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.MvPolynomial.Ideal | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 17
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn : ℕ\nr : R\nh : ¬r = 0\n⊢ C r ∈ idealOfVars σ R ^ n ↔ r = 0 ∨ n = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 68,
"column": 8
} | {
"line": 68,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i\n⊢ ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j • ⊤ ≤ I ^ i • ⊤",
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"Submodule",
"Ideal.smul_top_eq_map",
"IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (i : ℕ), ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ i)\n⊢ ∀ (i : ℕ), ∃ j, ↑(I ^ j • ⊤) * ↑(I ^ j • ⊤) ⊆ ↑(I ^ i • ⊤)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Ideal.smul_top_eq_map",
"instHSMul",
"NonUnitalCommRing.toNon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 104,
"column": 37
} | {
"line": 104,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i ↦ toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ↑(I ^ i) ⊆ U",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 131,
"column": 39
} | {
"line": 131,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ ↑(I ^ i • ⊤)\n⊢ a ∈ I ^ i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 17
} | [
{
"pp": "case mp.right\nR : Type u_1\ninst✝¹ : CommRing R\ntop : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nJ : Ideal R\nH : top = J.adicTopology\nthis : TopologicalSpace R := J.adicTopology\ns : Set R\nhs : s ∈ 𝓝 0\n⊢ ∃ n, ↑(J ^ n) ⊆ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 15
} | [
{
"pp": "case mp\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\nh : ∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)\n_h' : ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s\n⊢ IsOpen[inst✝¹] {0}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : WithIdeal R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : WithIdeal S\nf : R →+* S\nhf : Ideal.map f i ≤ i\nn : ℕ\nx✝ : True\n⊢ Ideal.map f (i ^ n) ≤ i ^ n",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n\nu : ℕ → R\nhu : Ca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 68,
"column": 31
} | {
"line": 68,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\ni : ℕ\nx✝ : True\nm : ℕ\nhm : i ≤ m\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f Filter.atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\ne : R ≃+* S\nthis✝ : WithIdeal R := { i := I }\nthis : WithIdeal S := { i := Ideal.map e I }\n⊢ IsUniformEmbedding ⇑↑(WithIdeal.uniformEquiv e ⋯)",
"usedConstants": [
"Equiv.instEquivLike",
"UniformEquiv.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 55
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nm✝ n✝ : ℕ\nh : m✝ ≤ n✝\nx✝ : MvPowerSeries σ R\n⊢ ((Ideal.Quotient.factorₐ (MvPolynomial σ R) ⋯).comp (truncTotalAlgHom σ R n✝)) x✝ = (truncTotalAlgHom σ R m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 42
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"usedConstants": [
... | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 42
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"usedConstants": [
... | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 42
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"usedConstants": [
... | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 102,
"column": 2
} | {
"line": 106,
"column": 97
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (toAdicCompletion σ R) ↑p = (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p",
"usedConstants": [
"MvPowerSeries.truncTotal",
"Iff.mpr",
"Finsupp.instAddZeroClass",
... | symm; ext n
suffices p - (truncTotal n) p ∈ MvPolynomial.idealOfVars σ R ^ n by
simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem]
exact (MvPolynomial.mem_pow_idealOfVars_iff' ..).mpr fun x hx ↦ by simp [coeff_truncTotal _ hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 102,
"column": 2
} | {
"line": 106,
"column": 97
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (toAdicCompletion σ R) ↑p = (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p",
"usedConstants": [
"MvPowerSeries.truncTotal",
"Iff.mpr",
"Finsupp.instAddZeroClass",
... | symm; ext n
suffices p - (truncTotal n) p ∈ MvPolynomial.idealOfVars σ R ^ n by
simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem]
exact (MvPolynomial.mem_pow_idealOfVars_iff' ..).mpr fun x hx ↦ by simp [coeff_truncTotal _ hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 145,
"column": 25
} | {
"line": 145,
"column": 36
} | [
{
"pp": "case h\nσ✝ : Type u_1\nR✝ : Type u_2\nn : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nx✝ : AdicCompletion (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)\nn✝ : ℕ\n⊢ ↑((↑↑(toAdicCompletion σ R).toRingHom).toFun (toAdicCompletionInv σ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 24,
"column": 69
} | {
"line": 24,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nh : I ≤ ⊥.jacobson\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ x ∈ ⨅ i, I ^ i • ⊤",
"usedConstants": [
"Eq.mpr",
"Sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 33,
"column": 65
} | {
"line": 33,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsNoetherianRing R\ninst✝² : Module.Finite R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : I ≠ ⊤\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ x ∈ ⨅ i, I ^ i • ⊤",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 22
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : Module.Finite R M\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : IsArtinianRing A\ninst✝ : IsLocalRing A\nf : ℕ → A\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f ... | by_cases h : m ≤ n | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 114,
"column": 6
} | {
"line": 115,
"column": 13
} | [
{
"pp": "case h₂.single_add\nk G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\na✝² : G\nb✝ : ↑A\nf✝ : G →₀ ↑A\na✝¹ : a✝² ∉ f✝.support\na✝ : b✝ ≠ 0\nh : ∀ ⦃x : ↑A⦄, (ModuleCat.Hom.hom (d₁₀ A)) f✝ = x → x ∈ Submodule.span k (Set.range fun gv ↦ (A.ρ gv.1) gv.2 - gv.2)\nx : ↑A\nhy : (ModuleCat.Hom.ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | [
{
"pp": "case H\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nI : Ideal S\ninst✝² : IsAdicComplete I S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf g : A →ₐ[R] S\nH : ∀ (n : ℕ), (Quotient.mkₐ R (I ^ n)).comp f = (Quotient.mkₐ R (I ^ n)).comp g\nx : A\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | [
{
"pp": "case a.refine_2\nR : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : CommRing S\nI : Ideal S\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → R →+* S ⧸ I ^ a n\nm n : ℕ\nhle : m ≤ n\nx : R\ns : ℕ\nhf : ∀ {m : ℕ} (x : R), ((factorPow I ⋯).comp (f (m + 1))) x = (f m) x\n⊢ (fun n ↦ (f n) x) s = (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Algebra | {
"line": 56,
"column": 7
} | {
"line": 56,
"column": 45
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsIntegral R A\nx✝ : A\n⊢ x✝ ∈ IsUnit.submonoid A ↔ x✝ ∈ A⁰",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"IsUnit",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 641,
"column": 2
} | {
"line": 641,
"column": 13
} | [
{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : G →₀ A\nhx : x ∈ cycles₁ (Rep.ofDistribMulAction k G A)\n⊢ IsCycle₁ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 875,
"column": 43
} | {
"line": 875,
"column": 54
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H0 A) → Prop\nx : ↑(H0 A)\nh : ∀ (x : ↑A), C ((ConcreteCategory.hom (H0π A)) x)\ny : ↑(cycles A 0)\n⊢ C ((ConcreteCategory.hom (π A 0)) y)",
"usedConstants": [
"ModuleCat",
"CategoryTheory.ConcreteCategory.hom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 936,
"column": 43
} | {
"line": 936,
"column": 60
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H1 A) → Prop\nx : ↑(H1 A)\nh : ∀ (x : ↥(cycles₁ A)), C ((ConcreteCategory.hom (H1π A)) x)\ny : ↑(cycles A 1)\n⊢ C ((ConcreteCategory.hom (π A 1)) y)",
"usedConstants": [
"ModuleCat",
"CategoryTheory.ConcreteCategory.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 972,
"column": 6
} | {
"line": 972,
"column": 29
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng h : G\na : ↑A\n⊢ (Multiplicative.toAdd\n (Multiplicative.ofAdd\n (↑(ModuleCat.Hom.hom (H1π A) ∘ₗ\n ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv ∘ₗ lsingle (g * h))).toIntLinearMap)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : ℕ\nh : a ≤ b\nx : AdicCompletion I ↥(I ^ b • ⊤)\nz : ↥(I ^ b • ⊤)\n⊢ (LinearMap.reduceModIdeal (I ^ a) (I ^ b • ⊤).subtype) (Submodule.Quotient.mk z) = 0",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1025,
"column": 8
} | {
"line": 1026,
"column": 57
} | [
{
"pp": "case H.h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\nx✝¹ : G\nx✝ : ↑A\n⊢ (TensorProduct.lift (mkH1OfIsTrivial A))\n ((QuotientAddGroup.lift (shortComplexH1 A).moduleCatToCycles.range.toAddSubgroup\n ((Finsupp.liftAddHom fun g ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b c : ℕ\nx : AdicCompletion I M\nh : c = b + a\nha : ↑x a = 0\n⊢ ∃ t, (powSMulQuotInclusion I M h ⊤) t = ↑x c",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1072,
"column": 2
} | {
"line": 1073,
"column": 5
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ↑x - ↑y ∈ boundaries₂ A",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Submodule",
"Rep.V",
"gr... | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1072,
"column": 2
} | {
"line": 1073,
"column": 5
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ↑x - ↑y ∈ boundaries₂ A",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Submodule",
"Rep.V",
"gr... | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1078,
"column": 44
} | {
"line": 1078,
"column": 61
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"usedConstants": [
"ModuleCat",
"CategoryTheory.ConcreteCategory.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 145,
"column": 2
} | {
"line": 149,
"column": 89
} | [
{
"pp": "case H.h.h\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : ι → R\ni✝ : ι\na✝ : AdicCauchySequence I M\nn✝ : ℕ\n⊢ ↑(((((lsum (AdicCompletion I R)) fun i ↦ (algebraMap R (AdicCompleti... | simp only [algebraMap_apply, Algebra.algebraMap_self, RingHom.id_apply, LinearMap.coe_comp,
coe_lsum, LinearMap.coe_smul, LinearMap.id_coe, LinearEquiv.coe_coe, Function.comp_apply,
finsuppLEquivDirectSum_symm_lof, Pi.smul_apply, id_eq, smul_zero, sum_single_index, smul_eval,
mapQ_eq_factor, factor_eq_facto... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 44
} | [
{
"pp": "k G H : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nhf : Function.Injective ⇑f\ninst✝ : Mono φ\ni : ℕ\n⊢ Mono ((chainsMap f φ).f i)",
"usedConstants": [
"Eq.mpr",
"Rep.V",
"Finsupp.module",
"Category... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 44
} | [
{
"pp": "k G H : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nhf : Function.Surjective ⇑f\ninst✝ : Epi φ\ni : ℕ\n⊢ Epi ((chainsMap f φ).f i)",
"usedConstants": [
"Eq.mpr",
"Rep.V",
"Finsupp.module",
"CategoryT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 159,
"column": 51
} | {
"line": 160,
"column": 23
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\n⊢ π A n ≫ map f φ n = cyclesMap f φ n ≫ π B n",
"usedConstants": [
"HomologicalComplex.homologyπ",
"Nat.instOne",
"CategoryTheory.CategoryStruct.to... | by
simp [map, cyclesMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 15
} | [
{
"pp": "case h.hf.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\ng✝ : Fin 0 → G\nx✝ : ↑(ModuleCat.of k ↑A)\n⊢ (ModuleCat.Hom.hom (ModuleCat.ofHom (lsingle g✝) ≫ (chainsMap f φ).f 0 ≫ (chainsIso₀ B).hom)) x✝ =\n (ModuleCat.Hom... | simp [chainsMap_f, Unique.eq_default (α := Fin 0 → G), Unique.eq_default (α := Fin 0 → H),
chainsIso₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 50
} | [
{
"pp": "case hf.h.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₂ f φ ≫ d₂₁ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₂₁ A ≫ chainsMap₁ f φ) ∘ₗ lsingle a✝) x✝",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 32
} | [
{
"pp": "case hf.h.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₁ f φ ≫ d₁₀ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₁₀ A ≫ Hom.toModuleCatHom φ) ∘ₗ lsingle a✝) x✝"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 63
} | [
{
"pp": "case hf.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nφ : A ⟶ res 1 B\nx : ↑(ModuleCat.of k ↥(cycles₁ A))\n⊢ ↑((ModuleCat.Hom.hom (mapCycles₁ 1 φ)) x) ∈ boundaries₁ B",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Rep.V",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 407,
"column": 2
} | {
"line": 409,
"column": 9
} | [
{
"pp": "k G H : Type u\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)\nx : G ⧸ S →₀ ↑(A.ofQuotient S)\nhx : x ∈ cycles₁ (A.ofQuotient S)\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 49,
"column": 14
} | {
"line": 49,
"column": 25
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : Nontrivial R\nha : IsGroupLikeElem R 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 21
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nhc : ∑ a ∈ s, c a • a =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 145,
"column": 55
} | {
"line": 145,
"column": 66
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nih :\n ∀ (f g : A × A ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\n⊢ LinearIndependent R GroupLike.val",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 544,
"column": 2
} | {
"line": 547,
"column": 73
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 574,
"column": 4
} | {
"line": 575,
"column": 36
} | [
{
"pp": "k G H : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ (A.quotientToCoinvariants S))\ny : ↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S)))\nhy : (ConcreteCategory.hom (mapCyc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 582,
"column": 2
} | {
"line": 584,
"column": 27
} | [
{
"pp": "case h\nk G H✝ : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H✝\nA : Rep k G\nB : Rep k H✝\nf : G →* H✝\nφ : A ⟶ res f B\nn : ℕ\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ (A.quotientToCoinvariants S))\ny : ↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S)))\nhy : (ConcreteCategory... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 607,
"column": 4
} | {
"line": 609,
"column": 45
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =\n 0\ny : ↑(ModuleCat.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 15
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : ¬p.coeff n = 0\n⊢ p.coeff n ∈ ↑p.coeffs",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset",
"Membership.mem",
"id",
"Polynomial.coeff",
"SetLike.coe",
"Finset.instSetLi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Congruence.Hom | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 33
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | exact this.symm h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Congruence.Hom | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 33
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | exact this.symm h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Congruence.Hom | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 33
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | exact this.symm h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 177,
"column": 2
} | {
"line": 178,
"column": 47
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\np q : R[X]\nP : Ideal R\nhpq : (p * q).contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P)\nhPprime : P.IsPrime\n⊢ p.contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P) ∨ q.contentIdeal ≤ RingHom.ker (Ideal.Quotient.mk P)",
"usedConstants": [
"Polynomial.contentIde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 185,
"column": 43
} | {
"line": 185,
"column": 59
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\np q : R[X]\nhp : p.contentIdeal = ⊤\nhq : q.contentIdeal = ⊤\n⊢ ⊤ = p.contentIdeal * q.contentIdeal",
"usedConstants": [
"Polynomial.contentIdeal",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"congrArg",
"CommSemirin... | by simp [hp, hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 686,
"column": 4
} | {
"line": 687,
"column": 11
} | [
{
"pp": "case hf.h.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₃ f φ ≫ d₃₂ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₃₂ A ≫ chainsMap₂ f φ) ∘ₗ lsingle a✝) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 686,
"column": 4
} | {
"line": 687,
"column": 66
} | [
{
"pp": "case hf.h.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₃ f φ ≫ d₃₂ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₃₂ A ≫ chainsMap₂ f φ) ∘ₗ lsingle a✝) x... | simpa [d₃₂, map_add, map_sub, ← map_inv]
using congr(Finsupp.single _ $((hom_comm_apply φ _ _).symm)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 691,
"column": 4
} | {
"line": 692,
"column": 11
} | [
{
"pp": "case hf.h.h\nk G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₂ f φ ≫ d₂₁ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₂₁ A ≫ chainsMap₁ f φ) ∘ₗ lsingle a✝) x✝",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 84
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn : ℕ\nhn : Fact (0 < n)\nval✝ inv✝ : R\nval_inv✝ : val✝ * inv✝ = 1\ninv_val✝ : inv✝ * val✝ = 1\nhx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 148,
"column": 23
} | {
"line": 148,
"column": 27
} | [
{
"pp": "case h.h\nA✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : Div... | hy.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 41
} | [
{
"pp": "case pos.refine_2\nA : Type u_3\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nS : Set A\nhS : I = span S\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, hI.dpow n a = hI'.dpow n a\nn : ℕ\na : A\nx✝ ha : a ∈ I\nm : ℕ\nb : A\nhb : b ∈ S\n⊢ (RingHom.id A) (hI.dpow m b) = hI'.dpow m ((RingHom.id A) b)",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 11
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nx✝ : A\nhx : x✝ ∈ I\n⊢ (if hx : x✝ ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 58
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx : A\nhn : n ≠ 0\nhx : x ∈ I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 55,
"column": 4
} | {
"line": 57,
"column": 11
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx y : A\nhx : x ∈ I\nhy : y ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 40
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\na x : A\nhx : x ∈ I\nhax : a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 62,
"column": 20
} | {
"line": 63,
"column": 31
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nm✝ n✝ : ℕ\nx✝ : A\nhx : x✝ ∈ I\n⊢ ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 33
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\nh : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)\n⊢ (fun g ↦ g ∘ ⇑(mkAlgHom R (Rel R M))) ⇑f =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 226,
"column": 50
} | {
"line": 230,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\n⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)",
"usedConstants": [
"Eq.mpr",
... | by
refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right mkAlgHom_surjective
simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 165,
"column": 4
} | {
"line": 166,
"column": 53
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhinj : Injective ⇑Coe.ringHom\nheq : Coe.ringHom ⟨dpow' p n ↑x, ⋯⟩ = (↑n !)⁻¹ʳ * Coe.ringHom x ^ n\n⊢ ⋯.choose = ⟨dpow' p n ↑x, ⋯⟩",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"usedConstants": [
"Semiring.toModule",
"congrArg",... | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"usedConstants": [
"Semiring.toModule",
"congrArg",... | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"usedConstants": [
"Semiring.toModule",
"congrArg",... | simp [if_pos hx, hJ.dpow_mem _ hn hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 9
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\ninst✝ : (x : A) → Decidable (x ∈ J)\nhJ : hI.IsSubDPIdeal J\n⊢ (dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A)",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 350,
"column": 21
} | {
"line": 350,
"column": 76
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ J' : hI.SubDPIdeal\nh : (fun J ↦ ⟨J.carrier, ⋯⟩) J = (fun J ↦ ⟨J.carrier, ⋯⟩) J'\n⊢ J = J'",
"usedConstants": [
"Eq.mpr",
"DividedPowers.SubDPIdeal",
"CommSemiring.toSemiring",
"id",
"_private.M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DualNumber | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 38
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x\n⊢ I = Ideal.span {ε}",
"usedConstants": [
"Semiring.toModule",
"Ring.toNonAssocRing",
"MulZeroClass.toMul",
"DistribMulAction.toDistribSM... | refine le_antisymm ?_ ?_ <;> intro x | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.DualNumber | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | [
{
"pp": "case refine_2\nK : Type u_2\ninst✝ : DivisionRing K\nI : Ideal K[ε]\nhb : I ≠ ⊥\nht : I ≠ ⊤\nhd : ∀ x ∈ I, ε ∣ x\nhd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x\np y : K[ε]\nhyI : y ∈ I\nhy0 : y ≠ 0\nr : K[ε]\nhr : ε = r * y\n⊢ p * r * y ∈ I",
"usedConstants": [
"Semigroup.toMul",
"Semiring.toMo... | exact Ideal.mul_mem_left _ _ hyI | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DualNumber | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 39
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx y : K[ε]\nthis : ∃ c, fst x * c.fst = fst y ∨ fst y * c.fst = fst x\n⊢ ∃ c, ε * x * c = ε * y ∨ ε * y * c = ε * x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"TrivSqZeroExt.mul",
"Semigroup.toMul",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.IsBaseChange | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 15
} | [
{
"pp": "case refine_1.a.h.h\nR : Type u_1\ninst✝²¹ : CommRing R\nS : Type u_2\ninst✝²⁰ : CommRing S\ninst✝¹⁹ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : AddCommGroup M₂\ninst✝¹⁶ : AddCommGroup M₃\ninst✝¹⁵ : AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.IsBaseChange | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 15
} | [
{
"pp": "case refine_2.a.h.h\nR : Type u_1\ninst✝²¹ : CommRing R\nS : Type u_2\ninst✝²⁰ : CommRing S\ninst✝¹⁹ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : AddCommGroup M₂\ninst✝¹⁶ : AddCommGroup M₃\ninst✝¹⁵ : AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 69,
"column": 27
} | {
"line": 69,
"column": 93
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nh : Infinite (R ⧸ Ideal.under R Q)\n⊢ Q = ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : S ⧸ Q\nhx : H.restrict x = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 54
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : S\nhx : x ∈ Ideal.comap φ Q\n⊢ x ∈ Q",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"Submodule",
... | rwa [← Ideal.Quotient.eq_zero_iff_mem, ← H.restrict_injective.eq_iff, map_zero, restrict_mk,
Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Frobenius | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 24
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ σ' : G\nH : IsArithFrobAt R σ Q\nH' : IsArithFrobAt R σ' Q\nx : S\n⊢ (σ * σ'⁻¹) • x - x ∈ Submodule.toA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Frobenius | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 40
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\nthis : Ideal.under R (Ideal.map (MulSemiringAction.toRingEq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 641,
"column": 2
} | {
"line": 641,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\n⊢ hI.IsSubDPIdeal (RingHom.ker (Ideal.Quotient.mk J) ⊓ I)",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"CommS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 90,
"column": 25
} | {
"line": 90,
"column": 36
} | [
{
"pp": "case add\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\ns : Set A\nh : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s\nx x✝ y✝ : A\nhx✝ : x✝ ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.TensorProduct | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ni j : ι\nx : A\nhx : x ∈ ↑(𝒜 i)\ny ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.TensorProduct | {
"line": 127,
"column": 22
} | {
"line": 127,
"column": 33
} | [
{
"pp": "case tmul\nι : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝¹² : DecidableEq ι\ninst✝¹¹ : AddMonoid ι\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S B\n𝒜 : ι → Submodule R A\nℬ : ι → Submo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 64
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : 0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 27
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) n).coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 109,
"column": 6
} | {
"line": 113,
"column": 55
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b))... | have : g = 0 ∧ (a.constantCoeff * b.constantCoeff) • (1 : V) ≠ 0 := by
simpa [hz, h] using hn
simp only [coe_image, Set.mem_image]
use (0, 0)
simp [this.2, this.1, h, hz, smul_smul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 109,
"column": 6
} | {
"line": 113,
"column": 55
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\ng : Γ\nh : ¬0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b))... | have : g = 0 ∧ (a.constantCoeff * b.constantCoeff) • (1 : V) ≠ 0 := by
simpa [hz, h] using hn
simp only [coe_image, Set.mem_image]
use (0, 0)
simp [this.2, this.1, h, hz, smul_smul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 73
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\nh : 0 < x.orderTop\ng : Γ\ni : ℕ\nhi : i ∈ image (fun i ↦ i.1 + i.2) (((powerSer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 164,
"column": 68
} | {
"line": 165,
"column": 52
} | [
{
"pp": "S : Type u_1\ninst✝ : CommRing S\nd e : ℕ\n⊢ invOneSubPow S (d + e) = invOneSubPow S d * invOneSubPow S e",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Units... | by
simp_rw [invOneSubPow_eq_inv_one_sub_pow, pow_add] | [anonymous] | Lean.Parser.Term.byTactic |
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