sup id)\nhbS : t.sup id ≤ sSup S\nhs₂ : t.sup id ≤ s.sup i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 25
} | {
"line": 61,
"column": 26
} | [
{
"pp": "case refine_1\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nΦ : LinearMap.BilinForm R M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nh : LinearMap.BilinForm.lieInvariant L Φ\nx : L\ny z... | [
"case refine_1\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nΦ : LinearMap.BilinForm R M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nh : LinearMap.BilinForm.lieInvariant L Φ\nx : L\ny z : M\n⊢ (⁅x,... | LieHom.lie_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 15
} | {
"line": 45,
"column": 16
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : IsIrreducible R L M\naux : ∀ (x y : M), x = y\nm : M\n⊢ m ∈ ⊥ ↔ m ∈ ⊤",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : IsIrreducible R L M\naux : ∀ (x y : M), x = y\nm : M\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 53
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case mp\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : ∀ φ ∈ (domRestrict B W).range, φ x = 0\ny : V\nhy : y ∈ W\n⊢ (B y) x = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Submod... | [] | exact hx (B.domRestrict W ⟨y, hy⟩) ⟨⟨y, hy⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 30
} | {
"line": 251,
"column": 31
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : (B.restrict p) ⟨z, hz⟩ = 0\nx y : M\nhx : x ∈ p\nhy : y ∈ q\n⊢ (B z)... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : (B.restrict p) ⟨z, hz⟩ = 0\nx y : M\nhx : x ∈ p\nhy : y ∈ q\n⊢ (B z) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 15
} | {
"line": 253,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : B z = 0\nx : M\nhx : x ∈ p\n⊢ ((B.restrict p) ⟨z, hz⟩) ⟨x, hx⟩ = 0 ⟨... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : B z = 0\nx : M\nhx : x ∈ p\n⊢ (B z) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 177,
"column": 22
} | {
"line": 177,
"column": 75
} | {
"line": 178,
"column": 2
} | [
{
"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\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\nI : ... | [] | by rw [← sup_inf_assoc_of_le _ hJI, this, top_inf_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 15
} | {
"line": 197,
"column": 16
} | [
{
"pp": "case refine_1\nK : 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\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieA... | [
"case refine_1\nK : 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\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 47
} | {
"line": 252,
"column": 48
} | [
{
"pp": "case inr.left\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\nI : LieIdeal R L\nhI✝ : I ≤ s.sup id\nS : Set (LieIdeal R L) := ⋯\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J ∈ s... | [
"case inr.left\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\nI : LieIdeal R L\nhI✝ : I ≤ s.sup id\nS : Set (LieIdeal R L) := ⋯\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J ∈ s\nhJI : ¬J ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Normalizer | {
"line": 86,
"column": 56
} | {
"line": 86,
"column": 81
} | {
"line": 88,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\n⊢ ⁅⊤, N⁆ ≤ N' ↔ N ≤ N'.normalizer",
"ppTerm": "?m.65",
"assig... | [] | by rw [lie_le_iff]; tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 15
} | {
"line": 301,
"column": 16
} | [
{
"pp": "case sSupIndep_isAtom\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSimple R L\n⊢ sSupIndep {I | IsAtom I}",
"ppTerm": "?sSupIndep_isAtom",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"sSupIndep",
"... | [
"case sSupIndep_isAtom\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSimple R L\n⊢ sSupIndep {⊤}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 49,
"column": 32
} | {
"line": 49,
"column": 43
} | {
"line": 49,
"column": 44
} | [
{
"pp": "R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : (i, j) ∈ Finset.antidiagonal p\n⊢ i + j = p",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : (i, j) ∈ Finset.antidiagonal p\n⊢ i + j = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Normalizer | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 19
} | {
"line": 191,
"column": 20
} | [
{
"pp": "case refine_2\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nh : ∀ m ∈ LieModule.maxTrivSubmodule R (↥H) (L ⧸ H.toLieSubmodule), m = 0\nx : L\nhx : x ∈ H.normalizer\ny : L ⧸ H.toLieSubmodule := (LieSubmodule.Quotient.mk' H.toLieSubmo... | [
"case refine_2\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nh : ∀ m ∈ LieModule.maxTrivSubmodule R (↥H) (L ⧸ H.toLieSubmodule), m = 0\nx : L\nhx : x ∈ H.normalizer\ny : L ⧸ H.toLieSubmodule := (LieSubmodule.Quotient.mk' H.toLieSubmodule) x\nhy ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 37
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Nontrivial L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\ne : H = ⊥\n⊢ False",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"LieRing.toAddCommGroup",
... | [
"R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Nontrivial L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\ne : H = ⊥\nx : L\nhx : x ≠ 0\n⊢ False"
] | obtain ⟨x, hx⟩ := exists_ne (0 : L) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 148,
"column": 10
} | {
"line": 148,
"column": 22
} | {
"line": 148,
"column": 22
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk✝ : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\nih : lowerCentralSeries R L (↥N) k = comap N.incl (lcs k N)\n⊢ ... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk✝ : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\nih : lowerCentralSeries R L (↥N) k = comap N.incl (lcs k N)\n⊢ lcs k N ≤ N"... | N.range_incl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | {
"line": 260,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero"... | [] | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | {
"line": 260,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero"... | [] | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | {
"line": 260,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero"... | [] | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 20
} | {
"line": 313,
"column": 21
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\nk✝ : ℕ\nN✝ : LieSubmodule R L M\nM₂✝ : Type w₁\ninst✝⁶ : AddCommGroup M₂✝\ninst✝⁵ : Module R M₂✝\ninst✝⁴ : LieRingModule L M... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\nk✝ : ℕ\nN✝ : LieSubmodule R L M\nM₂✝ : Type w₁\ninst✝⁶ : AddCommGroup M₂✝\ninst✝⁵ : Module R M₂✝\ninst✝⁴ : LieRingModule L M₂✝\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Engel | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 21
} | {
"line": 141,
"column": 22
} | [
{
"pp": "case h\nR : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nhI... | [
"case h\nR : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nhIM : LieModul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 520,
"column": 4
} | {
"line": 520,
"column": 15
} | {
"line": 520,
"column": 16
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R (↥(toEnd R L M).range) M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R ... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R (↥(toEnd R L M).range) M k) = ↑⊥\n⊢ lowerCentralSeries R L M k = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 520,
"column": 4
} | {
"line": 520,
"column": 15
} | {
"line": 520,
"column": 16
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R L M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R (↥(toEnd R L M).range... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R L M k) = ↑⊥\n⊢ lowerCentralSeries R L M k = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 590,
"column": 75
} | {
"line": 591,
"column": 88
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ LieModule.IsNilpotent L M ↔ ∃ k, ucs k ⊥ = ⊤",
"ppTerm": "?m.53",
"assigned": true,
"use... | [] | by
rw [LieModule.isNilpotent_iff R]; exact exists_congr fun k => by simp [ucs_eq_top_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 18
} | {
"line": 647,
"column": 19
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : Add... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Fixed | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 84
} | {
"line": 244,
"column": 0
} | [
{
"pp": "case inr\nG : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf g : Polynomial ↥(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (subfield ... | [] | rwa [← one_mul (minpoly G F x), hg3, mul_left_inj' (monic G F x).ne_zero] at hfg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Fixed | {
"line": 264,
"column": 4
} | {
"line": 265,
"column": 24
} | {
"line": 266,
"column": 6
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\ns : Finset F\nhs : LinearIndependent ↥(subfield G F) fun i ↦ ↑i\n⊢ #s ≤ Fintype.card G",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\ns : Finset F\nhs : LinearIndependent ↥(subfield G F) fun i ↦ ↑i\n⊢ #s ≤ Fintype.card G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 17
} | {
"line": 84,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : IntermediateField F E\nh : K.toSubalgebra = Algebra.adjoin F S\nx : E\n⊢ x ∈ K.toSubalgebra → x⁻¹ ∈ K.toSubalgebra",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Intermediat... | [] | exact K.inv_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 36
} | {
"line": 195,
"column": 37
} | [
{
"pp": "case h\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : EssFiniteType F E\ninst✝ : Algebra.IsAlgebraic F E\ns : Finset E\nhs : adjoin F ↑s = ⊤\n⊢ (adjoin F ↑s).toSubalgebra = ⊤",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"case h\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : EssFiniteType F E\ninst✝ : Algebra.IsAlgebraic F E\ns : Finset E\nhs : adjoin F ↑s = ⊤\n⊢ Algebra.adjoin F ↑s = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 55
} | {
"line": 43,
"column": 56
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Algebra F R\nx : R\nP : F[X]\nhP₁ : (Minpoly.toAdjoin F x) ((AdjoinRoot.mk (minpoly F x)) P) = 0\n⊢ (aeval x) P = 0",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"F : Type u_1\ninst✝² : Field F\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Algebra F R\nx : R\nP : F[X]\nhP₁ : (Minpoly.toAdjoin F x) ((AdjoinRoot.mk (minpoly F x)) P) = 0\n⊢ (aeval x) P = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 31
} | {
"line": 94,
"column": 0
} | [
{
"pp": "F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField K L ((mapAlg F K) f)\n⊢ (Polynomial.map (algebraMap K L) ((mapAlg F K) f)).Splits"... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 58
} | {
"line": 278,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nL : IntermediateField F K\nhalg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F ↥L\ni : E →ₐ[F] K := IsScalarT... | [
"F : Type u_1\ninst✝⁶ : Field F\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nL : IntermediateField F K\nhalg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F ↥L\ni : E →ₐ[F] K := IsScalarTower.toAlgHo... | apply_fun _ using Subalgebra.restrictScalars_injective F | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 54
} | {
"line": 147,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : IntermediateField F E\n⊢ (S ⊔ T).toSubfield = Subfield.closure (↑S.toSubfield ∪ ↑T.toSubfield)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Algebr... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : IntermediateField F E\n⊢ Subfield.closure (Set.range ⇑(algebraMap F E) ∪ (↑S ∪ ↑T)) = Subfield.closure (↑S ∪ ↑T)"
] | simp_rw [sup_def, adjoin_toSubfield, coe_toSubfield] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 164,
"column": 19
} | {
"line": 164,
"column": 44
} | {
"line": 164,
"column": 45
} | [
{
"pp": "K : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : (Polynomial.map (algebraMap K L) p).Splits\nhF : ∀ x ∈ p.rootSet L, x ∈ F\nthis :\n (Polynomial.map (algebraMap K L) p).Splits →\n (∀ a ∈ (Polynomial.map (algebraMap K L) p).roo... | [
"K : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : (Polynomial.map (algebraMap K L) p).Splits\nhF : ∀ x ∈ p.rootSet L, x ∈ F\nthis :\n (Polynomial.map (algebraMap K L) p).Splits →\n (∀ a ∈ (Polynomial.map (algebraMap K L) p).roots, a ∈ (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 47
} | {
"line": 118,
"column": 48
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) ((minpoly K x).comp (-X))).Splits",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ ((map (algebraMap K L) (minpoly K x)).comp (-X)).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 46
} | {
"line": 120,
"column": 4
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) ((-1) ^ (minpoly K x).natDegree)).Splits",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) ((-1) ^ (minpoly K x).natDegree)).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 48
} | {
"line": 125,
"column": 49
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x + (algebraMap K L) r))).Splits",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ ((map (algebraMap K L) (minpoly K x)).comp (X - C ((algebraMap K L) r))).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 44
} | {
"line": 130,
"column": 45
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x - (algebraMap K L) r))).Splits",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x + -(algebraMap K L) r))).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 29
} | {
"line": 135,
"column": 30
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K ((algebraMap K L) r + x))).Splits",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x + (algebraMap K L) r))).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 28
} | {
"line": 140,
"column": 29
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K ((algebraMap K L) r - x))).Splits",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K ((algebraMap K L) r - x))).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 13
} | {
"line": 292,
"column": 14
} | [
{
"pp": "F : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : Algebra F K\nι : Sort u_4\ninst✝ : Nonempty ι\nf : E →ₐ[F] K\ns : ι → IntermediateField F E\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.48",
"assigned": ... | [
"F : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : Algebra F K\nι : Sort u_4\ninst✝ : Nonempty ι\nf : E →ₐ[F] K\ns : ι → IntermediateField F E\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 13
} | {
"line": 366,
"column": 14
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : Subfield E\nHF : Set.range ⇑(algebraMap F E) ⊆ ↑K\nHS : S ⊆ ↑K\n⊢ (adjoin F S).toSubfield ≤ K",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraM... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : Subfield E\nHF : Set.range ⇑(algebraMap F E) ⊆ ↑K\nHS : S ⊆ ↑K\n⊢ Set.range ⇑(algebraMap F E) ⊆ ↑K ∧ S ⊆ ↑K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 712,
"column": 2
} | {
"line": 712,
"column": 37
} | {
"line": 712,
"column": 38
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": []... | [
"K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Extension | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 38
} | {
"line": 51,
"column": 39
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nL₁ L₂ L₃ : Lifts F E K\nh₁₂ : L₁.carrier ≤ L₂.carrier\nh₁₂' : ∀ (x : ↥L₁.carrier), L₂.emb ((inclusion h₁₂) x) = L₁.emb x\nh₂₃ : L₂.carrier ≤ L₃.carrier\n... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nL₁ L₂ L₃ : Lifts F E K\nh₁₂ : L₁.carrier ≤ L₂.carrier\nh₁₂' : ∀ (x : ↥L₁.carrier), L₂.emb ((inclusion h₁₂) x) = L₁.emb x\nh₂₃ : L₂.carrier ≤ L₃.carrier\nh₂₃' : ∀ (x ... | ← inclusion_inclusion h₁₂ h₂₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 114,
"column": 28
} | {
"line": 114,
"column": 71
} | {
"line": 114,
"column": 72
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < p.degree\nk : 𝕜\nhprod : C k - p = C (C k - p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) (C k - p).roots).prod\nh_ne : C k - p ≠ 0\nlead_ne : (C k - p).leadingC... | [
"𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < p.degree\nk : 𝕜\nhprod : C k - p = C (C k - p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) (C k - p).roots).prod\nh_ne : C k - p ≠ 0\nlead_ne : (C k - p).leadingCoeff ≠ 0\nle... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 97
} | {
"line": 70,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nS : Set E\nM : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulSemiringAction M E\ninst✝ : SMulCommClass M F E\nm : M\n⊢ (∀ x ∈ adjoin F S, m • x = x) ↔ ∀ x ∈ S, m • x = x",
"ppTerm": "?m.29",
"assigned": false,
... | [
"F : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nS : Set E\nM : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulSemiringAction M E\ninst✝ : SMulCommClass M F E\nm : M\n⊢ (∀ x ∈ adjoin F S, m • x = x) ↔ ∀ x ∈ S, m • x = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 44
} | {
"line": 127,
"column": 45
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nn : ℕ\n⊢ (fun x ↦ x ^ n) '' σ a ⊆ σ (a ^ n)",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nn : ℕ\n⊢ (fun x ↦ x ^ n) '' σ a ⊆ σ (a ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 9
} | {
"line": 137,
"column": 10
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nn : ℕ\nhn : 0 < n\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nn : ℕ\nhn : 0 < n\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 44
} | {
"line": 143,
"column": 45
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nha : (σ a).Nonempty\nn : ℕ\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nha : (σ a).Nonempty\nn : ℕ\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 45
} | {
"line": 168,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\np : A\nhp : IsIdempotentElem p\na✝ : Nontrivial A\na : 𝕜\nha : a ∈ (fun x ↦ eval x (X ^ 2 - X)) ⁻¹' spectrum 𝕜 ((aeval p) (X ^ 2 - X))\n⊢ a ^ 2 = a",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\np : A\nhp : IsIdempotentElem p\na✝ : Nontrivial A\na : 𝕜\nha : a ∈ (fun x ↦ eval x (X ^ 2 - X)) ⁻¹' spectrum 𝕜 ((aeval p) (X ^ 2 - X))\n⊢ a ^ 2 = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 198,
"column": 17
} | {
"line": 198,
"column": 41
} | {
"line": 198,
"column": 42
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 19
} | {
"line": 215,
"column": 4
} | [
{
"pp": "k : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\np : k'[X]\nhmp : p.Monic\nhp : Irreducible p\n⊢ (map e.symm.toRingHom p).degree ≠ 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"k : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\np : k'[X]\nhmp : p.Monic\nhp : Irreducible p\n⊢ p.degree ≠ 0"
] | rw [degree_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 36
} | {
"line": 162,
"column": 37
} | [
{
"pp": "K : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nι : Type u_5\nt : ι → IntermediateField K L\np : ι → K[X]\ns : Finset ι\nh0 : ∏ i ∈ s, p i ≠ 0\nF : IntermediateField K L := ⨆ i ∈ s, t i\nhF : ∀ i ∈ s, t i ≤ F\nh : ∀ i ∈ s, (Polynomial.map (algebraMap K ↥(t i)) (p i)... | [
"K : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nι : Type u_5\nt : ι → IntermediateField K L\np : ι → K[X]\ns : Finset ι\nh0 : ∏ i ∈ s, p i ≠ 0\nF : IntermediateField K L := ⨆ i ∈ s, t i\nhF : ∀ i ∈ s, t i ≤ F\nh : ∀ i ∈ s, (Polynomial.map (algebraMap K ↥(t i)) (p i)).Splits ∧ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 35
} | {
"line": 271,
"column": 36
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\nx : k[X]\nhu : x ∈ nonunits k[X]\nh0 : x ≠ 0\nw✝¹ w✝ : k[X]\nh : ∀ (a : K), (aeval a) (x * w✝¹) ≠ 0 ∨ (aeval a) (x * w✝) ≠ 0\n⊢ (map (algebraMap k K) x).degree ≠ 0",
"ppTerm": "?m.124",
"ass... | [
"k : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\nx : k[X]\nhu : x ∈ nonunits k[X]\nh0 : x ≠ 0\nw✝¹ w✝ : k[X]\nh : ∀ (a : K), (aeval a) (x * w✝¹) ≠ 0 ∨ (aeval a) (x * w✝) ≠ 0\n⊢ x.degree ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 40
} | {
"line": 319,
"column": 41
} | [
{
"pp": "M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx : M\n⊢ (p - C x).degree ≠ 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"CommSemirin... | [
"M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx : M\n⊢ p.degree ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 320,
"column": 17
} | {
"line": 320,
"column": 52
} | {
"line": 320,
"column": 53
} | [
{
"pp": "M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx y : M\nhy : eval y (p - C x) = 0\n⊢ (fun x ↦ eval x p) y = x",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"id",
"Field.toSemifield",
"Semifield.toDivis... | [
"M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx y : M\nhy : eval y (p - C x) = 0\n⊢ eval y p = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 325,
"column": 4
} | {
"line": 326,
"column": 11
} | {
"line": 326,
"column": 12
} | [
{
"pp": "case refine_2\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhp : Nat.Prime (finrank F E)\nK : IntermediateField F E\n⊢ K = ⊥ ∨ K = ⊤",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOrde... | [
"case refine_2\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhp : Nat.Prime (finrank F E)\nK : IntermediateField F E\n⊢ K.toSubalgebra = ⊥ ∨ K.toSubalgebra = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 36
} | {
"line": 180,
"column": 37
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ x... | [
"K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ xi) ∈ ⨆ μ, (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 52
} | {
"line": 68,
"column": 53
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → End R M\nμ : ι → R\ni : ι\nh✝ : ∀ (j : ι), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))\nthis : Nonempty ι\np : Submodule R M := (f i).maxGenEigenspace (μ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → End R M\nμ : ι → R\ni : ι\nh✝ : ∀ (j : ι), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))\nthis : Nonempty ι\np : Submodule R M := (f i).maxGenEigenspace (μ i)\nh : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 50
} | {
"line": 190,
"column": 4
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ x... | [
"K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ xi) ∈ ⨆ μ, (f... | rw [LinearMap.sub_apply, algebraMap_end_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 20
} | {
"line": 88,
"column": 21
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nχ₁ : ι → R\na✝ : χ₁ ∈ {χ | ⨅ i, (f i).maxGenEigenspace (χ i) ≠ ⊥}\nχ₂ : ι → R\nhχ₁₂ : ⨅ i, (f i).maxGenEigenspace (χ₁ i) = ⨅ i, (... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nχ₁ : ι → R\na✝ : χ₁ ∈ {χ | ⨅ i, (f i).maxGenEigenspace (χ i) ≠ ⊥}\nχ₂ : ι → R\nhχ₁₂ : ⨅ i, (f i).maxGenEigenspace (χ₁ i) = ⨅ i, (f i).maxGenE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 632,
"column": 39
} | {
"line": 632,
"column": 77
} | {
"line": 632,
"column": 78
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | [
"K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDegree ≠ 0\nh :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 634,
"column": 7
} | {
"line": 634,
"column": 45
} | {
"line": 634,
"column": 46
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | [
"K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDegree ≠ 0\nH₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 651,
"column": 59
} | {
"line": 651,
"column": 92
} | {
"line": 651,
"column": 93
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | [
"K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDegree ≠ 0\nH₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 75,
"column": 12
} | {
"line": 75,
"column": 23
} | {
"line": 75,
"column": 24
} | [
{
"pp": "case zero\nR : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ₁ χ₂ : ↥H → R\nx : L\nm : M\... | [
"case zero\nR : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ₁ χ₂ : ↥H → R\nx : L\nm : M\nhx : x ∈ ro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 749,
"column": 2
} | {
"line": 749,
"column": 13
} | {
"line": 749,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nE : Type u\ninst✝¹ : Field E\ninst✝ : Algebra F E\ns : Set E\n⊢ #↥(adjoin F s) ≤ max (max #F #↑s) ℵ₀",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Intermedi... | [
"F : Type u\ninst✝² : Field F\nE : Type u\ninst✝¹ : Field E\ninst✝ : Algebra F E\ns : Set E\n⊢ (#↥(adjoin F s) ≤ #F ∨ #↥(adjoin F s) ≤ #↑s) ∨ (↑(adjoin F s)).Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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