module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 605,
"column": 2
} | {
"line": 607,
"column": 63
} | {
"line": 608,
"column": 2
} | [
{
"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.hom... | [
"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.hom (mapCyc... | have H : d₁₀ A (Y - mapDomain S.subtype Z) = 0 := by
simpa [map_sub, sub_eq_zero, chains₁ToCoinvariantsKer, -LinearMap.sub_apply, d₁₀,
sum_mapDomain_index_inj] using! Subtype.ext_iff.1 hZ.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.GaussLemma | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 81
} | {
"line": 59,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nb : NNReal\nhb : 1 < b\np : R[X]\nhR : ¬IsField R\n⊢ p.contentIdeal = ⊤ ↔ ∀ (v : HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Polynomial.contentIdeal"... | [
"case e'_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nb : NNReal\nhb : 1 < b\np : R[X]\nhR : ¬IsField R\n⊢ (∀ (v : HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1) ↔\n ∀ (x : HeightOneSpectrum R), 1 ≤ gaussNorm (x.intAdicAbv hb) 1 p",
"case convert_2\nR : Type u_1\ninst✝¹ : CommR... | convert_to _ ↔ ∀ (x : HeightOneSpectrum R), 1 ≤ gaussNorm (x.intAdicAbv hb) 1 p | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convertTo_1 | Mathlib.Tactic.convertTo |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 94,
"column": 12
} | {
"line": 94,
"column": 14
} | {
"line": 95,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : ... | [
"case refine_1\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedM... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 47,
"column": 82
} | {
"line": 108,
"column": 77
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\n⊢ Subsingleton (N →ₗ[R] M) ↔ ∃ r ∈ annihilator R N, IsSMulRegula... | [] | by
refine ⟨fun hom0 ↦ ?_, fun ⟨r, mem_ann, reg⟩ ↦
linearMap_subsingleton_of_mem_annihilator reg mem_ann⟩
cases subsingleton_or_nontrivial M
· exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩
· by_contra! h
have hexist : ∃ p ∈ associatedPrimes R M, Module.annihilator R N ≤ p :... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Depth.Rees | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | {
"line": 112,
"column": 4
} | [
{
"pp": "case succ\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : L... | [
"case succ\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 79,
"column": 12
} | {
"line": 79,
"column": 14
} | {
"line": 80,
"column": 4
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\nhIJ : map f I ≤ J\nH : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)\nn : ℕ\nhn : n = 0\na✝ : A\n⊢ a✝ ∈ I → hJ.dpow n... | [
"case pos\nA : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\nhIJ : map f I ≤ J\nH : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)\nn : ℕ\nhn : n = 0\na✝ : A\nha : a✝ ∈ I\n⊢ hJ.dpow n (f a✝) ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 154,
"column": 54
} | {
"line": 154,
"column": 87
} | {
"line": 154,
"column": 88
} | [
{
"pp": "case succ\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 : Di... | [
"case succ\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 : DividedPowers ... | hJ.dpow_eval_zero n.succ_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 180,
"column": 22
} | {
"line": 180,
"column": 50
} | {
"line": 182,
"column": 0
} | [
{
"pp": "A✝ : 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 : DividedPowers... | [] | simp only [RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 180,
"column": 22
} | {
"line": 180,
"column": 50
} | {
"line": 182,
"column": 0
} | [
{
"pp": "A✝ : 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 : DividedPowers... | [] | simp only [RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 180,
"column": 22
} | {
"line": 180,
"column": 50
} | {
"line": 182,
"column": 0
} | [
{
"pp": "A✝ : 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 : DividedPowers... | [] | simp only [RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 583,
"column": 4
} | {
"line": 586,
"column": 50
} | {
"line": 587,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : R\nh2 : torsionBy R M b = a • torsionBy R M b → IsSMulRegular M b\nha : IsSMulRegular M a\nhb : IsSMulRegular (QuotSMulTop a M) b\n⊢ torsionBy R M b ≤ a • torsionBy R M b",
"ppTerm": "?m.108",
"a... | [] | refine le_of_eq_of_le ?_ <| smul_top_inf_eq_smul_of_isSMulRegular_on_quot <|
ha.of_injective _ <| ker_eq_bot.mp <| ker_liftQ_eq_bot' _ (lsmul R M b) rfl
rw [← (isSMulRegular_on_quot_iff_lsmul_comap_eq _ _).mp hb]
exact (inf_eq_right.mpr (ker_le_comap _)).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 583,
"column": 4
} | {
"line": 586,
"column": 50
} | {
"line": 587,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : R\nh2 : torsionBy R M b = a • torsionBy R M b → IsSMulRegular M b\nha : IsSMulRegular M a\nhb : IsSMulRegular (QuotSMulTop a M) b\n⊢ torsionBy R M b ≤ a • torsionBy R M b",
"ppTerm": "?m.108",
"a... | [] | refine le_of_eq_of_le ?_ <| smul_top_inf_eq_smul_of_isSMulRegular_on_quot <|
ha.of_injective _ <| ker_eq_bot.mp <| ker_liftQ_eq_bot' _ (lsmul R M b) rfl
rw [← (isSMulRegular_on_quot_iff_lsmul_comap_eq _ _).mp hb]
exact (inf_eq_right.mpr (ker_le_comap _)).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 102,
"column": 46
} | {
"line": 102,
"column": 61
} | {
"line": 102,
"column": 62
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ (↑p ^ (-↑x.valuation)) ^ n < ↑p ^ (-(↑n !).valuation)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"zpow_... | [
"case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ (↑p ^ (-↑x.valuation)) ^ ↑n < ↑p ^ (-(↑n !).valuation)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 138,
"column": 4
} | {
"line": 142,
"column": 85
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\nn : ℕ\na b : A\nha : a ∈ I\nhb : b ∈ I\nhab : a - b ∈ J\n⊢ hI.dpow n a - hI.dpow n b ∈ J",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have hab' : a - b ∈ I := I.sub_mem ha hb
rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
(hIJ.dpow_mem _ (ne_of_gt (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 138,
"column": 4
} | {
"line": 142,
"column": 85
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\nn : ℕ\na b : A\nha : a ∈ I\nhb : b ∈ I\nhab : a - b ∈ J\n⊢ hI.dpow n a - hI.dpow n b ∈ J",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have hab' : a - b ∈ I := I.sub_mem ha hb
rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
(hIJ.dpow_mem _ (ne_of_gt (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 17
} | {
"line": 179,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ K : Ideal A\nhJ : hI.IsSubDPIdeal J\nhK : hI.IsSubDPIdeal K\nn : ℕ\nhn : n ≠ 0\na : A\n⊢ a ∈ ↑J ∪ ↑K → hI.dpow n a ∈ span (↑J ∪ ↑K)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Submodule",
"... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ K : Ideal A\nhJ : hI.IsSubDPIdeal J\nhK : hI.IsSubDPIdeal K\nn : ℕ\nhn : n ≠ 0\na : A\nha : a ∈ ↑J ∪ ↑K\n⊢ hI.dpow n a ∈ span (↑J ∪ ↑K)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.TotallySplit | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 42
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case h\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Etale R S\ninst✝ : Module.Finite R S\nhn : Module.rankAtStalk S = ↑0\ne : R ⊗[R] S ≃ₐ[R] S := TensorProduct.lid R S\nthis : IsFiniteSplit R S\n⊢ IsFiniteSplit R (R ⊗[R] S)",
"ppTerm": "?h",
"assigned"... | [] | apply IsFiniteSplit.of_algEquiv e.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 16
} | {
"line": 454,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : A →+* B\nhf : Ideal.map f I ≤ J ∧ ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)\nn : ℕ\na b : A\n⊢ a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a -... | [
"A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : A →+* B\nhf : Ideal.map f I ≤ J ∧ ∀ {n : ℕ}, ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)\nn : ℕ\na b : A\nha : a ∈ I\n⊢ b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 467,
"column": 34
} | {
"line": 467,
"column": 36
} | {
"line": 467,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : hI.DPMorphism hJ\nx✝ : ℕ\nhn : x✝ ≠ 0\na : A\nha : f.toRingHom a = 0\nha' : a ∈ I\n⊢ hJ.dpow x✝ (f.toRingHom a) = 0 ∧ hI.dpow x✝ a ∈ I",
"ppTerm": "?m.70",
... | [
"A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : hI.DPMorphism hJ\nx✝ : ℕ\nhn : x✝ ≠ 0\na : A\nha : f.toRingHom a = 0\nha' : a ∈ I\n⊢ hJ.dpow x✝ 0 = 0 ∧ hI.dpow x✝ a ∈ I"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Frobenius | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 84
} | {
"line": 115,
"column": 2
} | [
{
"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✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk' : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ ... | [
"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✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk'✝ : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ k\nhk' : ↑k... | have hk' : ↑k ∉ Q := fun h ↦ hk' (Q.mem_of_dvd (Nat.cast_dvd_cast (hζ.2 m ‹_›)) h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 144,
"column": 10
} | {
"line": 144,
"column": 17
} | {
"line": 144,
"column": 17
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝ : CommRing S\nd✝ d : ℕ\na✝ : invOneSubPow S d = (Units.mkOfMulEqOne (1 - X) (mk 1) ⋯)⁻¹ ^ d\n⊢ invOneSubPow S (d + 1) = (Units.mkOfMulEqOne (1 - X) (mk 1) ⋯)⁻¹ ^ (d + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing... | [
"case succ\nS : Type u_1\ninst✝ : CommRing S\nd✝ d : ℕ\na✝ : invOneSubPow S d = (Units.mkOfMulEqOne (1 - X) (mk 1) ⋯)⁻¹ ^ d\n⊢ invOneSubPow S (d + 1) = (Units.mkOfMulEqOne (1 - X) (mk 1) ⋯ ^ (d + 1))⁻¹"
] | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 22
} | {
"line": 116,
"column": 12
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nx y : SummableFamily Γ R α\ng : Γ\na : α\n⊢ a ∈ {a | ((⇑x + ⇑y) a).coeff g ≠ 0} →\n a ∈ (Function.support fun a ↦ (x a).coeff g) ∪ Function.support fun a ↦ (y a).coe... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nx y : SummableFamily Γ R α\ng : Γ\na : α\nha : a ∈ {a | ((⇑x + ⇑y) a).coeff g ≠ 0}\n⊢ a ∈ (Function.support fun a ↦ (x a).coeff g) ∪ Function.support fun a ↦ (y a).coeff g"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 331,
"column": 8
} | {
"line": 332,
"column": 36
} | {
"line": 333,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommGroup R\ns✝ t : SummableFamily Γ R α\na : α\ns : SummableFamily Γ R α\n⊢ (⋃ a, (-s a).support).IsPWO",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"... | [] | simp_rw [support_neg]
exact s.isPWO_iUnion_support | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 331,
"column": 8
} | {
"line": 332,
"column": 36
} | {
"line": 333,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommGroup R\ns✝ t : SummableFamily Γ R α\na : α\ns : SummableFamily Γ R α\n⊢ (⋃ a, (-s a).support).IsPWO",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"... | [] | simp_rw [support_neg]
exact s.isPWO_iUnion_support | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 251,
"column": 10
} | {
"line": 251,
"column": 57
} | {
"line": 251,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.Xzero)\naux₀ : PowerSeries.HasSubst F.Xzero\n⊢ F.Xzero = PowerSeries.subst (PowerSeries.subst F.Xzero F.Xzero) F.Xzero.substInv",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq... | [
"R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.Xzero)\naux₀ : PowerSeries.HasSubst F.Xzero\n⊢ F.Xzero = PowerSeries.subst F.Xzero (PowerSeries.subst F.Xzero F.Xzero.substInv)"
] | ← PowerSeries.subst_comp_subst_apply aux₀ aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 302,
"column": 10
} | {
"line": 302,
"column": 57
} | {
"line": 302,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.zeroX)\naux₀ : PowerSeries.HasSubst F.zeroX\n⊢ F.zeroX = PowerSeries.subst (PowerSeries.subst F.zeroX F.zeroX) F.zeroX.substInv",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq... | [
"R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.zeroX)\naux₀ : PowerSeries.HasSubst F.zeroX\n⊢ F.zeroX = PowerSeries.subst F.zeroX (PowerSeries.subst F.zeroX F.zeroX.substInv)"
] | ← PowerSeries.subst_comp_subst_apply aux₀ aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 482,
"column": 4
} | {
"line": 483,
"column": 14
} | {
"line": 484,
"column": 2
} | [
{
"pp": "case refine_1.right\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α... | [] | · by_contra hi
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 673,
"column": 4
} | {
"line": 674,
"column": 93
} | {
"line": 676,
"column": 0
} | [
{
"pp": "case succ\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\nih : ∀ ⦃g : Γ⦄, g ∈ (x ^ n).support → g ∈ ↑(AddSubmonoid.closure x.support)\ng : Γ\nhn : g ∈ (x ^ (n + 1)).support\n⊢ g ∈ ↑(AddSubmonoid.cl... | [] | obtain ⟨i, hi, j, hj, rfl⟩ := support_mul_subset hn
exact SetLike.mem_coe.2 (AddSubmonoid.add_mem _ (ih hi) (AddSubmonoid.subset_closure hj)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 673,
"column": 4
} | {
"line": 674,
"column": 93
} | {
"line": 676,
"column": 0
} | [
{
"pp": "case succ\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\nih : ∀ ⦃g : Γ⦄, g ∈ (x ^ n).support → g ∈ ↑(AddSubmonoid.closure x.support)\ng : Γ\nhn : g ∈ (x ^ (n + 1)).support\n⊢ g ∈ ↑(AddSubmonoid.cl... | [] | obtain ⟨i, hi, j, hj, rfl⟩ := support_mul_subset hn
exact SetLike.mem_coe.2 (AddSubmonoid.add_mem _ (ih hi) (AddSubmonoid.subset_closure hj)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Henselian | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 46
} | {
"line": 79,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\nthis : IsUnit ((Ideal.Quotient.mk ⊥.jacobson) a)\n⊢ IsUnit a",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"Semiring.to... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R ⧸ ⊥.jacobson\nh1 : (Ideal.Quotient.mk ⊥.jacobson) a * y = 1\nh2 : y * (Ideal.Quotient.mk ⊥.jacobson) a = 1\n⊢ IsUnit a"
] | obtain ⟨⟨x, y, h1, h2⟩, rfl : x = _⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 84
} | {
"line": 55,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra.HasGoingUp R S\np q : Ideal R\ninst✝² : q.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhle : under R P ≤ q\nh : under R P ≠ q\n⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver ... | [] | exact Algebra.HasGoingUp.exists_ideal_ge_liesOver_of_lt P (lt_of_le_of_ne hle h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 55
} | {
"line": 61,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\n⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit",
"ppTerm": "?m.95",
"assigned": true,
"usedConstan... | [] | ext a b : 2
simpa [← (ℛ R b).eq] using congr(lsingle (R := R) (1 : G)
$(sum_mul_antipode_eq_algebraMap_counit (ℛ R b))) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 55
} | {
"line": 61,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\n⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit",
"ppTerm": "?m.95",
"assigned": true,
"usedConstan... | [] | ext a b : 2
simpa [← (ℛ R b).eq] using congr(lsingle (R := R) (1 : G)
$(sum_mul_antipode_eq_algebraMap_counit (ℛ R b))) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 56,
"column": 32
} | {
"line": 59,
"column": 55
} | {
"line": 61,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\n⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit",
"ppTerm": "?m.95",
"assigned": true,
"usedConstan... | [] | by
ext a b : 2
simpa [← (ℛ R b).eq] using congr(lsingle (R := R) (1 : G)
$(sum_mul_antipode_eq_algebraMap_counit (ℛ R b))) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 73,
"column": 10
} | {
"line": 73,
"column": 89
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\nf : A →ₐc[R] B\nhf : Function.Surjective ⇑f\nhS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R\n⊢ (↑f).toLinearMap ∘ₗ (toCon... | [] | rw [id_mul_antipode, algHom_comp_convOne, ← convOne_comp_coalgHom f.toCoalgHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 73,
"column": 10
} | {
"line": 73,
"column": 89
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\nf : A →ₐc[R] B\nhf : Function.Surjective ⇑f\nhS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R\n⊢ (↑f).toLinearMap ∘ₗ (toCon... | [] | rw [id_mul_antipode, algHom_comp_convOne, ← convOne_comp_coalgHom f.toCoalgHom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 73,
"column": 10
} | {
"line": 73,
"column": 89
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\nf : A →ₐc[R] B\nhf : Function.Surjective ⇑f\nhS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R\n⊢ (↑f).toLinearMap ∘ₗ (toCon... | [] | rw [id_mul_antipode, algHom_comp_convOne, ← convOne_comp_coalgHom f.toCoalgHom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Henselian | {
"line": 238,
"column": 8
} | {
"line": 238,
"column": 62
} | {
"line": 239,
"column": 8
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Pol... | [
"case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval... | rw [← add_assoc, hc, ← add_zero (c m), sub_eq_add_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 83
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case mp\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\na : WithIdealFilter F\ns : Set (WithIdealFilter F)\nhs : s ∈ 𝓝 a\n⊢ ∃ I ∈ F, a +ᵥ idealSet I ⊆ s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"AddGroupFilterBasis.nhds_hasBasis",
... | [
"case mp\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\na : WithIdealFilter F\ns : Set (WithIdealFilter F)\nhs : s ∈ 𝓝 a\nt : Set A\nht : t ∈ F.addGroupFilterBasis\nhts : (fun y ↦ a + y) '' t ⊆ s\n⊢ ∃ I ∈ F, a +ᵥ idealSet I ⊆ s"
] | rcases ((F.addGroupFilterBasis).nhds_hasBasis a).mem_iff.1 hs with ⟨t, ht, hts⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.KrullDimension.Module | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 60
} | {
"line": 58,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nthis : annihilator R R = ⊥\n⊢ supportDim R R = ringKrullDim R",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"WithBot",
"Module.supportDim_eq_ringKrullDim_quotient_annihilator",
... | [
"R : Type u_1\ninst✝ : CommRing R\nthis : annihilator R R = ⊥\n⊢ ringKrullDim (R ⧸ ⊥) = ringKrullDim R"
] | rw [supportDim_eq_ringKrullDim_quotient_annihilator, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 43,
"column": 8
} | {
"line": 43,
"column": 24
} | {
"line": 43,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nh : ¬IsField R\nh' : ringKrullDim R < 1\nh'' : ringKrullDim R ≤ 0\n⊢ IsField R",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nh : ¬IsField R\nh' : ringKrullDim R < 1\nh'' : ringKrullDim R ≤ ↑0\n⊢ IsField R"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 51
} | {
"line": 105,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nthis✝³ : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\np' : Ideal Rₚ := Ideal.map (algebraMap R Rₚ) p\np'_def : p' = Ideal.map (algebraMap R Rₚ) p\nthis✝² : p'.... | [] | apply height_eq_height_add_one_of_isMaximal p' P' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 94
} | {
"line": 405,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\ns : Set R\nhs : s ⊆ ↑(Ring.jacobson R)\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ Ideal.span s) + ↑s.encard",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemiring",
... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\ns : Set R\nhs : s ⊆ ↑(Ring.jacobson R)\n⊢ Ideal.span s ≤ Ring.jacobson R",
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\ns : Set R\nhs : s ⊆ ↑(Ring.jacobson R)\n⊢ ringKrullDim (R ⧸ Ideal.span s) + ↑(Sub... | refine le_trans (ringKrullDim_le_ringKrullDim_quotient_add_spanFinrank (Ideal.span s) ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LittleWedderburn | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 27
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case intro\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\n⊢ Subring.center D = ⊤",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ring.toNonAssocRing",
"congrArg",
"Subring.center",
"id",
... | [
"case intro\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\n⊢ Z = ⊤"
] | set Z := Subring.center D | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 44
} | {
"line": 86,
"column": 2
} | [
{
"pp": "case bot\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\na✝ : Subsingleton ↑M\n⊢ Subsingleton (LocalizedModule S ↑M)",
"ppTerm": "?bot... | [] | apply LocalizedModule.instSubsingleton _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 44
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case bot\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\na✝ : Subsingleton ↑M\n⊢ Subsingleton (LocalizedModule S ↑M)",
"ppTerm": "?bot",
"assigned": true,
... | [] | apply LocalizedModule.instSubsingleton _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.LaurentSeries | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 44
} | {
"line": 248,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx : R⸨X⸩\nn : ℤ\nh : 0 ≠ x.coeff n\n⊢ order x ≤ n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Int.instLinearOrder",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"SemilatticeInf.toPartialOrder",
"Ne.symm",
... | [] | exact order_le_of_coeff_ne_zero h.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LaurentSeries | {
"line": 556,
"column": 2
} | {
"line": 569,
"column": 31
} | {
"line": 570,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : HahnSeries.order f ≤ 0\n⊢ Valued.v f ≤ exp (-D)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Algebra.cast... | [
"case neg\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : 0 < HahnSeries.order f\n⊢ Valued.v f ≤ exp (-D)"
] | · obtain ⟨s, hs⟩ := Int.exists_eq_neg_ofNat ord_nonpos
rw [← f.single_order_mul_powerSeriesPart, hs, map_mul, valuation_single_zpow, neg_neg, mul_comm,
← le_mul_inv_iff₀, exp_neg, ← mul_inv, ← exp_add, ← exp_neg]
· by_cases! hDs : D + s ≤ 0
· apply le_trans ((PowerSeries.idealX K).valuation_le_one F... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LaurentSeries | {
"line": 1059,
"column": 13
} | {
"line": 1059,
"column": 51
} | {
"line": 1059,
"column": 51
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝ : Field K\na : K\n⊢ (LaurentSeriesRingEquiv K) ((algebraMap K K⸨X⸩) a) = (algebraMap K (RatFuncAdicCompl K)) a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Int.instAddCommMonoid",
"ZeroHom.funLike",
"Int.instIsStrictOrderedRi... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LaurentSeries | {
"line": 1155,
"column": 2
} | {
"line": 1155,
"column": 16
} | {
"line": 1157,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\n⊢ ∃ F_1, (ofPowerSeries ℤ K) F_1 = (ofPowerSeries ℤ K) F",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Int.instIsStrictOrderedRing",
"HahnSeries.instNonAssocSemiring",
"RingHom",
"SemilatticeInf.toPartialOrder",
... | [] | exact ⟨F, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.WittVector.Defs | {
"line": 283,
"column": 89
} | {
"line": 285,
"column": 51
} | {
"line": 287,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm n : ℕ\n⊢ constantCoeff (wittNSMul p m n) = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Int.instAddCommMonoid",
"RingHom.instRingHomClass",
"AddMonoidAlgebra.instAddMonoid",
"Nat.instM... | [] | by
apply constantCoeff_wittStructureInt p _ _ n
simp only [smul_zero, map_nsmul, constantCoeff_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 354,
"column": 64
} | {
"line": 358,
"column": 22
} | {
"line": 360,
"column": 0
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\n⊢ constantCoeff (wittStructureInt p Φ 0) = constantCoeff Φ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Int.cast",
"Eq.mpr",
"Nat.instMulZeroClass",
... | [] | by
have inj : Function.Injective (Int.castRingHom ℚ) := by intro m n; exact Int.cast_inj.mp
apply inj
rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat_zero,
constantCoeff_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 82,
"column": 2
} | {
"line": 85,
"column": 28
} | {
"line": 87,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\n⊢ (aeval x.coeff) (verschiebungPoly n) = x.verschiebungFun.coeff n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semi... | [] | rcases n with - | n
· simp only [verschiebungPoly, ite_true, map_zero, verschiebungFun_coeff_zero]
· rw [verschiebungPoly, verschiebungFun_coeff_succ, if_neg n.succ_ne_zero, aeval_X,
add_tsub_cancel_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 82,
"column": 2
} | {
"line": 85,
"column": 28
} | {
"line": 87,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\n⊢ (aeval x.coeff) (verschiebungPoly n) = x.verschiebungFun.coeff n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semi... | [] | rcases n with - | n
· simp only [verschiebungPoly, ite_true, map_zero, verschiebungFun_coeff_zero]
· rw [verschiebungPoly, verschiebungFun_coeff_succ, if_neg n.succ_ne_zero, aeval_X,
add_tsub_cancel_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfection | {
"line": 420,
"column": 10
} | {
"line": 420,
"column": 24
} | {
"line": 420,
"column": 24
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u₁\ninst✝⁴ : CommSemiring R\ninst✝³ : CharP R p\nP : Type u₃\ninst✝² : CommSemiring P\ninst✝¹ : CharP P p\ninst✝ : PerfectRing P p\nf : P →+* R\ng : P ≃+* Perfection R p\nhfg : (Perfection.lift p P R) f = ↑g\ny : ℕ → R\nhy : ∀ (n : ℕ), y (n + 1) ^ p = y n\nx... | [] | simp [hfg, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Perfection | {
"line": 420,
"column": 10
} | {
"line": 420,
"column": 24
} | {
"line": 420,
"column": 24
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u₁\ninst✝⁴ : CommSemiring R\ninst✝³ : CharP R p\nP : Type u₃\ninst✝² : CommSemiring P\ninst✝¹ : CharP P p\ninst✝ : PerfectRing P p\nf : P →+* R\ng : P ≃+* Perfection R p\nhfg : (Perfection.lift p P R) f = ↑g\ny : ℕ → R\nhy : ∀ (n : ℕ), y (n + 1) ^ p = y n\nx... | [] | simp [hfg, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Perfection | {
"line": 420,
"column": 10
} | {
"line": 420,
"column": 24
} | {
"line": 420,
"column": 24
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u₁\ninst✝⁴ : CommSemiring R\ninst✝³ : CharP R p\nP : Type u₃\ninst✝² : CommSemiring P\ninst✝¹ : CharP P p\ninst✝ : PerfectRing P p\nf : P →+* R\ng : P ≃+* Perfection R p\nhfg : (Perfection.lift p P R) f = ↑g\ny : ℕ → R\nhy : ∀ (n : ℕ), y (n + 1) ^ p = y n\nx... | [] | simp [hfg, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 184,
"column": 18
} | {
"line": 184,
"column": 55
} | {
"line": 186,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (ghostComponent (n + 1)) (verschiebung (mk p x)) = ↑p * (MvPolynomial.eval x) (wittPolynomial p ℤ n)",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonA... | [] | rw [ghostComponent_verschiebung]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 184,
"column": 18
} | {
"line": 184,
"column": 55
} | {
"line": 186,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (ghostComponent (n + 1)) (verschiebung (mk p x)) = ↑p * (MvPolynomial.eval x) (wittPolynomial p ℤ n)",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonA... | [] | rw [ghostComponent_verschiebung]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Identities | {
"line": 57,
"column": 73
} | {
"line": 62,
"column": 42
} | {
"line": 64,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\n⊢ (↑p ^ i).coeff i = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"HMul.hMul",
"WittVector.versc... | [] | by
induction i with
| zero => simp only [one_coeff_zero, pow_zero]
| succ i h =>
rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP,
verschiebung_coeff_succ, h, one_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Complete | {
"line": 91,
"column": 6
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"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⊢ (⇑verschiebung)^[n] (x.shift n) =\n (⇑verschiebung)^[n] ((⇑frobenius)^[n] ((⇑(frobeniusEquiv p k).symm)^[n] (x.shift n)))",
"ppTerm": "?m... | [] | congr
rw [← Function.comp_apply (f := frobenius^[n]), ← Function.Commute.comp_iterate]
· rw [← WittVector.frobeniusEquiv_apply, ← RingEquiv.coe_trans]
simp
· rw [Function.Commute, Function.Semiconj, ← WittVector.frobeniusEquiv_apply]
simp only [RingEquiv.apply_symm_apply, RingEquiv.sym... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Complete | {
"line": 91,
"column": 6
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"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⊢ (⇑verschiebung)^[n] (x.shift n) =\n (⇑verschiebung)^[n] ((⇑frobenius)^[n] ((⇑(frobeniusEquiv p k).symm)^[n] (x.shift n)))",
"ppTerm": "?m... | [] | congr
rw [← Function.comp_apply (f := frobenius^[n]), ← Function.Commute.comp_iterate]
· rw [← WittVector.frobeniusEquiv_apply, ← RingEquiv.coe_trans]
simp
· rw [Function.Commute, Function.Semiconj, ← WittVector.frobeniusEquiv_apply]
simp only [RingEquiv.apply_symm_apply, RingEquiv.sym... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 73
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ dickson 2 0 (n + 2) = X ^ (n + 2)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Polynomial.C",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | [] | simp only [dickson_add_two, C_0, zero_mul, sub_zero]
rw [dickson_two_zero (n + 1), pow_add X (n + 1) 1, mul_comm, pow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 73
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ dickson 2 0 (n + 2) = X ^ (n + 2)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Polynomial.C",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | [] | simp only [dickson_add_two, C_0, zero_mul, sub_zero]
rw [dickson_two_zero (n + 1), pow_add X (n + 1) 1, mul_comm, pow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 60,
"column": 57
} | {
"line": 60,
"column": 69
} | {
"line": 60,
"column": 69
} | [
{
"pp": "case succ\nn : ℕ\nih : hermite n = (fun p ↦ X * p - derivative p)^[n] 1\n⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"c... | [
"case succ\nn : ℕ\nih : hermite n = (fun p ↦ X * p - derivative p)^[n] 1\n⊢ X * hermite n - derivative (hermite n) = X * hermite n - derivative (hermite n)"
] | hermite_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 86
} | {
"line": 57,
"column": 4
} | [
{
"pp": "case succ\nn : ℕ\nx : ℝ\nih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))\nderiv_gaussian : deriv (fun y ↦ Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2))\n⊢ deriv^[n + 1] (fun y ↦ Real.exp (-(y ^ 2 / 2))) x =\n (-1) ^ ... | [
"case succ\nn : ℕ\nx : ℝ\nih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))\nderiv_gaussian : deriv (fun y ↦ Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2))\n⊢ (-1) ^ n *\n ((aeval x) (derivative (hermite n)) * Real.exp (-(x ^ 2 / ... | rw [Function.iterate_succ_apply', ih, deriv_const_mul_field, deriv_fun_mul, pow_succ (-1 : ℝ),
deriv_gaussian, hermite_succ, map_sub, map_mul, aeval_X, Polynomial.deriv_aeval] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 45
} | {
"line": 64,
"column": 2
} | [
{
"pp": "n : ℕ\nx : ℝ\n⊢ (aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun y ↦ Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2))",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Semiring.toModule",
"HMul.hMul",
"Real.dens... | [
"n : ℕ\nx : ℝ\n⊢ (aeval x) (hermite n) =\n (-1) ^ n * ((-1) ^ n * (aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2))) / Real.exp (-(x ^ 2 / 2))"
] | rw [deriv_gaussian_eq_hermite_mul_gaussian] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Morse | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 85
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulSemiringAction G S\ninst✝² : SMulCommClass G R S\nf : R[X]\ninst✝¹ : DecidableEq ↑(f.rootSet S)\nhf : (map (algebraMap R S) f).Splits\np... | [
"case neg\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulSemiringAction G S\ninst✝² : SMulCommClass G R S\nf : R[X]\ninst✝¹ : DecidableEq ↑(f.rootSet S)\nhf : (map (algebraMap R S) f).Splits\np : Ideal S\n... | simp only [Equiv.swap_apply_def, MulAction.toPermHom_apply, MulAction.toPerm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 65,
"column": 75
} | {
"line": 66,
"column": 15
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\n⊢ expand 1 ⋯ = AlgHom.id R R⟦X⟧",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.id",
"MvPowerSeries",
"... | [] | by
simp [expand] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 52,
"column": 65
} | {
"line": 59,
"column": 35
} | {
"line": 61,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ (coeff n) (X * largeSchroderSeries) = (n - 1).largeSchroder",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Finset.Nat.sum_antidiagonal_eq_sum_range_succ",
"Eq.mpr",
"NonAssocSemir... | [] | by
simp only [coeff_mul, coeff_largeSchroderSeries,
Nat.sum_antidiagonal_eq_sum_range_succ (coeff · X * largeSchroder ·),
succ_eq_add_one]
simp only [coeff_X, ite_mul, one_mul, zero_mul, sum_ite_eq', mem_range, lt_add_iff_pos_left,
ite_eq_left_iff, not_lt, nonpos_iff_eq_zero]
rintro rfl
simp_all onl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 127,
"column": 2
} | {
"line": 129,
"column": 18
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\np : Ideal R\nhp : p.IsPrime\neqbot : ¬p = ⊥\n⊢ IsRegularLocalRing (Localization.AtPrime p)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"IsDedekindDomain.toIsDomain",
"OreLocalization.instAlgeb... | [] | · have := IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain
R eqbot (Localization.AtPrime p)
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 49,
"column": 2
} | {
"line": 80,
"column": 26
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nn : ℕ\nh : ∀ (L : ModuleCat R), Module.Finite R ↑L → Subsingleton (Ext M L n)\n⊢ HasProjectiveDimensionLT M n",
"ppTerm": "?m.17",
"assigned": true,
"usedConst... | [] | induction n generalizing M with
| zero =>
have : Subsingleton (M ⟶ M) := Ext.homEquiv₀.subsingleton_congr.mp (h M ‹_›)
have : Limits.IsZero M := (Limits.IsZero.iff_id_eq_zero M).mpr (Subsingleton.eq_zero (𝟙 M))
exact this.hasProjectiveDimensionLT_zero
| succ n hn =>
rcases Module.exists_finite_pres... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 49,
"column": 2
} | {
"line": 80,
"column": 26
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nn : ℕ\nh : ∀ (L : ModuleCat R), Module.Finite R ↑L → Subsingleton (Ext M L n)\n⊢ HasProjectiveDimensionLT M n",
"ppTerm": "?m.17",
"assigned": true,
"usedConst... | [] | induction n generalizing M with
| zero =>
have : Subsingleton (M ⟶ M) := Ext.homEquiv₀.subsingleton_congr.mp (h M ‹_›)
have : Limits.IsZero M := (Limits.IsZero.iff_id_eq_zero M).mpr (Subsingleton.eq_zero (𝟙 M))
exact this.hasProjectiveDimensionLT_zero
| succ n hn =>
rcases Module.exists_finite_pres... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 49,
"column": 2
} | {
"line": 80,
"column": 26
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nn : ℕ\nh : ∀ (L : ModuleCat R), Module.Finite R ↑L → Subsingleton (Ext M L n)\n⊢ HasProjectiveDimensionLT M n",
"ppTerm": "?m.17",
"assigned": true,
"usedConst... | [] | induction n generalizing M with
| zero =>
have : Subsingleton (M ⟶ M) := Ext.homEquiv₀.subsingleton_congr.mp (h M ‹_›)
have : Limits.IsZero M := (Limits.IsZero.iff_id_eq_zero M).mpr (Subsingleton.eq_zero (𝟙 M))
exact this.hasProjectiveDimensionLT_zero
| succ n hn =>
rcases Module.exists_finite_pres... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 467,
"column": 4
} | {
"line": 487,
"column": 86
} | {
"line": 488,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng✝ : A⟦X⟧\nI✝ : Ideal A\ng : A[X]\nI : Ideal A\nH : g.IsDistinguishedAt I\ninst✝ : IsAdicComplete I A\nf : A[X] ⧸ Ideal.span {g}\n⊢ (⇑(Ideal.Quotient.mk (Ideal.span {g})) ∘ ⇑⋯.mod')\n ((↑↑(Ideal.quotientMapₐ (Ideal.span {↑g}) (Polynomial.coeToPowerSeries.algHom A... | [] | rcases subsingleton_or_nontrivial A with _ | _
· have : Subsingleton A[X] := inferInstance
have : Subsingleton (A[X] ⧸ Ideal.span {g}) := Quot.Subsingleton
exact Subsingleton.elim _ _
have hI : I ≠ ⊤ := by
rintro rfl
exact not_subsingleton _ ‹IsAdicComplete ⊤ A›.toIsHausdorff.subsingleto... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 467,
"column": 4
} | {
"line": 487,
"column": 86
} | {
"line": 488,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng✝ : A⟦X⟧\nI✝ : Ideal A\ng : A[X]\nI : Ideal A\nH : g.IsDistinguishedAt I\ninst✝ : IsAdicComplete I A\nf : A[X] ⧸ Ideal.span {g}\n⊢ (⇑(Ideal.Quotient.mk (Ideal.span {g})) ∘ ⇑⋯.mod')\n ((↑↑(Ideal.quotientMapₐ (Ideal.span {↑g}) (Polynomial.coeToPowerSeries.algHom A... | [] | rcases subsingleton_or_nontrivial A with _ | _
· have : Subsingleton A[X] := inferInstance
have : Subsingleton (A[X] ⧸ Ideal.span {g}) := Quot.Subsingleton
exact Subsingleton.elim _ _
have hI : I ≠ ⊤ := by
rintro rfl
exact not_subsingleton _ ‹IsAdicComplete ⊤ A›.toIsHausdorff.subsingleto... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 86
} | {
"line": 572,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\nf g : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) g ≠ 0\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\n⊢ f = g * IsWeierstrassDivisorAt.div ⋯ f + ↑(IsWeierstrassDivisorAt.mod ⋯ f)",
"ppTerm": "?m.38",
"assigned": true,
"usedCo... | [] | exact ((IsWeierstrassDivisor.of_map_ne_zero hg).isWeierstrassDivisionAt_div_mod f).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 46,
"column": 8
} | {
"line": 46,
"column": 30
} | {
"line": 46,
"column": 31
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nH : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range\nhker : RingHom.ker f ≤ nilradical R\nq q' : PrimeSpectrum S\nhqq' : comap f q = comap f q'\nx : S\nn : ℕ\nhn : n > 0\ny : R\nhy : f y = x ^ n\n⊢ f y ∈ q.asIdeal ↔ f y ∈ q'.asIdeal"... | [
"R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nH : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range\nhker : RingHom.ker f ≤ nilradical R\nq q' : PrimeSpectrum S\nhqq' : (comap f q).asIdeal = (comap f q').asIdeal\nx : S\nn : ℕ\nhn : n > 0\ny : R\nhy : f y = x ^ n\n⊢ f y ∈ q.asIdeal ↔ f y ∈ q'.... | PrimeSpectrum.ext_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 156,
"column": 12
} | {
"line": 156,
"column": 17
} | {
"line": 156,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [
"R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : IsScalarTower ... | infeq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 69,
"column": 65
} | {
"line": 69,
"column": 90
} | {
"line": 73,
"column": 0
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : ↥T\nm n : ℕ\n⊢ take m x = take n x ↔ min m (↑x).length = min n (↑x).length",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.Descriptive.Tree.0.Descriptive.Tree.take_eq_take._simp_1_1",
"congrArg",
"Membe... | [] | by simp [Subtype.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Lists | {
"line": 245,
"column": 2
} | {
"line": 257,
"column": 27
} | {
"line": 259,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Lists α → Sort u_2\nD : Lists' α true → Sort u_3\nC0 : (a : α) → C (atom a)\nC1 : (l : Lists' α true) → D l → C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists α) → (l : Lists' α true) → C a → D l → D (Lists'.cons a l)\n⊢ ((l : Lists α) → C l) ×' ((l : Lists' α true) → D l)",
"ppTerm"... | [] | suffices ∀ {b} (l : Lists' α b),
PProd (C ⟨_, l⟩)
(match b, l with
| true, l => D l
| false, _ => PUnit)
by exact ⟨fun ⟨b, l⟩ => (this _).1, fun l => (this l).2⟩
intro b l
induction l with
| atom => exact ⟨C0 _, ⟨⟩⟩
| nil => exact ⟨C1 _ D0, D0⟩
| cons' a l IH₁ IH =>
have ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Lists | {
"line": 245,
"column": 2
} | {
"line": 257,
"column": 27
} | {
"line": 259,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Lists α → Sort u_2\nD : Lists' α true → Sort u_3\nC0 : (a : α) → C (atom a)\nC1 : (l : Lists' α true) → D l → C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists α) → (l : Lists' α true) → C a → D l → D (Lists'.cons a l)\n⊢ ((l : Lists α) → C l) ×' ((l : Lists' α true) → D l)",
"ppTerm"... | [] | suffices ∀ {b} (l : Lists' α b),
PProd (C ⟨_, l⟩)
(match b, l with
| true, l => D l
| false, _ => PUnit)
by exact ⟨fun ⟨b, l⟩ => (this _).1, fun l => (this l).2⟩
intro b l
induction l with
| atom => exact ⟨C0 _, ⟨⟩⟩
| nil => exact ⟨C1 _ D0, D0⟩
| cons' a l IH₁ IH =>
have ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 22
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case refine_1.inl\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\nh : lfpApprox f x 0 = x\n⊢ f x = x ∨ 0 = 0",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"PartialOrder.toPreorder",
... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 22
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case refine_1.inl\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\nh : lfpApprox f x 0 = x\n⊢ f x = x ∨ 0 = 0",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"PartialOrder.toPreorder",
... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 22
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case refine_1.inl\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\nh : lfpApprox f x 0 = x\n⊢ f x = x ∨ 0 = 0",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"PartialOrder.toPreorder",
... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 153,
"column": 4
} | {
"line": 160,
"column": 75
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case refine_2.inr\nb o✝ o : Ordinal.{u_1}\nho : o ≠ 0\nIH : List.Pairwise (fun x1 x2 ↦ x1 > x2) (map Prod.fst (CNF b (o % b ^ log b o)))\nhb : 1 < b\n⊢ List.Pairwise (fun x1 x2 ↦ x1 > x2) (map Prod.fst (CNF b o))",
"ppTerm": "?refine_2.inr",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | · obtain hob | hbo := lt_or_ge o b
· rw [CNF.of_lt ho hob]
exact pairwise_singleton _ _
· rw [CNF.ne_zero ho, map_cons, pairwise_cons]
refine ⟨fun a H ↦ ?_, IH⟩
rw [mem_map] at H
rcases H with ⟨⟨a, a'⟩, H, rfl⟩
exact (fst_le_log H).trans_lt (log_mod_opow_log_lt_log_se... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Lists | {
"line": 287,
"column": 77
} | {
"line": 311,
"column": 26
} | {
"line": 313,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lists'.toList",
"congrArg",
"HEq.refl",
"List.Mem.tail",
"False.elim",
"forall_eq_or_imp._simp_1... | [] | by
let trans := fun l₁ : Lists α => ∀ ⦃l₂ l₃⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃
suffices PProd (∀ l₁, trans l₁) (∀ (l : Lists' α true), ∀ l' ∈ l.toList, trans l') by exact this.1
apply inductionMut
· intro a l₂ l₃ h₁ h₂
rwa [← equiv_atom.1 h₁] at h₂
· intro l₁ IH l₂ l₃ h₁ h₂
obtain - | l₂ := id h₁
· ex... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Lists | {
"line": 339,
"column": 44
} | {
"line": 339,
"column": 56
} | {
"line": 340,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ : Lists' α false\nl₂ : Lists' α true\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.239",
"assigned": true,
"usedConstants": [
"False",
"HEq.refl",
"False.elim",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Lists | {
"line": 340,
"column": 44
} | {
"line": 340,
"column": 56
} | {
"line": 341,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ : Lists' α true\nl₂ : Lists' α false\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.292",
"assigned": true,
"usedConstants": [
"False",
"HEq.refl",
"False.elim",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 218,
"column": 93
} | {
"line": 229,
"column": 20
} | {
"line": 231,
"column": 0
} | [
{
"pp": "b : Ordinal.{u_1}\nhb : b ≤ 1\no : Ordinal.{u_1}\n⊢ coeff b o = single 0 o",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"False",
"Finsupp.single_eq_same",
"Finsupp.ext",
"List.map_cons",
"congrArg",
... | [] | by
ext a
obtain rfl | ho := eq_or_ne o 0
· simp
· obtain rfl | ha := eq_or_ne a 0
· apply coeff_of_mem_CNF
rw [CNF.of_le_one hb ho]
simp
· rw [single_eq_of_ne ha]
apply coeff_of_notMem_CNF
rw [CNF.of_le_one hb ho]
simpa using ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 349,
"column": 12
} | {
"line": 349,
"column": 14
} | {
"line": 349,
"column": 15
} | [
{
"pp": "b o : Ordinal.{u_1}\na : Ordinal.{u_1} × Ordinal.{u_1}\n⊢ a ∈ CNF b o → ∀ (b_1 : Ordinal.{u_1}), b ^ a.1 * (coeff b o) a.1 + b_1 = b ^ a.1 * a.2 + b_1",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Membership.mem",
"List",
"List.instMembership",
"Ordinal.... | [
"b o : Ordinal.{u_1}\na : Ordinal.{u_1} × Ordinal.{u_1}\nha : a ∈ CNF b o\n⊢ ∀ (b_1 : Ordinal.{u_1}), b ^ a.1 * (coeff b o) a.1 + b_1 = b ^ a.1 * a.2 + b_1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.SetTheory.Ordinal.Topology | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 26
} | {
"line": 141,
"column": 0
} | [
{
"pp": "case refine_2\ns : Set Ordinal.{u}\nH :\n ∀ {o : Ordinal.{u}},\n o ≠ 0 →\n ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s\nι : Type u\nhι : Nonempty ι\nf : ι → Ordinal.{u}\nhf : ∀ (i : ι), f i ∈ s\n⊢ ∀ (i : Ordinal.{u}) (hi : i < ... | [] | exact fun i hi => hf _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 284,
"column": 54
} | {
"line": 317,
"column": 26
} | {
"line": 319,
"column": 0
} | [
{
"pp": "o₁ e₁ : ONote\nn₁ : ℕ+\na₁ : ONote\nh✝¹ : o₁ = e₁.oadd n₁ a₁\no₂ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh✝ : o₂ = e₂.oadd n₂ a₂\nh₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF\nh₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF\n⊢ ((namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).cmp (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)).Comp... | [] | by -- TODO: golf
rw [cmp]
have IHe := @cmp_compares _ _ h₁.fst h₂.fst
simp only [Ordering.Compares, gt_iff_lt] at IHe; revert IHe
cases cmp e₁ e₂
case lt => intro IHe; exact oadd_lt_oadd_1 h₁ IHe
case gt => intro IHe; exact oadd_lt_oadd_1 h₂ IHe
case eq =>
intro IHe; dsimp at IHe; subs... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 124,
"column": 40
} | {
"line": 124,
"column": 42
} | {
"line": 124,
"column": 42
} | [
{
"pp": "case mpr.inl\nf : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\na b : Ordinal.{u}\nha : veblenWith f b a = a\nhb : b < b + 1\n⊢ veblenWith f (↑⟨b, hb⟩) a = a",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"congrArg",
"Add... | [
"case mpr.inl\nf : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\na b : Ordinal.{u}\nha : veblenWith f b a = a\nhb : b < b + 1\n⊢ a = a"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 31
} | {
"line": 199,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\nhp : 0 < f 0\no : Ordinal.{u}\nho : IsSuccLimit o\na : Ordinal.{u}\nIH : ∀ b < o, veblenWith f b 0 ≤ a\n⊢ nfpFamily (fun x ↦ veblenWith f ↑x) 0 ≤ a",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"... | [
"f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\nhp : 0 < f 0\no : Ordinal.{u}\nho : IsSuccLimit o\na : Ordinal.{u}\nIH : ∀ b < o, veblenWith f b 0 ≤ a\nl : List ↑(Iio o)\n⊢ List.foldr (fun x ↦ veblenWith f ↑x) 0 l ≤ a"
] | apply nfpFamily_le fun l ↦ ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 67
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case inr.inl\nf : Ordinal.{u} → Ordinal.{u}\no₁ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ cmp (veblenWith f o₁ a) (veblenWith f o₁ b) =\n match cmp o₁ o₁ with\n | Ordering.eq => cmp a b\n | Ordering.lt => cmp a (veblenWith f o₁ b)\n | Ordering.gt => cmp (veblenWith f o₁ a) b",
"ppTerm": "?i... | [] | simp [(veblenWith_right_strictMono hf _).cmp_map_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 67
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case inr.inl\nf : Ordinal.{u} → Ordinal.{u}\no₁ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ cmp (veblenWith f o₁ a) (veblenWith f o₁ b) =\n match cmp o₁ o₁ with\n | Ordering.eq => cmp a b\n | Ordering.lt => cmp a (veblenWith f o₁ b)\n | Ordering.gt => cmp (veblenWith f o₁ a) b",
"ppTerm": "?i... | [] | simp [(veblenWith_right_strictMono hf _).cmp_map_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 67
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case inr.inl\nf : Ordinal.{u} → Ordinal.{u}\no₁ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ cmp (veblenWith f o₁ a) (veblenWith f o₁ b) =\n match cmp o₁ o₁ with\n | Ordering.eq => cmp a b\n | Ordering.lt => cmp a (veblenWith f o₁ b)\n | Ordering.gt => cmp (veblenWith f o₁ a) b",
"ppTerm": "?i... | [] | simp [(veblenWith_right_strictMono hf _).cmp_map_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.ZFC.Rank | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 41
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case a\nx y y✝ : ZFSet.{u_1}\n⊢ y✝ ∈ x ∨ y✝ ∈ y → y✝.rank < x.rank ∨ y✝.rank < y.rank",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"ZFSet",
"PartialOrder.toPreorder",
"Membership.mem",
"SemilatticeInf... | [] | apply Or.imp <;> apply rank_lt_of_mem | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 541,
"column": 19
} | {
"line": 541,
"column": 47
} | {
"line": 542,
"column": 2
} | [
{
"pp": "o : ONote\nx✝ : NF 0\nh₂ : o.NF\n⊢ (0 * o).NF",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"ONote.NF",
"ONote.instMul",
"HMul.hMul",
"ONote.oadd",
"ONote.instZero",
"ONote.zero",
"Eq.ndrec",
"Zero.toOfNat0",
"Eq.refl",
... | [] | by cases o <;> exact NF.zero | [anonymous] | Lean.Parser.Term.byTactic |
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