module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 44
} | {
"line": 136,
"column": 45
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nhs : ⊤ ≤ span K s\n⊢ range Subtype.val = ⋯.extend ⋯",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend",
"Eq.mpr",
"congrArg",
"A... | [
"K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nhs : ⊤ ≤ span K s\n⊢ {x | x ∈ ⋯.extend ⋯} = ⋯.extend ⋯"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 300,
"column": 4
} | {
"line": 301,
"column": 40
} | {
"line": 303,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\np : Submodule K V\nv : V\nf : ↥p →ₗ[K] V'\nhv : v ∉ p\ny : V'\ng : V →ₗ[K] V'\nhg :\n g ∘ₗ ({ domain := p, toFun := f }.supSpa... | [] | have := LinearPMap.supSpanSingleton_apply_self ⟨p, f⟩ y hv
simpa using! congr($hg _).trans this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 300,
"column": 4
} | {
"line": 301,
"column": 40
} | {
"line": 303,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\np : Submodule K V\nv : V\nf : ↥p →ₗ[K] V'\nhv : v ∉ p\ny : V'\ng : V →ₗ[K] V'\nhg :\n g ∘ₗ ({ domain := p, toFun := f }.supSpa... | [] | have := LinearPMap.supSpanSingleton_apply_self ⟨p, f⟩ y hv
simpa using! congr($hg _).trans this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 360,
"column": 4
} | {
"line": 362,
"column": 12
} | {
"line": 363,
"column": 4
} | [
{
"pp": "case hs\nK : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nv : V\nhfv : f v ≠ 0\nb₁ : Basis (↑(Basis.ofVectorSpaceIndex K ↥f.ker)) K ↥f.ker := ⋯\ns : Set V := ⋯\nhs : span K s = f.ker\nn : Set V := ⋯\n⊢ LinearIndepOn K _root_.id s",
"ppTerm": ... | [
"case hx\nK : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nv : V\nhfv : f v ≠ 0\nb₁ : Basis (↑(Basis.ofVectorSpaceIndex K ↥f.ker)) K ↥f.ker := ⋯\ns : Set V := ⋯\nhs : span K s = f.ker\nn : Set V := ⋯\n⊢ v ∉ span K s"
] | · apply LinearIndepOn.image
· exact b₁.linearIndependent.linearIndepOn_id
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 412,
"column": 4
} | {
"line": 414,
"column": 12
} | {
"line": 415,
"column": 4
} | [
{
"pp": "case hs\nK : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\nthis : LinearIndepOn K _root_.id {v}\nb₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := ⋯\nw : V\nhw : f w = 1\ns : Set V := ⋯\nhs : span K s = f.ker\nhvs : ↑v... | [
"case hx\nK : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\nthis : LinearIndepOn K _root_.id {v}\nb₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := ⋯\nw : V\nhw : f w = 1\ns : Set V := ⋯\nhs : span K s = f.ker\nhvs : ↑v ∈ s\nn : Se... | · apply LinearIndepOn.image
· exact b₁.linearIndependent.linearIndepOn_id
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 385,
"column": 2
} | {
"line": 386,
"column": 82
} | {
"line": 387,
"column": 2
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\ninst✝ : FiniteDimensional K ↥p\nf : V →ₗ[K] V\nh : ∀ x ∈ p, f x ∈ p\nh' : Disjoint p f.ker\nx : V\nhx : x ∈ comap f p\n⊢ x ∈ p ⊔ f.ker",
"ppTerm": "?m.75",
"assigned": true,
"us... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\ninst✝ : FiniteDimensional K ↥p\nf : V →ₗ[K] V\nh : ∀ x ∈ p, f x ∈ p\nh' : Disjoint p f.ker\nx : V\nhx : x ∈ comap f p\ny : V\nhy : y ∈ p\nhxy : (f.restrict h) ⟨y, hy⟩ = ⟨f x, hx⟩\n⊢ x ∈ p ⊔ f.ker"
] | obtain ⟨⟨y, hy⟩, hxy⟩ :=
surjective_of_injective ((injective_restrict_iff_disjoint h).mpr h') ⟨f x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 17
} | {
"line": 244,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F ↥S ≤ 1\ninst✝ : Free F ↥S\n⊢ S = ⊥",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Lattice.toSem... | [
"F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F ↥S ≤ 1\ninst✝ : Free F ↥S\na✝ : Nontrivial E\n⊢ S = ⊥"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 310,
"column": 76
} | {
"line": 310,
"column": 95
} | {
"line": 312,
"column": 0
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Monoid M\nf : R →+* S\n⊢ map (↑f) 1 = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRing... | [] | ext; simp [one_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 310,
"column": 76
} | {
"line": 310,
"column": 95
} | {
"line": 312,
"column": 0
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Monoid M\nf : R →+* S\n⊢ map (↑f) 1 = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRing... | [] | ext; simp [one_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 17
} | {
"line": 302,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\ninst✝ : Free F E\n⊢ finrank F E = 1 → Function.Bijective ⇑(algebraMap F E)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Nontrivial",
"False",
... | [
"F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\ninst✝ : Free F E\na✝ : Nontrivial E\n⊢ finrank F E = 1 → Function.Bijective ⇑(algebraMap F E)"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 275,
"column": 31
} | {
"line": 275,
"column": 40
} | {
"line": 277,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Monoid M\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : M →* A\na : M\nb : R\nf : M →₀ R\na✝¹ : ... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 275,
"column": 31
} | {
"line": 275,
"column": 40
} | {
"line": 277,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Monoid M\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : M →* A\na : M\nb : R\nf : M →₀ R\na✝¹ : ... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 275,
"column": 31
} | {
"line": 275,
"column": 40
} | {
"line": 277,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Monoid M\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : M →* A\na : M\nb : R\nf : M →₀ R\na✝¹ : ... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 23
} | {
"line": 82,
"column": 23
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\nn : ℕ\n⊢ eval₂ f x (X ^ n) = x ^ n",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Polynomial.X_p... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\nn : ℕ\n⊢ eval₂ f x ((monomial n) 1) = x ^ n"
] | X_pow_eq_monomial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 649,
"column": 31
} | {
"line": 649,
"column": 40
} | {
"line": 651,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddMonoid M\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : Multiplicative M →* A\na : M\nb : R\n... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 649,
"column": 31
} | {
"line": 649,
"column": 40
} | {
"line": 651,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddMonoid M\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : Multiplicative M →* A\na : M\nb : R\n... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 649,
"column": 31
} | {
"line": 649,
"column": 40
} | {
"line": 651,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u_1\nS : Type u_2\nA : Type u_4\nM : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddMonoid M\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\nf✝ : Multiplicative M →* A\na : M\nb : R\n... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.FreeAlgebra | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 47
} | {
"line": 520,
"column": 2
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = (algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1\nhf0 : f0 (ι R x) = 0\n⊢ False",
"ppTerm": "?m.60",
"assigned": true,
... | [
"R : Type u_1\nX : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = (algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1\nhf0 : f0 (ι R x) = 0\nhf1 : f1 (ι R x) = 1\n⊢ False"
] | have hf1 : f1 (ι R x) = 1 := lift_ι_apply _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 520,
"column": 4
} | {
"line": 520,
"column": 61
} | {
"line": 521,
"column": 4
} | [
{
"pp": "case h'\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T... | [
"case h'\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T G\nf g : E ... | simp only [neg_domain, neg_apply, neg_eq_iff_add_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.CharP.Defs | {
"line": 283,
"column": 22
} | {
"line": 283,
"column": 45
} | {
"line": 284,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ ↑1 * r = 0 * r",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"AddMonoid.... | [] | rw [CharP.cast_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CharP.Defs | {
"line": 283,
"column": 22
} | {
"line": 283,
"column": 45
} | {
"line": 284,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ ↑1 * r = 0 * r",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"AddMonoid.... | [] | rw [CharP.cast_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Defs | {
"line": 283,
"column": 22
} | {
"line": 283,
"column": 45
} | {
"line": 284,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ ↑1 * r = 0 * r",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"AddMonoid.... | [] | rw [CharP.cast_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 607,
"column": 2
} | {
"line": 609,
"column": 36
} | {
"line": 611,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : DirectedOn (... | [] | · intro p hpc
refine ⟨le_sSup <| Set.mem_image_of_mem domain hpc, fun x y hxy => Eq.symm ?_⟩
exact f_eq ⟨p, hpc⟩ _ _ hxy.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 827,
"column": 2
} | {
"line": 828,
"column": 28
} | {
"line": 829,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\nx : E\nh : x ∈ f.domain\n⊢ ∃ y, (x, y) ∈ f.graph",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Subm... | [
"case mpr\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\nx : E\nh : ∃ y, (x, y) ∈ f.graph\n⊢ x ∈ f.domain"
] | · use f ⟨x, h⟩
exact f.mem_graph ⟨x, h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 902,
"column": 11
} | {
"line": 902,
"column": 13
} | {
"line": 902,
"column": 14
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.1 = 0 → x.2 = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ : F\n⊢ ∀ (y₂ : F),... | [
"case refine_2\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.1 = 0 → x.2 = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\n⊢ (a, y₁) ∈ g → (a, y₂... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 270,
"column": 54
} | {
"line": 272,
"column": 47
} | {
"line": 274,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝² : Semiring R\ninst✝¹ : SemilatticeSup B\ninst✝ : OrderBot B\nD : A → B\ns : R[A]\nhs : s.support.Nonempty\n⊢ supDegree (WithBot.some ∘ D) s = ↑(supDegree D s)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Wi... | [] | by
unfold AddMonoidAlgebra.supDegree
rw [← Finset.coe_sup' hs, Finset.sup'_eq_sup] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 44
} | {
"line": 333,
"column": 2
} | [
{
"pp": "case h₀\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : SemilatticeSup B\ninst✝⁴ : OrderBot B\nD : A → B\np q : R[A]\ninst✝³ : AddZeroClass A\ninst✝² : Add B\nhadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Funct... | [] | exact (add_lt_add_of_lt_of_le (((Finset.le_sup ha).trans hp).lt_of_ne <| hD.ne_iff.2 hne)
<| (Finset.le_sup ha').trans hq).ne he | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 433,
"column": 10
} | {
"line": 433,
"column": 64
} | {
"line": 434,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\ni : ℕ\nhi : (p ^ i).coeff (i * p.natDegree) = p.leadingCoeff ^ i\nhp1 : p.leadingCoeff ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ (p ^ i).natDegree < i * p.natDegree",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"instMulNat",... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\ni : ℕ\nhi : (p ^ i).coeff (i * p.natDegree) = p.leadingCoeff ^ i\nhp1 : p.leadingCoeff ^ i = 0\nhp2 : ¬p ^ i = 0\nh : (p ^ i).natDegree = i * p.natDegree\n⊢ p ^ i = 0"
] | refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Degree.SmallDegree | {
"line": 90,
"column": 74
} | {
"line": 93,
"column": 70
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u\na b c : R\ninst✝ : Semiring R\n⊢ (C a * X ^ 2 + C b * X + C c).degree ≤ 2",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"WithBo... | [] | by
simpa only [add_assoc] using!
degree_add_le_of_degree_le (degree_C_mul_X_pow_le 2 a)
(le_trans degree_linear_le <| WithBot.coe_le_coe.mpr one_le_two) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Adjoin.Polynomial.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 34
} | {
"line": 72,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type z\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\na : ↥R[x]\n⊢ ∃ p, (aeval x) p = ↑a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Algebra.adjoin",
"Membership.mem",
"SetLike.coe_m... | [
"R : Type u\nA : Type z\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\na : ↥R[x]\nthis : ↑a ∈ R[x]\n⊢ ∃ p, (aeval x) p = ↑a"
] | have : (a : A) ∈ R[x] := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Notation.Lemmas | {
"line": 24,
"column": 9
} | {
"line": 24,
"column": 32
} | {
"line": 26,
"column": 0
} | [
{
"pp": "case isTrue\nα : Type u_1\ninst✝² : One α\np : Prop\ninst✝¹ : Decidable p\na : p → α\nb : ¬p → α\ninst✝ : LE α\nha : ∀ (h : p), 1 ≤ a h\nhb : ∀ (h : ¬p), 1 ≤ b h\nh✝ : p\n⊢ 1 ≤ a h✝",
"ppTerm": "?isTrue",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"ha",
"h✝"
]... | [] | exacts [ha ‹_›, hb ‹_›] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Notation.Lemmas | {
"line": 28,
"column": 9
} | {
"line": 28,
"column": 32
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case isTrue\nα : Type u_1\ninst✝² : One α\np : Prop\ninst✝¹ : Decidable p\na : p → α\nb : ¬p → α\ninst✝ : LE α\nha : ∀ (h : p), a h ≤ 1\nhb : ∀ (h : ¬p), b h ≤ 1\nh✝ : p\n⊢ a h✝ ≤ 1",
"ppTerm": "?isTrue",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"ha",
"h✝"
]... | [] | exacts [ha ‹_›, hb ‹_›] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Notation.Lemmas | {
"line": 32,
"column": 9
} | {
"line": 32,
"column": 32
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case isTrue\nα : Type u_1\ninst✝² : One α\np : Prop\ninst✝¹ : Decidable p\na : p → α\nb : ¬p → α\ninst✝ : LT α\nha : ∀ (h : p), 1 < a h\nhb : ∀ (h : ¬p), 1 < b h\nh✝ : p\n⊢ 1 < a h✝",
"ppTerm": "?isTrue",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"ha",
"h✝"
]... | [] | exacts [ha ‹_›, hb ‹_›] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Notation.Lemmas | {
"line": 36,
"column": 9
} | {
"line": 36,
"column": 32
} | {
"line": 38,
"column": 0
} | [
{
"pp": "case isTrue\nα : Type u_1\ninst✝² : One α\np : Prop\ninst✝¹ : Decidable p\na : p → α\nb : ¬p → α\ninst✝ : LT α\nha : ∀ (h : p), a h < 1\nhb : ∀ (h : ¬p), b h < 1\nh✝ : p\n⊢ a h✝ < 1",
"ppTerm": "?isTrue",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"ha",
"h✝"
]... | [] | exacts [ha ‹_›, hb ‹_›] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 392,
"column": 35
} | {
"line": 394,
"column": 47
} | {
"line": 396,
"column": 0
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\np : R[X]\n⊢ (p.comp (-X)).comp (-X) = p",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.comp_assoc",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Polynomial.instNeg",
"CommRing.toNonUnitalCo... | [] | by
rw [comp_assoc]
simp only [neg_comp, X_comp, neg_neg, comp_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 509,
"column": 34
} | {
"line": 509,
"column": 49
} | {
"line": 509,
"column": 49
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R S\ninst✝ : Algebra T S\nφ : R →+* T\nh : (algebraMap T S).comp φ = algebraMap R S\np : R[X]\na : S\n⊢ (algebraMap ?m.47 ?m.48).comp ?m.54 = RingHom.comp ?m.55 (algebraMap R S)",... | [
"R : Type u\nS : Type v\nT : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R S\ninst✝ : Algebra T S\nφ : R →+* T\nh : (algebraMap T S).comp φ = algebraMap R S\np : R[X]\na : S\n⊢ (algebraMap ?m.47 S).comp ?m.54 = algebraMap R S"
] | RingHom.id_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 612,
"column": 2
} | {
"line": 617,
"column": 49
} | {
"line": 618,
"column": 2
} | [
{
"pp": "case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ eval z f\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∈ f.support\n⊢ p ∣ f.coeff i * z ^ i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"Dvd.dvd",
... | [
"case neg\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ eval z f\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∉ f.support\n⊢ p ∣ f.coeff i * z ^ i"
] | · rw [eval, eval₂_eq_sum, sum_def] at dvd_eval
rw [← Finset.insert_erase hi, Finset.sum_insert (Finset.notMem_erase _ _)] at dvd_eval
refine (dvd_add_left ?_).mp dvd_eval
apply Finset.dvd_sum
intro j hj
exact dvd_terms j (Finset.ne_of_mem_erase hj) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finsupp.Order | {
"line": 147,
"column": 59
} | {
"line": 147,
"column": 91
} | {
"line": 148,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : 0 ≤ g\n⊢ 0 ≤ mapDomain f g",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Finsupp.mapDomain",
"AddMono... | [] | by simpa using mapDomain_mono hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Order | {
"line": 148,
"column": 59
} | {
"line": 148,
"column": 91
} | {
"line": 150,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : g ≤ 0\n⊢ mapDomain f g ≤ 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Finsupp.mapDomain",
"AddMono... | [] | by simpa using mapDomain_mono hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 294,
"column": 78
} | {
"line": 295,
"column": 92
} | {
"line": 297,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nα : Type u_2\ns : Finset α\nf : α → σ →₀ ℕ\ng : α → R\n⊢ (monomial (∑ i ∈ s, f i)) (∏ i ∈ s, g i) = ∏ i ∈ s, (monomial (f i)) (g i)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | [] | by
simp_rw [monomial_sum_index, map_prod, ← Finset.prod_mul_distrib, C_mul_monomial, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 20
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case mul_X\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n (∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * (monomial s) a) = eval₂ f g p * f a * s.prod fun n e ↦ g n ^ e) →\n ... | [
"case mul_X\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : σ\nih : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * (monomial s) a) = eval₂ f g p * f a * s.prod fun n e ↦ g n ^ e\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f... | intro p n ih s a | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 332,
"column": 16
} | {
"line": 332,
"column": 25
} | {
"line": 333,
"column": 6
} | [
{
"pp": "case zero\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : n ∉ p.support\n_he : e ≠ 0\nih : motive ((m... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 332,
"column": 16
} | {
"line": 332,
"column": 25
} | {
"line": 333,
"column": 6
} | [
{
"pp": "case zero\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : n ∉ p.support\n_he : e ≠ 0\nih : motive ((m... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 332,
"column": 16
} | {
"line": 332,
"column": 25
} | {
"line": 333,
"column": 6
} | [
{
"pp": "case zero\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : n ∉ p.support\n_he : e ≠ 0\nih : motive ((m... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 54
} | {
"line": 187,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nd : τ →₀ ℕ\nhd : coeff d ((rename f) φ) ≠ 0\nhi : i ∈ d.support\n⊢ ∃ a ∈ φ.degrees, f a = i",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"... | [
"R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) ((rename f) φ) ≠ 0\nhi : i ∈ (Finsupp.mapDomain f x).support\n⊢ ∃ a ∈ φ.degrees, f a = i"
] | obtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 280,
"column": 25
} | {
"line": 280,
"column": 62
} | {
"line": 280,
"column": 63
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : σ\nk : ℕ\n⊢ degreeOf i ((monomial (Finsupp.single i k)) 1) = k",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : σ\nk : ℕ\n⊢ (Finsupp.single i k) i = k"
] | degreeOf_monomial_eq _ _ one_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 985,
"column": 2
} | {
"line": 986,
"column": 71
} | {
"line": 988,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\ni : σ →₀ ℕ\nx : S\n⊢ (monomial i) x ∈ coeffsIn σ M ↔ x ∈ M",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"N... | [] | simp only [mem_coeffsIn, coeff_monomial]
exact ⟨fun h ↦ by simpa using h i, fun hs j ↦ by split <;> simp [hs]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 985,
"column": 2
} | {
"line": 986,
"column": 71
} | {
"line": 988,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\ni : σ →₀ ℕ\nx : S\n⊢ (monomial i) x ∈ coeffsIn σ M ↔ x ∈ M",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"N... | [] | simp only [mem_coeffsIn, coeff_monomial]
exact ⟨fun h ↦ by simpa using h i, fun hs j ↦ by split <;> simp [hs]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 148,
"column": 25
} | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\n⊢ ((monomial n) 1).eraseLead = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Polynomial.eraseLead_monomial",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Lin... | [
"R : Type u_1\ninst✝ : Semiring R\nn : ℕ\n⊢ 0 = 0"
] | eraseLead_monomial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 152,
"column": 31
} | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℕ\n⊢ ((monomial n) r).eraseLead = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Polynomial.eraseLead_monomial",
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Polynomial.mon... | [
"R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℕ\n⊢ 0 = 0"
] | eraseLead_monomial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 375,
"column": 44
} | {
"line": 376,
"column": 64
} | {
"line": 378,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nn : ℕ\n⊢ (X ^ n).natTrailingDegree = n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NeZero.one",
"Semiring.toModule",
"Polynomial.X_pow_eq_... | [] | by
rw [X_pow_eq_monomial, natTrailingDegree_monomial one_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 46
} | {
"line": 220,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\n⊢ f.eraseLead.natDegree = f.natDegree - 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_pos_of_nextCoeff_ne_zero",
"instOfNatNat",
"Nat",
"LT.lt",
"instLTNat",
... | [
"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\nthis : 0 < f.natDegree\n⊢ f.eraseLead.natDegree = f.natDegree - 1"
] | have := natDegree_pos_of_nextCoeff_ne_zero h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 22
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\nthis : 0 < f.natDegree\n⊢ 0 < f.natDegree",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"this"
],
"usedGoals": []
},
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 46
} | {
"line": 252,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\n⊢ f.eraseLead.leadingCoeff = f.nextCoeff",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_pos_of_nextCoeff_ne_zero",
"instOfNatNat",
"Nat",
"LT.lt",
"instLTNat",
... | [
"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\nthis : 0 < f.natDegree\n⊢ f.eraseLead.leadingCoeff = f.nextCoeff"
] | have := natDegree_pos_of_nextCoeff_ne_zero h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 22
} | {
"line": 256,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.nextCoeff ≠ 0\nthis : 0 < f.natDegree\n⊢ 0 < f.natDegree",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"this"
],
"usedGoals": []
},
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 270,
"column": 4
} | {
"line": 282,
"column": 58
} | {
"line": 283,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : P.eraseLead = 0\n⊢ ((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | rw [he, mul_zero]
by_cases he₂ : ((X - C x) * P).eraseLead = 0
· simp [he₂]
suffices #((X - C x) * P).support ≤ 2 by
rw [← card_support_eq_zero]
linarith [eraseLead_support_card_lt he₂,
eraseLead_support_card_lt (mul_ne_zero (X_sub_C_ne_zero x) hp)]
have h₂ : #(X - C x).support = 2 :... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 270,
"column": 4
} | {
"line": 282,
"column": 58
} | {
"line": 283,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : P.eraseLead = 0\n⊢ ((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | rw [he, mul_zero]
by_cases he₂ : ((X - C x) * P).eraseLead = 0
· simp [he₂]
suffices #((X - C x) * P).support ≤ 2 by
rw [← card_support_eq_zero]
linarith [eraseLead_support_card_lt he₂,
eraseLead_support_card_lt (mul_ne_zero (X_sub_C_ne_zero x) hp)]
have h₂ : #(X - C x).support = 2 :... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 102,
"column": 10
} | {
"line": 102,
"column": 21
} | {
"line": 102,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nN i : ℕ\nf : AddMonoidAlgebra R ℕ\n⊢ (embDomain (revAt N) f) i = (embDomain (revAt N) f) ((revAt N) ((revAt N) i))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Polynomial.revAt",
"congrA... | [
"R : Type u_1\ninst✝ : Semiring R\nN i : ℕ\nf : AddMonoidAlgebra R ℕ\n⊢ (embDomain (revAt N) f) i = (embDomain (revAt N) f) i"
] | revAt_invol | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 369,
"column": 6
} | {
"line": 369,
"column": 13
} | {
"line": 370,
"column": 6
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nfu : ℕ → ℕ\nfu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0\nfc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m\nφ_k : ∀ {f : R[X]}, f.... | [
"case neg.zero\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nfu : ℕ → ℕ\nφ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n\nf g : R[X]\nfg : f.natDegree < g.natDegr... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 369,
"column": 4
} | {
"line": 371,
"column": 30
} | {
"line": 373,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nfu : ℕ → ℕ\nfu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0\nfc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m\nφ_k : ∀ {f : R[X]}, f.... | [] | · cases k
· nomatch FG
· rwa [φ_k FG, zero_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 251,
"column": 77
} | {
"line": 251,
"column": 88
} | {
"line": 251,
"column": 88
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : ¬f = 0\n⊢ f.coeff ((revAt f.natDegree) ((revAt f.natDegree) f.natTrailingDegree)) ≠ 0",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.revAt",
"congrArg",
"id",
"Ne",
... | [
"case neg.a\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : ¬f = 0\n⊢ f.coeff f.natTrailingDegree ≠ 0"
] | revAt_invol | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 48
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial (Option σ) R\ni : ℕ\nm : σ →₀ ℕ\n⊢ m ∈ (((optionEquivLeft R σ) f).coeff i).support ↔ optionElim i m ∈ f.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroCla... | [] | simp [← optionEquivLeft_coeff_some_coeff_none] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 48
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial (Option σ) R\ni : ℕ\nm : σ →₀ ℕ\n⊢ m ∈ (((optionEquivLeft R σ) f).coeff i).support ↔ optionElim i m ∈ f.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroCla... | [] | simp [← optionEquivLeft_coeff_some_coeff_none] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 48
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial (Option σ) R\ni : ℕ\nm : σ →₀ ℕ\n⊢ m ∈ (((optionEquivLeft R σ) f).coeff i).support ↔ optionElim i m ∈ f.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroCla... | [] | simp [← optionEquivLeft_coeff_some_coeff_none] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 269,
"column": 44
} | {
"line": 269,
"column": 57
} | {
"line": 269,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < t.card\na✝ : Nontrivial R\n⊢ Multiset ?m.52",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"t"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 51
} | {
"line": 232,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nn : ℕ\nh✝ : Submodule.span R s ≤ degreeLE R ↑n\n⊢ ∃ n, Submodule.span R s ≤ degreeLT R n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"WithBot",... | [] | exact ⟨n + 1, by rwa [degreeLT_succ_eq_degreeLE]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Monic | {
"line": 561,
"column": 15
} | {
"line": 561,
"column": 42
} | {
"line": 561,
"column": 42
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\nhp : h.unit⁻¹ • p * q = 0\nthis : h.unit⁻¹ • (p * q) = h.unit⁻¹ • p * q\n⊢ q = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul"... | [
"case mp\nR : Type u\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\nhp : q = 0\nthis : h.unit⁻¹ • (p * q) = h.unit⁻¹ • p * q\n⊢ q = 0",
"case mp.h\nR : Type u\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\nhp : h.unit⁻¹ • p * q = 0\nthis : h.unit⁻¹ • (p * q) = h.unit⁻¹ • ... | Monic.mul_right_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.Tower | {
"line": 37,
"column": 2
} | {
"line": 42,
"column": 13
} | {
"line": 43,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\n⊢ Subalgebra.restrictScalars C (adjoin D S) =\n Subalgebra.restrictScalars C ... | [
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\n⊢ Set.range ⇑(algebraMap D E) = Set.range ⇑(algebraMap (↥(Subalgebra.map (IsScalarTower.toAl... | suffices
Set.range (algebraMap D E) =
Set.range (algebraMap ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) E) by
ext x
change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S)
rw [this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Ring.GeomSum | {
"line": 344,
"column": 75
} | {
"line": 345,
"column": 49
} | {
"line": 347,
"column": 0
} | [
{
"pp": "m k x : ℕ\nhmk : m ∣ k\n⊢ x ^ m - 1 ∣ x ^ k - 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Dvd.dvd",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"Eq.mp",
"instSubNat",
... | [] | by
simpa using pow_sub_pow_dvd_pow_sub_pow x 1 hmk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Adjoin.Tower | {
"line": 115,
"column": 10
} | {
"line": 115,
"column": 24
} | {
"line": 115,
"column": 24
} | [
{
"pp": "case inr.inr\nA : Type w\nB : Type u₁\nC : Type u_1\ninst✝⁶ : CommSemiring A\ninst✝⁵ : CommSemiring B\ninst✝⁴ : Semiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\nx : Finset C\nhx : Algebra.adjoin A ↑x = ⊤\ny : Finset C\nhy : span B ↑y = ⊤\nf : C ... | [
"case inr.inr\nA : Type w\nB : Type u₁\nC : Type u_1\ninst✝⁶ : CommSemiring A\ninst✝⁵ : CommSemiring B\ninst✝⁴ : Semiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\nx : Finset C\nhx : Algebra.adjoin A ↑x = ⊤\ny : Finset C\nhy : span B ↑y = ⊤\nf : C → C → B\nh✝ ... | ← hf (yi * yj) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FiniteType | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 32
} | {
"line": 130,
"column": 33
} | [
{
"pp": "case mp\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Finset A\nhs : adjoin R ↑s = ⊤\n⊢ ↑(adjoin R (Set.range Subtype.val)) = Set.univ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | [
"case mp\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Finset A\nhs : adjoin R ↑s = ⊤\n⊢ ↑(adjoin R {x | x ∈ s}) = Set.univ"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 853,
"column": 12
} | {
"line": 853,
"column": 71
} | {
"line": 854,
"column": 12
} | [
{
"pp": "case h.inr.convert_2\nR : Type u\ninst✝ : CommRing R\ninst : IsNoetherianRing R\nI : Ideal R[X]\nM : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯\nhm : M ∈ Set.range I.leadingCoeffNth\nN : ℕ\nHN : I.leadingCoeffNth N = M\ns : Finset R[X]\nhs : Submodule.span R ↑s = I.degreeLE ↑N\nhm2 : ∀ (k :... | [
"case pos\nR : Type u\ninst✝ : CommRing R\ninst : IsNoetherianRing R\nI : Ideal R[X]\nM : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯\nhm : M ∈ Set.range I.leadingCoeffNth\nN : ℕ\nHN : I.leadingCoeffNth N = M\ns : Finset R[X]\nhs : Submodule.span R ↑s = I.degreeLE ↑N\nhm2 : ∀ (k : ℕ), I.leadingCoeffNth k... | by_cases hpq : p - q * Polynomial.X ^ (k - q.natDegree) = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.FiniteType | {
"line": 194,
"column": 43
} | {
"line": 194,
"column": 69
} | {
"line": 194,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ ↑(adjoin R (Set.range Subtype.val)) = Set.univ",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"S... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ ↑(adjoin R {x | x ∈ s}) = Set.univ"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FiniteType | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 29
} | {
"line": 264,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nh : (g.comp f).FiniteType\n⊢ g.FiniteType",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nh : (g.comp f).FiniteType\nalgInst✝² : Algebra A B := f.toAlgebra\nalgInst✝¹ : Algebra B C := g.toAlgebra\nalgInst✝ : Algebra A C := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalarTow... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 939,
"column": 2
} | {
"line": 939,
"column": 49
} | {
"line": 940,
"column": 2
} | [
{
"pp": "σ : Type v\nR : Type u_2\ninst✝ : CommSemiring R\nm n : ℕ\nF : MvPolynomial σ R\nhF : F.totalDegree ≤ m\nf : σ → R[X]\nhf : ∀ (i : σ), (f i).natDegree ≤ n\nd : σ →₀ ℕ\nhd : d ∈ (AddMonoidAlgebra.coeff F).support\n⊢ (Polynomial.C ((AddMonoidAlgebra.coeff F) d) * d.prod fun n e ↦ f n ^ e).natDegree ≤ m *... | [
"σ : Type v\nR : Type u_2\ninst✝ : CommSemiring R\nm n : ℕ\nF : MvPolynomial σ R\nhF : F.totalDegree ≤ m\nf : σ → R[X]\nhf : ∀ (i : σ), (f i).natDegree ≤ n\nd : σ →₀ ℕ\nhd : d ∈ (AddMonoidAlgebra.coeff F).support\n⊢ (d.prod fun n e ↦ f n ^ e).natDegree ≤ m * n"
] | apply (Polynomial.natDegree_C_mul_le _ _).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Nat.Prime.Basic | {
"line": 25,
"column": 91
} | {
"line": 26,
"column": 78
} | {
"line": 28,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ Prime (a * b) ↔ Prime a ∧ b = 1 ∨ Prime b ∧ a = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Nat.Prime.Basic.0.Nat.prime_mul_iff._simp_1_2",
"Nat.Prime",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Na... | [] | by
simp only [irreducible_mul_iff, ← irreducible_iff_nat_prime, Nat.isUnit_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Prime.Basic | {
"line": 35,
"column": 2
} | {
"line": 39,
"column": 23
} | {
"line": 41,
"column": 0
} | [
{
"pp": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p ↔ p = a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"Nat.instMonoid",
"False.elim",
"Nat.instMulOneClass",
"Ne",
"instMulNat",
"instOfNatNat",
... | [] | refine ⟨?_, by rintro rfl; rfl⟩
rintro ⟨j, rfl⟩
rcases prime_mul_iff.mp hp with (⟨_, rfl⟩ | ⟨_, rfl⟩)
· exact mul_one _
· exact (a1 rfl).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Basic | {
"line": 35,
"column": 2
} | {
"line": 39,
"column": 23
} | {
"line": 41,
"column": 0
} | [
{
"pp": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p ↔ p = a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"Nat.instMonoid",
"False.elim",
"Nat.instMulOneClass",
"Ne",
"instMulNat",
"instOfNatNat",
... | [] | refine ⟨?_, by rintro rfl; rfl⟩
rintro ⟨j, rfl⟩
rcases prime_mul_iff.mp hp with (⟨_, rfl⟩ | ⟨_, rfl⟩)
· exact mul_one _
· exact (a1 rfl).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Basic | {
"line": 182,
"column": 22
} | {
"line": 182,
"column": 89
} | {
"line": 182,
"column": 89
} | [
{
"pp": "x✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\n⊢ p = a * y",
"ppTerm": "?m.183",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Semigroup... | [] | by rwa [ha, sq, mul_assoc, mul_right_inj' hp.ne_zero, eq_comm] at h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Dynamics.PeriodicPts.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 73,
"column": 54
} | {
"line": 75,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → α\nx : α\ng : α → α\nh : Function.Commute f g\nhco : (minimalPeriod f x).Coprime (minimalPeriod g x)\n⊢ minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"Nat.Coprime.sym... | [] | apply h.minimalPeriod_of_comp_dvd_mul.antisymm
suffices ∀ {f g : α → α},
Commute f g →
Coprime (minimalPeriod f x) (minimalPeriod g x) →
minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from
hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)
intro f g h hco
refin... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.PeriodicPts.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 73,
"column": 54
} | {
"line": 75,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → α\nx : α\ng : α → α\nh : Function.Commute f g\nhco : (minimalPeriod f x).Coprime (minimalPeriod g x)\n⊢ minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"Nat.Coprime.sym... | [] | apply h.minimalPeriod_of_comp_dvd_mul.antisymm
suffices ∀ {f g : α → α},
Commute f g →
Coprime (minimalPeriod f x) (minimalPeriod g x) →
minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from
hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)
intro f g h hco
refin... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.PeriodicPts.Lemmas | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 28
} | {
"line": 126,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → α\ng : β → β\nx : α × β\n⊢ minimalPeriod f x.1 ∣ minimalPeriod (Prod.map f g) x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.lcm",
"Eq.mpr",
"Function.minimalPeriod_prodMap",
"Dvd.dvd",
"congrArg",
"id... | [
"α : Type u_1\nβ : Type u_2\nf : α → α\ng : β → β\nx : α × β\n⊢ minimalPeriod f x.1 ∣ (minimalPeriod f x.1).lcm (minimalPeriod g x.2)"
] | rw [minimalPeriod_prodMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Dynamics.PeriodicPts.Lemmas | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 28
} | {
"line": 129,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → α\ng : β → β\nx : α × β\n⊢ minimalPeriod g x.2 ∣ minimalPeriod (Prod.map f g) x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.lcm",
"Eq.mpr",
"Function.minimalPeriod_prodMap",
"Dvd.dvd",
"congrArg",
"id... | [
"α : Type u_1\nβ : Type u_2\nf : α → α\ng : β → β\nx : α × β\n⊢ minimalPeriod g x.2 ∣ (minimalPeriod f x.1).lcm (minimalPeriod g x.2)"
] | rw [minimalPeriod_prodMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.Cycle | {
"line": 336,
"column": 46
} | {
"line": 336,
"column": 68
} | {
"line": 337,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nh : l ≠ []\nhn : l.Nodup\n⊢ l.length - 1 + 1 = l.length",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.List.Cycle.0.List.next_getLast_eq_head._proof_1_1"
],
"usedFVars": [
"α",
"l",
... | [] | grind [length_pos_iff] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Data.List.Cycle | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 14
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + 1 + ([].length - 1)) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Cycle | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 14
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + 1 + ([].length - 1)) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Cycle | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 14
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + 1 + ([].length - 1)) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Basic | {
"line": 265,
"column": 14
} | {
"line": 265,
"column": 57
} | {
"line": 265,
"column": 58
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\ng h k : G\na b c : α\nhg : b = g • a\n⊢ Subgroup.map (↑(MulAut.conj g)) (stabilizer G a) = stabilizer G (g • a)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"G : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MulAction G β\ng h k : G\na b c : α\nhg : b = g • a\n⊢ Subgroup.map (↑(MulAut.conj g)) (stabilizer G a) =\n Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (stabilizer G a)"
] | stabilizer_smul_eq_stabilizer_map_conj g a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Cycle | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 14
} | {
"line": 369,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + ([].length - 1) + 1) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Cycle | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 14
} | {
"line": 369,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + ([].length - 1) + 1) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Cycle | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 14
} | {
"line": 369,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nh : [].Nodup\nhn : n < [].length\nhx : [][n] ∈ []\n⊢ [][(n + ([].length - 1) + 1) % [].length] = [][n]",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"False",
"False.elim",
"HSub.hSub",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Cycle | {
"line": 543,
"column": 92
} | {
"line": 544,
"column": 32
} | {
"line": 546,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Cycle α\n⊢ s.reverse.Subsingleton ↔ s.Subsingleton",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"congrArg",
"Cycle.Subsingleton",
"_private.Mathlib.Data.List.Cycle.0.Cycle.subsingleton_reverse_iff._simp_1_1",
"instOfNatNat",
"LE.le... | [] | by
simp [length_subsingleton_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Cycle | {
"line": 824,
"column": 65
} | {
"line": 824,
"column": 77
} | {
"line": 824,
"column": 78
} | [
{
"pp": "case cons.cons.succ.cons\nα : Type u_1\nr : α → α → Prop\nc✝ : Cycle α\nd : ℕ\nhd :\n ∀ (a : α) (l : List α) (b : α) (m : List α),\n (a :: l).rotate d = b :: m → (IsChain r (a :: l ++ [a]) ↔ IsChain r (b :: m ++ [b]))\na b : α\nm : List α\nc : α\ns : List α\nhn : ((c :: s ++ [a]).rotate 0).rotate d... | [
"case cons.cons.succ.cons\nα : Type u_1\nr : α → α → Prop\nc✝ : Cycle α\nd : ℕ\nhd :\n ∀ (a : α) (l : List α) (b : α) (m : List α),\n (a :: l).rotate d = b :: m → (IsChain r (a :: l ++ [a]) ↔ IsChain r (b :: m ++ [b]))\na b : α\nm : List α\nc : α\ns : List α\nhn : (c :: s ++ [a]).rotate d = b :: m\n⊢ IsChain r ... | rotate_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Cycle | {
"line": 837,
"column": 34
} | {
"line": 837,
"column": 51
} | {
"line": 837,
"column": 51
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\na : α\n⊢ IsChain r [a, a] ↔ r a a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"List.isChain_pair",
"Eq.mpr",
"congrArg",
"id",
"List.IsChain",
"List.cons",
"Iff",
"propext",
"Eq",
"List... | [
"α : Type u_1\nr : α → α → Prop\na : α\n⊢ r a a ↔ r a a"
] | List.isChain_pair | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Quotient | {
"line": 359,
"column": 13
} | {
"line": 359,
"column": 15
} | {
"line": 359,
"column": 16
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : Group α\ninst✝² : MulAction α β\nx : β\ninst✝¹ : IsPretransitive α β\ninst✝ : IsCancelSMul α β\nH : Subgroup α\nq : orbitRel.Quotient (↥H) β\ny₁ : β\n⊢ ∀ (b : β), (orbitRel (↥H) β) y₁ b → ↑⋯.choose = ↑⋯.choose",
"ppTerm": "?m.45",
"assigned": true,
... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : Group α\ninst✝² : MulAction α β\nx : β\ninst✝¹ : IsPretransitive α β\ninst✝ : IsCancelSMul α β\nH : Subgroup α\nq : orbitRel.Quotient (↥H) β\ny₁ y₂ : β\n⊢ (orbitRel (↥H) β) y₁ y₂ → ↑⋯.choose = ↑⋯.choose"
] | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Nat.PrimeFin | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 9
} | {
"line": 126,
"column": 2
} | [
{
"pp": "k n : ℕ\nhk : k ≠ 0\n⊢ (n ^ k).primeFactors = n.primeFactors",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finset",
"Nat.instMonoid",
"Ne",
"instOfNatNat",
"Monoid.toPow",
"Nat.casesAuxOn",
"instHAdd",
"HPow.hPow",
"HAdd.hAdd... | [
"case zero\nn : ℕ\nhk : 0 ≠ 0\n⊢ (n ^ 0).primeFactors = n.primeFactors",
"case succ\nn n✝ : ℕ\nhk : n✝ + 1 ≠ 0\n⊢ (n ^ (n✝ + 1)).primeFactors = n.primeFactors"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.GroupTheory.GroupAction.Quotient | {
"line": 439,
"column": 4
} | {
"line": 441,
"column": 90
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case a\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (MulAction.toPermHom G (G ⧸ H)).ker ≤ H.normalCore",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"MonoidHom.instFunLike",
"InvOneClass.toO... | [] | refine (Subgroup.normal_le_normalCore.mpr fun g hg => ?_)
rw [← H.inv_mem_iff, ← mul_one g⁻¹, ← QuotientGroup.eq, ← mul_one g]
exact (MulAction.Quotient.smul_mk H g 1).symm.trans (Equiv.Perm.ext_iff.mp hg (1 : G)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Quotient | {
"line": 439,
"column": 4
} | {
"line": 441,
"column": 90
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case a\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (MulAction.toPermHom G (G ⧸ H)).ker ≤ H.normalCore",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"MonoidHom.instFunLike",
"InvOneClass.toO... | [] | refine (Subgroup.normal_le_normalCore.mpr fun g hg => ?_)
rw [← H.inv_mem_iff, ← mul_one g⁻¹, ← QuotientGroup.eq, ← mul_one g]
exact (MulAction.Quotient.smul_mk H g 1).symm.trans (Equiv.Perm.ext_iff.mp hg (1 : G)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factors | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 22
} | {
"line": 216,
"column": 2
} | [
{
"pp": "n k : ℕ\nh : k ≠ 0\n⊢ n.primeFactorsList <+ (n * k).primeFactorsList",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"instMulNat",
"Nat.casesAuxOn",
"Nat",
"List.Sublist",
"Nat.primeFactorsList",
"instHMul"
],
"usedFVar... | [
"case zero\nk : ℕ\nh : k ≠ 0\n⊢ primeFactorsList 0 <+ (0 * k).primeFactorsList",
"case succ\nk : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+ ((hn + 1) * k).primeFactorsList"
] | rcases n with - | hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Nat.Factors | {
"line": 253,
"column": 4
} | {
"line": 263,
"column": 29
} | {
"line": 265,
"column": 0
} | [
{
"pp": "case succ\na : ℕ\nha : Prime a\nn : ℕ\nih : ∀ {b : ℕ}, b ≠ 0 → (replicate n a <+~ b.primeFactorsList ↔ a ^ n ∣ b)\nb : ℕ\nhb : b ≠ 0\n⊢ replicate (n + 1) a <+~ b.primeFactorsList ↔ a ^ (n + 1) ∣ b",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Nat.pow_succ'",
"instPo... | [] | constructor
· rw [List.subperm_iff]
rintro ⟨u, hu1, hu2⟩
rw [← Nat.prod_primeFactorsList hb, ← hu1.prod_eq, ← prod_replicate]
exact hu2.prod_dvd_prod
· rintro ⟨c, rfl⟩
rw [Ne, pow_succ', mul_assoc, mul_eq_zero, _root_.not_or] at hb
rw [pow_succ', mul_assoc, replicate_succ,
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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