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.RingTheory.Localization.AtPrime.Extension | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\np : Ideal R\ninst✝¹⁴ : p.IsPrime\nRₚ : Type u_3\ninst✝¹³ : CommRing Rₚ\ninst✝¹² : Algebra R Rₚ\ninst✝¹¹ : IsLocalization.AtPrime Rₚ p\ninst✝¹⁰ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 20
} | [
{
"pp": "n : Type u_3\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ni : n\nS : Type (max u_4 u_3) := MvPolynomial { j // j ≠ i } R\ne : MvPolynomial n R ≃ₐ[R] S[X] :=\n (renameEquiv R (Equiv.optionSubtypeNe i).symm).trans (optionEquivLeft R { b // b ≠ i })\nhe : (↑e.symm).comp Polynomial.CAlgHom = re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 98,
"column": 33
} | {
"line": 98,
"column": 44
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 49
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 119,
"column": 38
} | {
"line": 119,
"column": 63
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 17
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 61
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na b : MvPolynomial n R\nhab : p = a * b\nhle : a.totalDegree ≤ b.totalDegree\nha₀ : a ≠ 0\nhb₀ : b ≠ 0\n⊢ a.totalDegree + b.tot... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 32
} | [
{
"pp": "case hp'.a\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc_nonempty : c.support.Nonempty\nhc_gcd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nr : R\nhr : ∀ (i : n →₀ ℕ), r ∣ coeff i (sumSMulX c)\ni : n\n⊢ r ∣ c i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 24
} | [
{
"pp": "case isPrimitive.a\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc : c.support.Nontrivial\nh_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nι : n ↪ n ⊕ n →₀ ℕ := { toFun := fun i ↦ Finsupp.single (Sum.inl i) 1 + Finsupp.single (Sum.inr i) 1, inj' := ⋯ }\naux : su... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 28
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : d ∈ (fun x ↦ p • x) '' Function.support φ\n⊢ d ∈ Function.support ((expand p hp) φ)",
"usedConstants": []
}
] | obtain ⟨n, hn₁, hn₂⟩ := hd | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 20
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nhφ : φ = 0\n⊢ ((expand p hp) φ).order = p • φ.order",
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"MvPow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 245,
"column": 4
} | {
"line": 250,
"column": 9
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx : R⸨X⸩\nn : ℤ\nh : ¬order x ≤ n\n⊢ ((ofPowerSeries ℤ R) x.powerSeriesPart).coeff (n - order x) = x.coeff n",
"usedConstants": [
"Eq.mpr",
"HahnSeries.order",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RelEmbedding.mk",
"In... | rw [ofPowerSeries_apply, embDomain_notin_range]
· contrapose! h
exact order_le_of_coeff_ne_zero h.symm
· contrapose h
simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LaurentSeries | {
"line": 245,
"column": 4
} | {
"line": 250,
"column": 9
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx : R⸨X⸩\nn : ℤ\nh : ¬order x ≤ n\n⊢ ((ofPowerSeries ℤ R) x.powerSeriesPart).coeff (n - order x) = x.coeff n",
"usedConstants": [
"Eq.mpr",
"HahnSeries.order",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RelEmbedding.mk",
"In... | rw [ofPowerSeries_apply, embDomain_notin_range]
· contrapose! h
exact order_le_of_coeff_ne_zero h.symm
· contrapose h
simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 200,
"column": 4
} | {
"line": 213,
"column": 20
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ∀ (i : σ), p ∣ d i\n⊢ MvPolynomial.coeff d ((trunc' R (p • n)) ((expand p hp) φ)) =\n MvPolynomial.coeff d ((MvPolynomial.expand p) ((trunc' R n) φ)... | obtain ⟨m, hm⟩ : ∃ m, p • m = d := ⟨d.mapRange (fun a ↦ a / p) (by simp),
by ext i; simp [(Nat.mul_div_cancel' (h i))]⟩
by_cases h_le : m ≤ n
· rw [← hm, coeff_trunc', if_pos (nsmul_le_nsmul_right h_le p), coeff_expand_smul,
MvPolynomial.coeff_expand_smul _ hp, coeff_trunc', if_pos h_le]
· hav... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 200,
"column": 4
} | {
"line": 213,
"column": 20
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ∀ (i : σ), p ∣ d i\n⊢ MvPolynomial.coeff d ((trunc' R (p • n)) ((expand p hp) φ)) =\n MvPolynomial.coeff d ((MvPolynomial.expand p) ((trunc' R n) φ)... | obtain ⟨m, hm⟩ : ∃ m, p • m = d := ⟨d.mapRange (fun a ↦ a / p) (by simp),
by ext i; simp [(Nat.mul_div_cancel' (h i))]⟩
by_cases h_le : m ≤ n
· rw [← hm, coeff_trunc', if_pos (nsmul_le_nsmul_right h_le p), coeff_expand_smul,
MvPolynomial.coeff_expand_smul _ hp, coeff_trunc', if_pos h_le]
· hav... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 15
} | [
{
"pp": "σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ := Fintype.card σ\nthis : (-1) ^ (k + 1) * ∑ a ∈ antidiagonal k, (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0\n⊢ ∑ a ∈ antidiagonal (Fintype.card σ), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\np : MvPowerSeries σ R\nx : σ →₀ ℕ\n⊢ ∀ b ∈ ⋯.toFinset, b ≠ x → (coeff b) p = 0",
"usedConstants": [
"Finsupp.mapDomain_tendstoCofinite",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 13
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\nf : σ → τ\ninst✝¹ : TendstoCofinite f\ninst✝ : CommSemiring R\ni : σ\n⊢ (rename f) (X i) = X (f i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 99
} | [
{
"pp": "k : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\n⊢ IsGeometricallyReduced k A ↔ IsReduced (AlgebraicClosure k ⊗[k] A)",
"usedConstants": [
"AlgEquiv.instEquivLike",
"isReduced_of_injective",
"Algebra.to_smulCommClass",
"IsDomain.to_noZeroDi... | let e (p : Ideal k) [p.IsPrime] : AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=
have := p.algEquivResidueFieldOfField.isAlgebraic
IsAlgClosure.equiv k _ _
refine ⟨fun ⟨h⟩ ↦ ?_, fun h ↦ ⟨fun p hp ↦ ?_⟩⟩
· exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e ⊥) AlgEquiv.refl).injective... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 99
} | [
{
"pp": "k : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\n⊢ IsGeometricallyReduced k A ↔ IsReduced (AlgebraicClosure k ⊗[k] A)",
"usedConstants": [
"AlgEquiv.instEquivLike",
"isReduced_of_injective",
"Algebra.to_smulCommClass",
"IsDomain.to_noZeroDi... | let e (p : Ideal k) [p.IsPrime] : AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=
have := p.algEquivResidueFieldOfField.isAlgebraic
IsAlgClosure.equiv k _ _
refine ⟨fun ⟨h⟩ ↦ ?_, fun h ↦ ⟨fun p hp ↦ ?_⟩⟩
· exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e ⊥) AlgEquiv.refl).injective... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 28
} | [
{
"pp": "case H.h\nσ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nx✝ : MvPowerSeries σ R\ny : σ →₀ ℕ\n⊢ (coeff y) ((rename id) x✝) = (coeff y) ((AlgHom.id R (MvPowerSeries σ R)) x✝)",
"usedConstants": [
"Finsupp.mapDomain_tendstoCofinite",
"Eq.mpr",
"Semiring.toModule",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | [
{
"pp": "case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\na₁✝ a₂✝ : MvPowerSeries σ R\nh : (rename ⇑e) a₁✝ = (rename ⇑e) a₂✝\nx : σ →₀ ℕ\n⊢ (coeff x) a₁✝ = (coeff x) a₂✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 243,
"column": 17
} | {
"line": 243,
"column": 28
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nγ : Type u_3\nR : Type u_4\nS : Type u_5\nf : σ → τ\ng : τ → γ\ninst✝² : TendstoCofinite f\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\ne✝ e : σ ↪ τ\n⊢ killComplFun e 1 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 247,
"column": 18
} | {
"line": 247,
"column": 29
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nγ : Type u_3\nR : Type u_4\nS : Type u_5\nf : σ → τ\ng : τ → γ\ninst✝² : TendstoCofinite f\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\ne✝ e : σ ↪ τ\n⊢ ∀ (r : R), killComplFun e ((algebraMap R (MvPowerSeries τ R)) r) = (algebraMap R (MvPowerSeries σ R)) r",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\nr : R\n⊢ (killCompl e) (C r) = C r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\nt : τ\nh : single t 1 ∉ Set.range (embDomain e)\n⊢ (killCompl e) (X t) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 661,
"column": 4
} | {
"line": 661,
"column": 69
} | [
{
"pp": "case h.refine_2\nK : Type u_2\ninst✝ : Field K\nuK : UniformSpace K\nd : ℤ\nS : Set (K × K)\nhS : S ∈ uniformity K\nγ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯\nx✝ : K⸨X⸩ × K⸨X⸩\nhP : x✝ ∈ {P | Valued.v (P.2 - P.1) < ↑γ}\n⊢ (x✝.1.coeff d, x✝.2.coeff d) ∈ S",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.OfPrime | {
"line": 50,
"column": 4
} | {
"line": 59,
"column": 95
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nhsup : (I ⊔ span {a}).FG\nhcolon : (Submodule.colon I ↑(span {a})).FG\nw✝¹ : ℕ\nf : Fin w✝¹ → R\nhf : span (Set.range f) = I ⊔ span {a}\nw✝ : ℕ\ni : Fin w✝ → R\nhi : Submodule.span R (Set.range i) = Submodule.colon I ↑(span {a})\nr p ... | · rw [Submodule.mem_sup]
obtain ⟨s, H⟩ := mem_span_range_iff_exists_fun.1 (hf ▸ Ideal.mem_sup_left hy)
simp_rw [← Hf] at H
ring_nf at H
rw [sum_add_distrib, ← sum_mul, add_comm] at H
refine ⟨(∑ k, s k * p k), sum_mem _ (fun _ _ ↦ mul_mem_left _ _ mem_span_range_self),
(∑ k, s k * r... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LaurentSeries | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 71
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nℱ : Filter K⸨X⸩\nhℱ : Cauchy ℱ\nentourage : Set (K⸨X⸩ × K⸨X⸩) := {P | Valued.v.restrict (P.2 - P.1) < 1}\nζ : (MonoidWithZeroHom.ValueGroup₀ Valued.v)ˣ := Units.mk0 1 ⋯\nS : Set K⸨X⸩\nhS : S ∈ ℱ\nT : Set K⸨X⸩\nhT : T ∈ ℱ\nH : S ×ˢ T ⊆ entourage\nf : K⸨X⸩\nhf : f ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 37
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nℱ : Filter K⸨X⸩\nhℱ : Cauchy ℱ\nD : ℤ\nφ : ℤ → Set K⸨X⸩ := fun d ↦ {f | coeff hℱ d = f.coeff d}\na✝ : K⸨X⸩\nhf : a✝ ∈ ⋂ n ∈ Set.Iio D, φ n\n⊢ a✝ ∈ {x | ∀ d < D, coeff hℱ d = x.coeff d}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 818,
"column": 4
} | {
"line": 818,
"column": 15
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : 1 < η\n⊢ (PowerSeries.idealX K).intValuation (F - ↑0) < ↑η",
"usedConstants": [
"MvPowerSeries.instAddCommGroup",
"Units.val",
"Eq.mpr",
"Int.instAddCommMonoid",
"Multiplicativ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 80
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0\nd : ℕ\nhd : Multiplicative.toAdd (unzero ⋯) = -↑d\n⊢ ∀ n < d + 1, (PowerSeries.coeff n) (F - ↑((trunc (d + 1)) F)) = 0",
"usedConstants": [
"Eq.mpr",
"PowerSeries.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 857,
"column": 10
} | {
"line": 857,
"column": 27
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nγ : (WithZero (Multiplicative ℤ))ˣ\nF : K⟦X⟧ := f.powerSeriesPart\nhF : F = f.powerSeriesPart\nord_nonpos : HahnSeries.order f < 0\nη : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (HahnSeries.order f)) ⋯\nhη : η = Units.mk0 (exp (HahnSeries.order f)... | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R\nhx : IsUnit x\n⊢ ord R x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 87,
"column": 27
} | {
"line": 87,
"column": 38
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\n⊢ (T1 f (-1)).comp (T1 f 1) = AlgHom.id k (MvPolynomial (Fin (n + 1)) k)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 43
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\nvlt : ∀ (i : Fin (n + 1)), v i < up\nwlt : ∀ (i : Fin (n + 1)), w i < up\nne : v ≠ w\nh : ∑ x, r x * v x = ∑ x, r x * w x\n⊢ ofDigits up (ofFn ⇑v) = ofDigits up (ofFn ⇑w)",
"usedConstants": [
"Fins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 24
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv : Fin (n + 1) →₀ ℕ\na : k\nha : a ≠ 0\nh : ∀ (i : Fin n), (Polynomial.C (MvPolynomial.X i) + Polynomial.X ^ r i.succ) ^ v i.succ ≠ 0\n⊢ 0 +\n ((Polynomial.X ^ v 0).natDegree +\n ∑ i, ((Polynomial.C (MvPolynomial.X i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 153,
"column": 4
} | {
"line": 154,
"column": 11
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x' ∈ f.support,\n ((MvPolynomial.finSuccEquiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 160,
"column": 28
} | {
"line": 160,
"column": 43
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 162,
"column": 9
} | {
"line": 162,
"column": 68
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 41
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 214,
"column": 13
} | {
"line": 216,
"column": 26
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\nI : Ideal (MvPolynomial (Fin (n + 1)) k)\n⊢ Ideal.map ((T f).symm.toRingEquiv.toRingHom.comp ↑(T f)) I = Ideal.map (↑(T f).symm) (Ideal.map (T f) I)",
"usedConstants": [
"Finsupp.instAddZeroClass",... | by
rw [← Ideal.map_map, Ideal.map_coe, Ideal.map_coe]
exact congrArg _ rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Teichmuller | {
"line": 47,
"column": 35
} | {
"line": 47,
"column": 46
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : CharP (R ⧸ I) p\nx : Perfection (R ⧸ I) p\n⊢ ∀ (n : ℕ), x.teichmullerAux n ≡ x.teichmullerAux (n + 1) [SMOD I ^ n • ⊤]",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Teichmuller | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 70
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\n⊢ (Ideal.Quotient.mk I) ((teichmuller p I) x) = (coeff (R ⧸ I) p 0) x",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"... | have := teichmuller_sModEq <| Ideal.Quotient.mk_out <| coeff _ p 0 x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Teichmuller | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 38
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\nthis : (teichmuller p I) x ≡ Quotient.out ((coeff (R ⧸ I) p 0) x) ^ p ^ 0 [SMOD I]\n⊢ (Ideal.Quotient.mk I) ((teichmuller p I) x) = (coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 123,
"column": 48
} | {
"line": 129,
"column": 37
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ constantCoeff (W_ R n) = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRingHomClass",
"wittPolynomial",
"pow_pos",
"Nat.instMulZeroClass",
"Nat.Prime",
... | by
simp only [wittPolynomial, map_sum, constantCoeff_monomial]
rw [sum_eq_zero]
rintro i _
rw [if_neg]
rw [Finsupp.single_eq_zero]
exact ne_of_gt (pow_pos hp.1.pos _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 50
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Invertible ↑p\nk : ℕ\n⊢ X k - ∑ i ∈ range k, C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (k - i) +\n ∑ x ∈ range k, C ↑p ^ x * (bind₁ (xInTermsOfW p R)) (X x) ^ p ^ (k - x) =\n X k",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.i... | simp only [C_pow, bind₁_X_right, sub_add_cancel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Perfection | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\np : ℕ\nf : Perfection M p\nn m : ℕ\n⊢ (coeffMonoidHom M p n) ((⇑(powMulEquiv (Perfection M p) p).symm)^[m] f) = (coeffMonoidHom M p (n + m)) f",
"usedConstants": [
"Function.iterate_succ_apply'",
"Eq.mpr",
"Nat.recAux",
"MulEquiv.instEquiv... | induction m generalizing n with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Perfection | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 39
} | [
{
"pp": "case a\nR : Type u_1\ninst✝¹ : CommSemiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx✝ : Perfection R p\n⊢ (pthRoot R p) x✝ = ↑(frobeniusEquiv (Perfection R p) p).symm x✝",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Perfection",
"f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 234,
"column": 81
} | {
"line": 260,
"column": 29
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\n⊢ C ↑(p ^ n) ∣\n (bind₁ fun b ↦ (rename fun i ↦ (b, i)) (W_ ℤ n)) Φ -\n ∑ i ∈ Finset.range n, C (... | by
rcases n with - | n
· simp
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_wittPolynomial Φ n IH
apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key
conv_lhs at key => simp only [map_bind₁, map_rename, map_expand, map_wittPolynomial]
-- clean up and massage
rw [C_dv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 312,
"column": 4
} | {
"line": 313,
"column": 33
} | [
{
"pp": "case h.a.py₁\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nφ : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n (map (Int.castRingHom ℚ)) ((bind₁ φ) (W_ ℤ n)) =\n (map (Int.castRingHom ℚ)) ((bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℤ n)) Φ)\n⊢ (bind₁ fun k ↦ (map (Int.castRingHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Basic | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 15
} | [
{
"pp": "case refine_1\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nn : ℕ\nh : ((map f) x).coeff n = coeff 0 n\n⊢ f (x.coeff n) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Basic | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 15
} | [
{
"pp": "case refine_2.h\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nh : ∀ (n : ℕ), f (x.coeff n) = 0\nn : ℕ\n⊢ ((map f) x).coeff n = coeff 0 n",
"usedConstants": [
"WittVector.instZero",
"Eq.mpr",
"congrA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 58
} | [
{
"pp": "case h.a\np : ℕ\nidx : Type u_1\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n ((fun fam ↦ (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n ↦\n (bind₁ f) (wittPolynomial p ℤ n)) =\n (fun fam ↦ (bind₁ (⇑(MvPolynomial.map (Int.cast... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 58
} | [
{
"pp": "case h.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial ℕ ℤ\nn : ℕ\nh :\n ((fun fam ↦ (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n ↦\n (bind₁ f) (wittPolynomial p ℤ n)) =\n (fun fam ↦ (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xIn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 34
} | [
{
"pp": "case pos\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : i = 1\n⊢ (↑p).coeff i = 1",
"usedConstants": [
"Eq.mpr",
"WittVector.instNatCast",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 30
} | [
{
"pp": "case neg\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : ¬i = 1\n⊢ (↑p).coeff i = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 47
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\nh : ↑p = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\n⊢ ↑p ≠ 0",
"usedConstants": [
"WittVector.instZero",
"WittVector.instCommRing",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"FractionRing",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 17
} | [
{
"pp": "case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → (x * ↑p ^ n).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 361,
"column": 2
} | {
"line": 366,
"column": 22
} | [
{
"pp": "p : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\ng : R →+* S\nx y : 𝕎 R\n⊢ (WittVector.map g) (f x y) = f ((WittVector.map g) x) ((WittVector.map g) y)",
"usedConstants": [
"Fins... | obtain ⟨φ, hf⟩ := hf
ext n
simp +unfoldPartialApp only [map_coeff, hf, map_aeval, peval, uncurry]
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
ext ⟨i, k⟩
fin_cases i <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 361,
"column": 2
} | {
"line": 366,
"column": 22
} | [
{
"pp": "p : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\ng : R →+* S\nx y : 𝕎 R\n⊢ (WittVector.map g) (f x y) = f ((WittVector.map g) x) ((WittVector.map g) y)",
"usedConstants": [
"Fins... | obtain ⟨φ, hf⟩ := hf
ext n
simp +unfoldPartialApp only [map_coeff, hf, map_aeval, peval, uncurry]
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
ext ⟨i, k⟩
fin_cases i <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Identities | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 17
} | [
{
"pp": "case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → ((⇑verschiebung)^[n] x).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Morita.Matrix | {
"line": 167,
"column": 25
} | {
"line": 167,
"column": 55
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Morita.Matrix | {
"line": 194,
"column": 51
} | {
"line": 194,
"column": 62
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni : ι\nX : ModuleCat R\nthis :\n ↑(toModuleCatFromModuleCatLinearEquiv R ((toMatrixModCat R ι).obj X) i).symm ∘ₗ\n LinearMap.mapMatrixModule ι (Hom.hom ((unitIso R i).inv.app X)) =\n LinearMap.id\n⊢ Hom.hom\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Truncated | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 30
} | [
{
"pp": "case h\np n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx y : TruncatedWittVector p n R\nh : ∀ (n_1 : ℕ), x.out.coeff n_1 = y.out.coeff n_1\ni : Fin n\n⊢ coeff i x = coeff i y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nn : ℕ\nx : R\n⊢ ((teichmuller p) x * ↑p ^ n).coeff n = x ^ p ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Complete | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 17
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ x = verschiebung (x.shift 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 100,
"column": 4
} | {
"line": 106,
"column": 66
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nx : 𝕎 R\nn i : ℕ\nhi : i < n + 1\n⊢ x.coeff i =\n ∑ s ∈ Finset.Iic n, ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ s) (x.coeff s)) * ↑p ^ s).coeff i",
"usedConstants": [
... | rw [Finset.sum_eq_add_sum_diff_singleton_of_mem (Finset.mem_Iic.mpr (Nat.lt_succ_iff.mp hi))]
let g := fun x : ℕ ↦ (0 : R)
rw [Finset.sum_congr rfl (g := g)]
· simp [g]
· intro b hb
simp only [Finset.mem_sdiff, Finset.mem_Iic, Finset.mem_singleton] at hb
exact teichmuller_mul_pow_coeff_of_ne... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 100,
"column": 4
} | {
"line": 106,
"column": 66
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nx : 𝕎 R\nn i : ℕ\nhi : i < n + 1\n⊢ x.coeff i =\n ∑ s ∈ Finset.Iic n, ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ s) (x.coeff s)) * ↑p ^ s).coeff i",
"usedConstants": [
... | rw [Finset.sum_eq_add_sum_diff_singleton_of_mem (Finset.mem_Iic.mpr (Nat.lt_succ_iff.mp hi))]
let g := fun x : ℕ ↦ (0 : R)
rw [Finset.sum_congr rfl (g := g)]
· simp [g]
· intro b hb
simp only [Finset.mem_sdiff, Finset.mem_Iic, Finset.mem_singleton] at hb
exact teichmuller_mul_pow_coeff_of_ne... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Complete | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 17
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\nh : ∀ m < n, x.coeff m = 0\n⊢ x = (⇑verschiebung)^[n] (x.shift n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Complete | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 15
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx✝ : 𝕎 k\nn : ℕ\nh : ∀ (n : ℕ), x✝ ≡ 0 [SMOD Ideal.span {↑p} ^ n]\nthis : ∀ m < n + 1, x✝.coeff m = 0\n⊢ x✝.coeff n = coeff 0 n",
"usedConstants": [
"Eq.mpr",
"congrA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : R♭\n⊢ (ghostComponentModPPow n) ((teichmuller p) ((PreTilt.coeff n) x)) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 214,
"column": 4
} | {
"line": 216,
"column": 38
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p →+* K := (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])).comp C\nthis : CharP K p\n⊢ ∃ K x, ∃ (_ : CharP K p), Infinite K",
"usedConstants": [
"Infinite.of_injective",
"ZMod.commRing",
"Algebra... | haveI : Infinite K :=
Infinite.of_injective (algebraMap (Polynomial (ZMod p)) (FractionRing (Polynomial (ZMod p))))
(IsFractionRing.injective _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 39
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommM... | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommM... | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommM... | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfectoid.BDeRham | {
"line": 67,
"column": 10
} | {
"line": 67,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete (span {↑p}) R\n⊢ IsUnit (((algebraMap R (Localization.Away ↑p)).comp (fontaineTheta R p)) ↑p)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi✝ : Irreducible (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 130,
"column": 2
} | {
"line": 136,
"column": 82
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\n⊢ (p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d",
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalComm... | delta hilbertPoly
induction d with
| zero => simp only [add_zero]
| succ d _ =>
simp only
rw [← sum_def _ fun _ r => r • _]
exact sum_add_index _ _ _ (fun _ => zero_smul ..) (fun _ _ _ => add_smul ..) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 130,
"column": 2
} | {
"line": 136,
"column": 82
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\n⊢ (p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d",
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalComm... | delta hilbertPoly
induction d with
| zero => simp only [add_zero]
| succ d _ =>
simp only
rw [← sum_def _ fun _ r => r • _]
exact sum_add_index _ _ _ (fun _ => zero_smul ..) (fun _ _ _ => add_smul ..) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nht : ∀ (r : R), (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a\n⊢ b = c",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 69,
"column": 98
} | {
"line": 80,
"column": 64
} | [
{
"pp": "n : ℕ\nhn1 : n ≠ 1\n⊢ Irreducible (X ^ n - X - 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Polynomial.map_one",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Units.val",
"Eq.mpr",
"Polynomial.... | by
by_cases hn0 : n = 0
· rw [hn0, pow_zero, sub_sub, add_comm, ← sub_sub, sub_self, zero_sub]
exact Associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := by
simp only [trinomial, C_neg, C_1]; ring
have hn : 1 < n := Nat.one_lt_iff_ne_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 79
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)",
"usedConstants": [
"Polynomial.instOne",
"Polynomial.instNeg",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"AlgH... | have := congr(aeval x $(neg_one_pow_mul_shiftedLegendre_comp_one_sub_X_eq n)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 26
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\nthis : (aeval x) ((-1) ^ n * (shiftedLegendre n).comp (1 - X)) = (aeval x) (shiftedLegendre n)\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ b = -a * (x1 + x2)",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ c = a * x1 * x2",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\nhg : ∀ i ≤ n, (coeff i) g ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nhg : ∀ (i : ℕ), (coeff i) g ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\ninst✝ : I.IsTwoSided\nhf : ∀ i ≤ n, (coeff i) f ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\ninst✝ : I.IsTwoSided\nhf : ∀ (i : ℕ), (coeff i) f ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 72,
"column": 60
} | {
"line": 74,
"column": 84
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\ninst✝ : I.IsTwoSided\nhf : ∀ (i : ℕ), (coeff i) f ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"usedConstants": [
"Ideal.one_eq_top",
"Submodule.mem_top._simp_1",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
... | by
simpa only [Ideal.IsTwoSided.mul_one] using
coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (J := 1) (g := g) hf (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 29
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nP : Submodule R M\nhP : P.FG\nt : ↥P ⊗[R] N\nh : (rTensor N P.subtype) t = (rTensor N P.subtype) 0\n⊢ ∃ Q, ∃ (hPQ : P ≤ Q), Q.FG ∧ (rTensor N (in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 44
} | [
{
"pp": "case right\nR : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nP : Submodule R M\nhP : P.FG\nt : ↥P ⊗[R] N\nP' : Submodule R M\nhP' : P'.FG\nt' : ↥P' ⊗[R] N\nh✝ :\n (rTensor N (P ⊔ P').subtype) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 96,
"column": 21
} | {
"line": 96,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nz : R\nh : IsUnit z\n⊢ (∀ b ∈ ∅, Prime b) ∧ Associated ∅.prod z",
"usedConstants": [
"CommMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nthis : a ∈ closure {r | IsUnit r ∨ Prime r}\n⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a",
"usedCo... | induction this using closure_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
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