module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 508,
"column": 8
} | {
"line": 508,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\na b : ValueGroup₀ v\nhab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose\nx : K := ⋯.choose\nhx_def : x = ⋯.choose\nhx : (restrict₀ v) ⋯.choose = a := Exists.choos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nalg : Algebra.IsAlgebraic (↥(adjoin R (Subtype.val '' range (Set.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 597,
"column": 19
} | {
"line": 597,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nγ : Γ₀\na✝ : K\nha : v ↑a✝ = γ\n⊢ v ?m.261 = γ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 447,
"column": 2
} | {
"line": 447,
"column": 50
} | [
{
"pp": "ι : Type u\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nthis : lift.{u, max u v} (trdeg S (MvPolynomial ι S)) = lift.{max u v, u} #ι\n⊢ trdeg S (MvPolynomial ι S) = lift.{v, u} #ι",
"usedConstants": [
"Nat.instMulZeroClass",
"Cardinal",
"Cardinal.lift_lift",
"AddMon... | rwa [lift_id', ← lift_lift.{u}, lift_id] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ trdeg R R[X] = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Defs | {
"line": 102,
"column": 7
} | {
"line": 102,
"column": 18
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\n⊢ (trace R R) 1 = LinearMap.id 1",
"usedConstants": [
"LinearMap.id",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalCommRing",
"CommS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 40
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ¬IsIntegral K x\n⊢ (Algebra.trace K ↥K⟮x⟯) (gen K x) = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instSMulOfMul",
... | trace_eq_zero_of_not_exists_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 59
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁵ : CommRing F\ninst✝⁴ : Ring E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Ring K\ninst✝¹ : Algebra F K\ne : K ≃ₐ[F] E\ninst✝ : IsPurelyInseparable F K\nx : E\nh : (minpoly F (e.symm x)).Separable\n⊢ x ∈ (algebraMap F E).range",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 24
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nh : separableClosure F E = ⊥\nx : E\nhs : IsSeparable F x\n⊢ x ∈ (algebraMap F E).range",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 29
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\n⊢ IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range",
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Algebra.algebraMa... | isPurelyInseparable_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\nh : ∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)\nx : E\ng : F[X]\nh1 : g.Separable\nn : ℕ\nh2 : (expand F (q ^ n)) g = minpoly F x\n⊢ (aeval (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : IsIntegral K x\n⊢ (Algebra.trace K ↥K⟮x⟯) (AdjoinSimple.gen K x) = -(minpoly K x).nextCoeff",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 23
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nq : ℕ\nh2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range\nh1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).ra... | refine ⟨n + m, z, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 15
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nq : ℕ\ninst✝² : ExpChar F q\ninst✝¹ : IsPurelyInseparable F E\ninst✝ : FiniteDimensional F E\nthis :\n ∀ (F E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (q : ℕ) [ExpChar F q]\n [IsPurelyInsep... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 353,
"column": 8
} | {
"line": 353,
"column": 31
} | [
{
"pp": "case pos\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : Algebra.IsAlgebraic F E\n⊢ IsPurelyInseparable F E",
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Algebra.algebraMap",
"Ring.toNonAssocRing",
... | isPurelyInseparable_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 29
} | [
{
"pp": "case pos\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : Algebra.IsAlgebraic F E\nx : E\nhsep : IsSeparable F x\nthis : F⟮x⟯ = ⊥ ∧ finSepDegree (↥F⟮x⟯) E = 1\n⊢ x ∈ (algebraMap F E).range",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 195,
"column": 45
} | {
"line": 195,
"column": 79
} | [
{
"pp": "case neg\nA : Type u_7\nB : Type u_8\nC : Type u_9\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Algebra B C\ne : A ≃+* B\nhe : (algebraMap B C).comp ↑e = algebraMap A C\nx : C\nh : ¬∃ s, Nonempty (Basis (↥s) B C)\n⊢ e ((trace A C) x) = 0 x",
"usedCon... | trace_eq_zero_of_not_exists_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 309,
"column": 41
} | {
"line": 309,
"column": 52
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nhg₂ : (expand K (p ^ 0)) g = minpoly K x\n⊢ g = minpoly K x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 318,
"column": 45
} | {
"line": 318,
"column": 63
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis✝ : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpoly K x\nh : p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.IntegralClosure | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 72
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : Algebra A K\ninst✝⁶ : IsFractionRing A K\nL : Type u_3\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A L\ninst✝² : IsScalarTower A K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : IsDomain A\nthis : DecidableEq L := Cla... | exact (algebraMap K L).injective.comp (IsFractionRing.injective A K) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SeparableDegree | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 69
} | [
{
"pp": "case isAlgebraic\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\nx✝ : Algebra E K := (↑i).toAlgebra\nx : K\nh : IsAlgebraic E (↑i (↑i.symm x))\n⊢ IsAlgebraic E x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 41
} | [
{
"pp": "case refine_2\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\nι : Type w\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 60
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 29
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 13
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 34
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsAlgebraic E K\n⊢ finSepDegree F E * finSepDegree E K = finSepDegree F K",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 354,
"column": 38
} | {
"line": 354,
"column": 54
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf : F[X]\nhf : f ≠ 0\n⊢ f.natSepDegree = f.natDegree ↔ Fintype.card ↥(f.aroots f.SplittingField).toFinset = f.natDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"Finset",
"Polynomial.SplittingField",
"Classical.... | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 151,
"column": 14
} | {
"line": 151,
"column": 56
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\nx : ↥(separableClosure F E)\n⊢ IsSeparable F x",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 156,
"column": 45
} | {
"line": 156,
"column": 87
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nL : IntermediateField F E\nhs : ∀ (x : ↥L), IsSeparable F x\nx : E\nh : x ∈ L\n⊢ x ∈ separableClosure F E",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 200,
"column": 15
} | {
"line": 200,
"column": 47
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsSepClosed E\nh : separableClosure F E = ⊥\np : F[X]\nx✝ : p.Monic\nhirr : Irreducible p\nhsep : p.Separable\nx : F\nhx : (aeval ((Algebra.ofId F E).toRingHom x)) p = 0\n⊢ eval x p = 0",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 436,
"column": 6
} | {
"line": 437,
"column": 29
} | [
{
"pp": "case pos.inr\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : f = 0 ∨ g = 0\nthis :\n ∀ (f g : F[X]),\n f = 0 ∨ g = 0 → f = 0 → ((f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprime f g)\nhf : ¬f = 0\n⊢ (f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 40
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\n⊢ finInsepDegree F E = finInsepDegree F K",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 455,
"column": 23
} | {
"line": 455,
"column": 63
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicCl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 456,
"column": 17
} | {
"line": 456,
"column": 57
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicCl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 22
} | [
{
"pp": "case neg.refine_2.inr\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nx : AlgebraicClosure F\nhf : f ≠ 0 ∧ (aeval x) f = 0\nhg : g ≠ 0 ∧ (aeval x) g = 0\nu v : F[X]\nhfg : u * f + v * g = 1\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 40
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\n⊢ finInsepDegree F ↥⊥ = finInsepDegree F E",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 483,
"column": 2
} | {
"line": 484,
"column": 27
} | [
{
"pp": "case prime\nF : Type u\ninst✝ : Field F\nf : F[X]\nq n : ℕ\nhprime : Nat.Prime q\nhchar✝ : CharP F q\nthis : Fact (Nat.Prime q)\n⊢ ((expand F (q ^ n)) f).natSepDegree = f.natSepDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Polynomial.roots",
"Algebra.algebraMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 606,
"column": 2
} | {
"line": 607,
"column": 24
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\np : F[X]\nhM : p.Monic\nhI : Irreducible p\nhf✝ : p ∣ f\nhD : p.natSepDegree = 1\nn : ℕ\ny : F\nH✝ : n = 0 ∨ y ∉ (frobenius F q).range\nhp : p = X ^ q ^ n - C y\nhF : FiniteMultiplicity p f\nhne : multiplicity p f ≠ 0\nc ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 592,
"column": 81
} | {
"line": 607,
"column": 27
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\nh : f.natSepDegree = 1\n⊢ ∃ m n y, m ≠ 0 ∧ (n = 0 ∨ y ∉ (frobenius F q).range) ∧ f = (X ^ q ^ n - C y) ^ m",
"usedConstants": [
"IsCoprime.pow_left",
"Iff.mpr",
"Eq.mpr",
"Polynomial.C",
... | by
obtain ⟨p, hM, hI, hf⟩ := exists_monic_irreducible_factor _ <| not_isUnit_of_natDegree_pos _
<| Nat.pos_of_ne_zero <| (natSepDegree_ne_zero_iff _).1 (h.symm ▸ Nat.one_ne_zero)
have hD := (h ▸ natSepDegree_le_of_dvd p f hf hm.ne_zero).antisymm <|
Nat.pos_of_ne_zero <| (natSepDegree_ne_zero_iff _).2 hI.nat... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparableDegree | {
"line": 789,
"column": 2
} | {
"line": 789,
"column": 44
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nL : IntermediateField F E\ninst✝ : Algebra.IsSeparable F ↥L\nx : E\nh : x ∈ L\n⊢ IsSeparable F x",
"usedConstants": [
"Field.toDivisionRing",
"id",
"DivisionRing.toRing",
"Field.toCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 816,
"column": 4
} | {
"line": 816,
"column": 54
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\nE' : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Finset.DenselyOrdered | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 44
} | [
{
"pp": "case dense\nX : Type u_1\ninst✝¹ : LinearOrder X\ninst✝ : LocallyFiniteOrder X\ns : Set (WithBot X)\nH : DenselyOrdered ↑s\nx : X\nhx : ↑x ∈ s\ny : X\nhy : ↑y ∈ s\nhxy : x < y\nthis : ⟨↑x, hx⟩ < ⟨↑y, hy⟩\nz : { x // x ∈ s }\nhz : ⟨↑x, hx⟩ < z\nhz' : z < ⟨↑y, hy⟩\n⊢ ∃ a, (↑a ∈ s ∧ x < a) ∧ a < y",
"... | simp only [← Subtype.coe_lt_coe] at hz hz' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 128,
"column": 66
} | {
"line": 131,
"column": 39
} | [
{
"pp": "Γ : Type u_1\ninst✝² : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝¹ : Ring A\nv : Valuation A Γ\ninst✝ : v.IsRankOneDiscrete\n⊢ zpowers (generator' v) = ⊤",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedComm... | by
rw [← map_subtype_inj, MonoidHom.map_zpowers,
subtype_apply, ← MonoidHom.range_eq_map, Subgroup.subtype_range]
apply generator_zpowers_eq_valueGroup | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 12
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝² : Field K\nv : Valuation K Γ\ninst✝¹ : IsCyclic ↥(valueGroup v)\ninst✝ : Nontrivial ↥(valueGroup v)\nI : Ideal ↥v.valuationSubring\n⊢ ∀ (P : Ideal ↥v.valuationSubring), P.IsPrime → Submodule.IsPrincipal P",
"usedConstants... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 435,
"column": 15
} | {
"line": 435,
"column": 65
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\n⊢ IsLocalRing.maximalIdeal A ≠ ⊥",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Com... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 22
} | [
{
"pp": "case h.right\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : Valuation K (WithZero (Multiplicative ℤ)) := ⋯\nπ : K := ⋯\nhπ : v π = ↑(ofAdd (-1))\n⊢ Units.mk0 (v π) ⋯ ≠ 1",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 43
} | [
{
"pp": "case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : v.IsEquiv w\nγ : (ValueGroup₀ Valued.v)ˣ\nr s : R\nhr₀ : 0 <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 54
} | [
{
"pp": "case h.refine_2\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : A\nh : x ∈ IsUnit.submonoid A\n⊢ (valuation K (maximalIdeal A)) ((algebraMap A K) x) = 1",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 624,
"column": 46
} | {
"line": 624,
"column": 70
} | [
{
"pp": "K : Type u_7\ninst✝¹ : Field K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nγ : (ValueGroup₀ v)ˣ\nr : K\nhr : (restrict₀ v) r = ↑γ\n⊢ embedding ((restrict₀ v) 0) < v r",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrdered... | ← embedding_restrict₀ r, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 644,
"column": 4
} | {
"line": 644,
"column": 47
} | [
{
"pp": "case ih\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀'\nK : Type u_7\ninst✝ : Field K\nv : Valuation K Γ₀\nw : Valuation K Γ₀'\nh : v.IsEquiv w\na : WithVal v\n⊢ Valued.v ↑a ≤ 1 ↔ Valued.v ((mapEquiv h.uniformEquiv) ↑a) ≤ 1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 174,
"column": 14
} | {
"line": 175,
"column": 11
} | [
{
"pp": "case h.h₁\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := ⋯\nl : List (ValueGroup R K) := ⋯\nlmax : ValueGroup R K := ⋯\nhlmax_mem : lmax ∈ l\nhlma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 174,
"column": 14
} | {
"line": 175,
"column": 11
} | [
{
"pp": "case h.h₂\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := ⋯\nl : List (ValueGroup R K) := ⋯\nlmax : ValueGroup R K := ⋯\nhlmax_mem : lmax ∈ l\nhlma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 174,
"column": 14
} | {
"line": 175,
"column": 11
} | [
{
"pp": "case h.h₃\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := ⋯\nl : List (ValueGroup R K) := ⋯\nlmax : ValueGroup R K := ⋯\nhlmax_mem : lmax ∈ l\nhlma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 174,
"column": 14
} | {
"line": 175,
"column": 11
} | [
{
"pp": "case h.h₄\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := ⋯\nl : List (ValueGroup R K) := ⋯\nlmax : ValueGroup R K := ⋯\nhlmax_mem : lmax ∈ l\nhlma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 174,
"column": 14
} | {
"line": 175,
"column": 11
} | [
{
"pp": "case h.h₆\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := ⋯\nl : List (ValueGroup R K) := ⋯\nlmax : ValueGroup R K := ⋯\nhlmax_mem : lmax ∈ l\nhlma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 43
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : x = 0\n⊢ (if x * y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * y})).factors))) =\n (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count... | · rw [hx, zero_mul, if_pos rfl, zero_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 28
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsDomain R\ninst✝¹ : ValuationRing R\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nl₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]\nl : List (ValueGroup R K) := List.map (fun a ↦ (valuation R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 15
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero G\ninst✝ : MulArchimedean G\nb a : Gˣ\nha : ↑a < 1\nha' : ↑a ≠ 0\n⊢ ∃ n, ↑a ^ n < ↑b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 37
} | [
{
"pp": "e : ℤ ≃+ ℤ\nhe : ¬IsOfFinAddOrder (e 1)\n⊢ AddSubgroup.zmultiples (e 1) = AddSubgroup.zmultiples 1",
"usedConstants": [
"Int.zmultiples_one",
"Eq.mpr",
"congrArg",
"id",
"Int",
"AddSubgroup",
"instOfNat",
"AddSubgroup.zmultiples",
"Int.instAdd",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 54
} | [
{
"pp": "case refine_1\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : x = 1\na : G\nha : a ∈ closure {x}\n⊢ (fun x_1 ↦ ⟨1, ⋯⟩) ((fun x_1 ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 61
} | [
{
"pp": "case refine_2\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : x = 1\na : G'\nha : a ∈ closure {y}\n⊢ (fun x_1 ↦ ⟨1, ⋯⟩) ((fun x_1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 154,
"column": 10
} | {
"line": 154,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : ¬x + y = 0\nnmin : ℕ :=\n min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)\n ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y}... | · exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 241,
"column": 25
} | {
"line": 241,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nW : WeierstrassCurve K\ninst✝ : IsMinimal R W\n⊢ IsIntegral R W",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nhW : IsMinimal R W\nh :\n ¬(valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (int... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 169,
"column": 34
} | {
"line": 169,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\na b : G\nh : b = |a|ₘ ∧ 1 < b\nthis : ∀ {a : G}, b = |a|ₘ ∧ 1 < b → 1 ≤ a → IsLeast {y | y ∈ closure {a} ∧ 1 < y} b\nha : ¬1 ≤ a\n⊢ 1 ≤ a⁻¹",
"usedConstants": [
"Eq.mpr",
"I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nhv : Irreducible (Associates.mk v.asIdeal)\nhlt : v.asIdeal ^ 2 < v.asIdeal\n⊢ ∃ π, v.intValuation π = exp (-1)",
"usedConstants": [
"Semiring.toModule",
"SetLike.exists_of_lt",
"CommSemiring.t... | obtain ⟨π, mem, notMem⟩ := SetLike.exists_of_lt hlt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 190,
"column": 67
} | {
"line": 190,
"column": 78
} | [
{
"pp": "G✝ : Type u_1\ninst✝⁷ : CommGroup G✝\ninst✝⁶ : LinearOrder G✝\ninst✝⁵ : IsOrderedMonoid G✝\ninst✝⁴ : MulArchimedean G✝\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nx : G\nh : IsLeast {y | 0 < y} x\n⊢ IsLeast {y | y ∈ ⊤ ∧ 0 < y} x"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 38
} | [
{
"pp": "case neg.refine_2\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nx y : G\nhxy : x < y\nH : ¬IsLeast {y | 0 < y} (y - x)\nz : G\nhz : 0 < z ∧ z < y - x\n⊢ x + z < y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 26
} | [
{
"pp": "case inl\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nthis : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G\nh : G ≃+o ℤ\n⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G",
"usedConstants": [
"Eq.mpr",
"False",
"OrderAddMonoidIs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 254,
"column": 34
} | {
"line": 254,
"column": 54
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\nφ : MvPowerSeries σ R\na : R\n⊢ a * (coeff (m + n - m)) φ = a * (coeff n) φ",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"Finsupp.partialorder",
"Nat.instOrderedSub... | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 273,
"column": 18
} | {
"line": 273,
"column": 29
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (1 * φ) = (coeff n) φ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 276,
"column": 18
} | {
"line": 276,
"column": 29
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (φ * 1) = (coeff n) φ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 308,
"column": 6
} | {
"line": 308,
"column": 44
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (φ * ψ) = (coeff n) (ψ * φ)",
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 360,
"column": 27
} | {
"line": 360,
"column": 72
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : IsEmpty σ\np : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (C (p 0)) = (coeff n) p",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Mv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 412,
"column": 44
} | {
"line": 412,
"column": 55
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\na : R\n⊢ (coeff n) (φ * C a) = (coeff n) φ * a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 416,
"column": 44
} | {
"line": 416,
"column": 55
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\na : R\n⊢ (coeff n) (C a * φ) = a * (coeff n) φ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 13
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn d : R\nhd : d ∈ nonZeroDivisors R\nhx : IsLocalization.mk' K n ⟨d, hd⟩ * (algebraMap R K) ↑(n, ⟨d, hd⟩).2 = (algebraMap R K) (n, ⟨d, hd⟩).1\nhn0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 419,
"column": 40
} | {
"line": 419,
"column": 51
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\ns : σ\nh : single s 1 ≤ 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 467,
"column": 7
} | {
"line": 467,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDedekindDomain R\nK : Type u_2\nS : Type u_3\ninst✝³ : Field K\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\n⊢ x ∈ (subalgebra.ofField K v.asIdeal.primeCompl ⋯).toSubring ∨\n x⁻¹ ∈ (subalgebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\na b : R\n⊢ (monomial m) a * (monomial n) b = (monomial (m + n)) (a * b)",
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Nat.instMulZeroClass",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nH : X = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff n) (φ * X) = (coeff n) (ψ * X)\nn : ℕ\n⊢ (coeff n) φ = (coeff n) ψ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff n) (X * φ) = (coeff n) (X * ψ)\nn : ℕ\n⊢ (coeff n) φ = (coeff n) ψ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 537,
"column": 2
} | {
"line": 539,
"column": 27
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\na b : R\nhv : v.intValuation b ≤ v.intValuation a\nγ : Multiplicative ℤ\nha : a = 0\n⊢ ∃ y, v.intValuation (b - y * a) < ↑γ",
"usedConstants": [
"Int.instAddCommMonoid",
"Multiplicative.lin... | · subst ha
rw [map_zero, le_zero_iff] at hv
exact ⟨0, by simp [hv]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 613,
"column": 8
} | {
"line": 613,
"column": 19
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\n⊢ (coeff m) φ = (coeff (single s n, m - single s n).2) fun m ↦ (coeff (m + single s n)) φ",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 555,
"column": 4
} | {
"line": 556,
"column": 90
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\na b : R\nhv : v.intValuation b ≤ v.intValuation a\nγ : Multiplicative ℤ\nha : ¬a = 0\nhvaz : v.intValuation a ≠ 0\nhγz : ↑γ ≠ 0\nn : ℕ\nhna : exp (-↑n) < v.intValuation a\nhnγ : exp (-↑n) < ↑γ\nhvn : emult... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff n) (φ * X ^ k) = (coeff n) (ψ * X ^ k)\nn : ℕ\n⊢ (coeff n) φ = (coeff n) ψ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 618,
"column": 10
} | {
"line": 623,
"column": 42
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j).1 + (i, j).2 = m\nhne : (i, j) ≠ (single s n, m - single s n)\nhi : (i, j).1 = single... | exfalso
apply hne
rw [← hij, ← hi, Prod.mk_inj]
refine ⟨rfl, ?_⟩
ext t
simp only [add_tsub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 618,
"column": 10
} | {
"line": 623,
"column": 42
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j).1 + (i, j).2 = m\nhne : (i, j) ≠ (single s n, m - single s n)\nhi : (i, j).1 = single... | exfalso
apply hne
rw [← hij, ← hi, Prod.mk_inj]
refine ⟨rfl, ?_⟩
ext t
simp only [add_tsub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff n) (X ^ k * φ) = (coeff n) (X ^ k * ψ)\nn : ℕ\n⊢ (coeff n) φ = (coeff n) ψ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 73
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : (WithZero (Multiplicative ℤ))ˣ\nx✝ : ∃ a, ¬a = 0 ∧ ∃ x, Valued.v a * ↑n = Valued.v x\na : adicCompletion K v\nha0 : ¬a =... | rwa [← ne_eq, ← (valuation K v).ne_zero_iff, hb, Valuation.ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 45
} | [
{
"pp": "case h.e'_2.a.mpr.a\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nφ : R⟦X⟧\nh : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0\nm : ℕ\nhm : m < n\n⊢ (single () m) () < n",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"Nat.instMulZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 13
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\n⊢ f ≠ 0 ↔ ∃ n d, (coeff d) f ≠ 0 ∧ (weight w) d = n",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"MvPowerS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 644,
"column": 10
} | {
"line": 644,
"column": 21
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : n ≤ m s\n⊢ (m - single s n + single s n) s = m s",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonically... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ Valued.v ((WithVal.equiv (valuation K v)).symm ((algebraMap R K) r)) ≤ 1",
"usedConstants": [
"Int.instAddCommGro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 52
} | [
{
"pp": "case coe.h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\na✝ : ℕ\nh : ∀ (d : σ →₀ ℕ), ↑((weight w) d) < ↑a✝ → (coeff d) f = 0\n⊢ ∀ (d : σ →₀ ℕ), (weight w) d < a✝ → (coeff d) f = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 212,
"column": 18
} | {
"line": 212,
"column": 54
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\nn : ℕ\nh : weightedOrder w f = ↑n\nd : σ →₀ ℕ\nhd : (coeff d) f ≠ 0 ∧ ↑((weight w) d) = weightedOrder w f\n⊢ (coeff d) f ≠ 0 ∧ (weight w) d = n",
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 694,
"column": 8
} | {
"line": 694,
"column": 67
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommSemiring R\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq σ\nf : ι → MvPowerSeries σ R\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)\nu v u' v' : σ →₀ ℕ\nhuv ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 50
} | [
{
"pp": "R : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nf : ι → ℕ\ng : ι → R\ns : Finset ι\n⊢ ∏ i ∈ s, (monomial (f i)) (g i) = (monomial (∑ i ∈ s, f i)) (∏ i ∈ s, g i)",
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Nat.instMulZeroClass",
"Semiring.toModule",
"MvPowerSeries.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 24
} | [
{
"pp": "R : Type u_2\ninst✝ : CommSemiring R\nm : ℕ\na : R\nn : ℕ\n⊢ (monomial m) a ^ n = (monomial (n * m)) (a ^ n)",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"CommSemiring.toSemiring",
"LinearMap.instFunLike",
"id",
"instMulNat",
"MvPowerSeries.instMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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