module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 33
} | {
"line": 368,
"column": 4
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set hI.SubDPIdeal\n⊢ ⨅ x, ⊤.carrier ⊓ ⨅ (_ : x ∈ S), x.carrier = ⨅ x, I ⊓ ↑(⨅ (_ : x ∈ S), ⟨x.carrier, ⋯⟩)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"DividedPowers.SubDPIdea... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set hI.SubDPIdeal\nJ : hI.SubDPIdeal\n⊢ ⊤.carrier ⊓ ⨅ (_ : J ∈ S), J.carrier = I ⊓ ↑(⨅ (_ : J ∈ S), ⟨J.carrier, ⋯⟩)"
] | apply iInf_congr (fun J ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.TotallySplit | {
"line": 175,
"column": 6
} | {
"line": 176,
"column": 94
} | {
"line": 177,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [
"n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝⁴ : CommRing... | have : Module.Finite (S × U) U :=
Module.Finite.of_surjective (Algebra.linearMap (S × U) U) (RingHom.snd S U).surjective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.TotallySplit | {
"line": 183,
"column": 4
} | {
"line": 186,
"column": 11
} | {
"line": 189,
"column": 4
} | [
{
"pp": "case succ.inr\nn : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S ... | [
"case succ.inr\nn : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\nin... | have : Module.rankAtStalk (R := S) U = n := by
ext p
simp only [Pi.natCast_def, Nat.cast_id]
grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Etale.Finite | {
"line": 181,
"column": 10
} | {
"line": 181,
"column": 45
} | {
"line": 181,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R Ω✝\nS : Type w\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Ω✝\ninst✝² : IsScalarTower R S Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nX : FintypeCat\nx : ↑(O... | [
"R : Type u\ninst✝⁹ : CommRing R\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R Ω✝\nS : Type w\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Ω✝\ninst✝² : IsScalarTower R S Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nX : FintypeCat\nx : ↑(Opposite.unop... | FintypeCat.equivEquivIso_apply_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 593,
"column": 20
} | {
"line": 598,
"column": 90
} | {
"line": 599,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn✝ : ℕ\nx✝ y✝ : B\nhx : x✝ ∈ J\nhy : y✝ ∈ J\n⊢ dpow hI f n✝ (x✝ + y✝) = ∑ k ... | [] | by
obtain ⟨a, ha, rfl⟩ := (mem_map_iff_of_surjective f hf).mp (hIJ ▸ hx)
obtain ⟨b, hb, rfl⟩ := (mem_map_iff_of_surjective f hf).mp (hIJ ▸ hy)
rw [← map_add, dpow_apply' hI hIf (I.add_mem ha hb), hI.dpow_add ha hb, map_sum,
Finset.sum_congr rfl]
exact fun k _ ↦ by rw [dpow_apply' hI hIf ha, dpow_a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Frobenius | {
"line": 63,
"column": 81
} | {
"line": 65,
"column": 11
} | {
"line": 67,
"column": 0
} | [
{
"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\nx : S\n⊢ (Ideal.Quotient.mk Q) (φ x) = (Ideal.Quotient.mk Q) x ^ Nat.card (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants... | [] | by
rw [← map_pow, Ideal.Quotient.eq]
exact H x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Frobenius | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 53
} | {
"line": 239,
"column": 0
} | [
{
"pp": "case intro\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nG : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulSemiringAction G S\ninst✝⁴ : SMulCommClass G R S\nQ : Ideal S\ninst✝³ : Finite G\ninst✝² : Algebra.IsInvariant R S G\ninst✝¹ : Q.IsPrime\ninst✝ : Finite (S... | [] | exact DFunLike.congr_fun hσ (Ideal.Quotient.mk Q x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 16
} | {
"line": 113,
"column": 6
} | [
{
"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))... | [
"case h\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)) n).coeff g ≠ ... | use (0, 0) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 39
} | {
"line": 258,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : AddCommMonoid R\nι : Type u_7\ninst✝ : Unique ι\nx : SummableFamily Γ R ι\n⊢ x.hsum = x default",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"congrArg",
"AddMonoid.toAddZeroClas... | [] | ext g
simp only [coeff_hsum, finsum_unique] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 39
} | {
"line": 258,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : AddCommMonoid R\nι : Type u_7\ninst✝ : Unique ι\nx : SummableFamily Γ R ι\n⊢ x.hsum = x default",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"congrArg",
"AddMonoid.toAddZeroClas... | [] | ext g
simp only [coeff_hsum, finsum_unique] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 69
} | {
"line": 457,
"column": 2
} | [
{
"pp": "Γ : 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\nt : SummableFamily ... | [
"Γ : 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\nt : SummableFamily Γ' V β\ng : ... | simp_rw [sum_vAddAntidiagonal_eq, Finset.smul_sum, Finset.sum_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 276,
"column": 61
} | {
"line": 292,
"column": 54
} | {
"line": 294,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\n⊢ PowerSeries.subst F.zeroX F.zeroX = F.zeroX",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"CommRing",
"Trans.trans",
"MvPowerSeries.instZero",
"Fintype.elems",
... | [] | by
calc
_ = F.toPowerSeries.subst ![0, F.toPowerSeries.subst ![0, PowerSeries.X]] := by
have : PowerSeries.HasSubst (subst ![0, PowerSeries.X (R := R)] F.toPowerSeries) := by
refine PowerSeries.HasSubst.of_constantCoeff_zero' ?_
rw [PowerSeries.constantCoeff, PowerSeries.X, constantCoeff_sub... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 701,
"column": 4
} | {
"line": 703,
"column": 61
} | {
"line": 704,
"column": 2
} | [
{
"pp": "case pos.refine_1\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n,... | [] | obtain ⟨hi, _, rfl⟩ := mem_antidiagonal.1 (mem_coe.1 hij)
exact lt_add_of_pos_left ij.2 <| lt_of_lt_of_le ((zero_lt_orderTop_iff h0).mp hx) <|
order_le_of_coeff_ne_zero <| Function.mem_support.mp hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 701,
"column": 4
} | {
"line": 703,
"column": 61
} | {
"line": 704,
"column": 2
} | [
{
"pp": "case pos.refine_1\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n,... | [] | obtain ⟨hi, _, rfl⟩ := mem_antidiagonal.1 (mem_coe.1 hij)
exact lt_add_of_pos_left ij.2 <| lt_of_lt_of_le ((zero_lt_orderTop_iff h0).mp hx) <|
order_le_of_coeff_ne_zero <| Function.mem_support.mp hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Henselian | {
"line": 240,
"column": 8
} | {
"line": 240,
"column": 12
} | {
"line": 241,
"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] := ⋯\nc : ℕ → R := ⋯\nhc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Henselian | {
"line": 230,
"column": 6
} | {
"line": 244,
"column": 25
} | {
"line": 246,
"column": 6
} | [
{
"pp": "R : 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.eva... | [
"R : 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 b f * (Pol... | have aux : ∀ m n, m ≤ n → c m ≡ c n [SMOD (I ^ m • ⊤ : Ideal R)] := by
intro m n hmn
rw [← Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one]
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
clear hmn
induction k with
| zero => rw [add_zero]
| succ k ih => ?_
rw ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.UFD | {
"line": 41,
"column": 48
} | {
"line": 41,
"column": 98
} | {
"line": 42,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : WfDvdMonoid R\nx : R\nhx : Prime x\np : Ideal R\ninst✝³ : p.IsPrime\nhxp : x ∉ p\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nhp : Submodule.IsPrincipal (map (algebraMap R S) p)\nthis : Disjoi... | [] | simp [Ideal.map_eq_bot_iff_of_injective hi, hpbot] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 64,
"column": 27
} | {
"line": 64,
"column": 55
} | {
"line": 65,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\nhp : IsLocalRing.maximalIdeal R ∈ I.minimalPrimes\nq : Ideal R\nh₁ : q.IsPrime\nh₂ : q < IsLocalRing.maximalIdeal R\nthis : q.height = 0\n⊢ q.height < ↑1",
"ppTerm":... | [] | rw [this]; exact zero_lt_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 64,
"column": 27
} | {
"line": 64,
"column": 55
} | {
"line": 65,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\nhp : IsLocalRing.maximalIdeal R ∈ I.minimalPrimes\nq : Ideal R\nh₁ : q.IsPrime\nh₂ : q < IsLocalRing.maximalIdeal R\nthis : q.height = 0\n⊢ q.height < ↑1",
"ppTerm":... | [] | rw [this]; exact zero_lt_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Pure | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 36
} | {
"line": 93,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\nx : R\nhx : x ∈ I\nh : x ∈ I * span {x}\n⊢ ∃ y ∈ I, x = x * y",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Mem... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\nx : R\nhx : x ∈ I\nh : ∃ z ∈ I, z * x = x\n⊢ ∃ y ∈ I, x = x * y"
] | Ideal.mem_mul_span_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Pure | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 56
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\nx : R\nhx : x ∈ I\np : ↑(zeroLocus ↑I)\n⊢ (algebraMap R (Localization.AtPrime (↑p).asIdeal)) x = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Ideal.exists_eq_m... | [] | exact Ideal.le_ker_atPrime_of_forall_exists_eq_mul
(fun x hx ↦ Ideal.exists_eq_mul_of_pure hx) p.2 hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
"line": 83,
"column": 10
} | {
"line": 83,
"column": 33
} | {
"line": 83,
"column": 33
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace G\ninst✝ : TotallyDisconnectedSpace G\nH : ClosedSubgroup G\ng : G\nhg : g ∈ sInf {N | IsOpen ↑N ∧ ↑H ≤ N}\nhg_not : g ∉ ↑H\nU : Set G := (g • ↑H)ᶜ\nUOpen : IsOpen U\neinU : 1 ∈ U\nN : Open... | [
"G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace G\ninst✝ : TotallyDisconnectedSpace G\nH : ClosedSubgroup G\ng : G\nhg : g ∈ sInf {N | IsOpen ↑N ∧ ↑H ≤ N}\nhg_not : g ∉ ↑H\nU : Set G := (g • ↑H)ᶜ\nUOpen : IsOpen U\neinU : 1 ∈ U\nN : OpenNormalSubgro... | ← eq_mul_inv_iff_mul_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 38
} | {
"line": 30,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝ : IsPrincipalIdealRing (R ⧸ P)\nthis : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime\n⊢ (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal",
"ppTerm": "?m.105",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝ : IsPrincipalIdealRing (R ⧸ P)\nthis : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime\n⊢ Ideal.map (Ideal.Quotient.mk P) I ≠ ⊥"
] | refine IsPrime.to_maximal_ideal ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 32,
"column": 2
} | {
"line": 33,
"column": 38
} | {
"line": 34,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝ : IsPrincipalIdealRing (R ⧸ P)\nthis : (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal\n⊢ I.IsMaximal",
"ppTerm": "?m.86",
"assigned": true,
"usedCon... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝¹ : IsPrincipalIdealRing (R ⧸ P)\nthis✝ : (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal\nthis : (Ideal.comap (Ideal.Quotient.mk P) (Ideal.map (Ideal.Quotient.mk P) I)).IsMa... | have := Ideal.comap_isMaximal_of_surjective (Ideal.Quotient.mk P) Ideal.Quotient.mk_surjective
(K := I.map (Ideal.Quotient.mk P)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 10
} | {
"line": 325,
"column": 6
} | [
{
"pp": "case e'_4\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ ↑(Submodule.spanFinrank I) = Cardinal.toENat (Submodule.spanRank I)",
"ppT... | [
"case e'_4\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ Cardinal.toENat (Submodule.spanRank I) = ↑(Submodule.spanFinrank I)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 344,
"column": 33
} | {
"line": 350,
"column": 9
} | {
"line": 351,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis✝ : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nthis : Set.SurjOn ⇑(Q... | [] | by
refine le_trans h (hs ▸ ?_)
norm_cast
have : (Submodule.FG.finite_generators hI).toFinset.card = I.spanFinrank := by
rw [← Set.ncard_eq_toFinset_card (hs := Submodule.FG.finite_generators hI)]
exact Submodule.FG.generators_ncard hI
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 12
} | {
"line": 373,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nh : I ≤ Ring.jacobson R\na✝ : Nontrivial R\n⊢ ∀ ⦃m : Ideal R⦄, m.IsMaximal → ↑m.height ≤ ringKrullDim (R ⧸ I) + ↑(Submodule.spanFinrank I)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"CommSemirin... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nh : I ≤ Ring.jacobson R\na✝ : Nontrivial R\nm : Ideal R\nhm : m.IsMaximal\n⊢ ↑m.height ≤ ringKrullDim (R ⧸ I) + ↑(Submodule.spanFinrank I)"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 442,
"column": 33
} | {
"line": 447,
"column": 49
} | {
"line": 448,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [] | by
rw [← heq, ← heq']
apply le_trans h
norm_cast
refine le_trans (Finset.card_union_le _ _) (add_le_add Finset.card_image_le ?_)
rw [← himgo, Finset.card_image_of_injOn hinj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Lasker | {
"line": 195,
"column": 8
} | {
"line": 195,
"column": 19
} | {
"line": 195,
"column": 20
} | [
{
"pp": "case neg\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nS : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl\nf : M →ₗ[R] Loc... | [
"case neg\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nS : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl\nf : M →ₗ[R] LocalizedModule... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalIso | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 25
} | {
"line": 99,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh✝ : Ideal.span (Set.range f) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\... | [] | exact .of_algEquiv e₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 17
} | {
"line": 61,
"column": 18
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMax... | [
"case h\nR : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Alge... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 65
} | {
"line": 65,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → ... | [] | simpa using ⟨_, _, x.2, rfl⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 65
} | {
"line": 65,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → ... | [] | simpa using ⟨_, _, x.2, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 65
} | {
"line": 65,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → ... | [] | simpa using ⟨_, _, x.2, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 8
} | {
"line": 130,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Etale R S\nβ : S\nhadj : R[β] = ⊤\nh : (minpoly R β).Monic\n⊢ (aeval ((residue S) β))... | [
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Etale R S\nβ : S\nhadj : R[β] = ⊤\nh : (minpoly R β).Monic\n⊢ (map (residue R) (minpoly R β)).nat... | · rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.EulerIdentity | {
"line": 49,
"column": 42
} | {
"line": 53,
"column": 44
} | {
"line": 55,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nn : ℕ\ni : σ\nh : φ.IsHomogeneous n\n⊢ ((pderiv i) φ).IsHomogeneous (n - 1)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"MvPoly... | [] | by
obtain _ | n := n
· rw [← totalDegree_zero_iff_isHomogeneous, totalDegree_eq_zero_iff_eq_C] at h
rw [h, pderiv_C]; apply isHomogeneous_zero
· exact IsWeightedHomogeneous.pderiv h rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.Expand | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 8
} | {
"line": 39,
"column": 4
} | [
{
"pp": "case succ\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\np : ℕ\ninst✝ : ExpChar R p\nf : MvPolynomial σ R\nk : ℕ\nn_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k)) f) = f ^ p ^ k\n⊢ (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1))) f) = f ^ p ^ (k + 1)",
"ppTerm": "?succ",
... | [
"case succ\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\np : ℕ\ninst✝ : ExpChar R p\nf : MvPolynomial σ R\nk : ℕ\nn_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k)) f) = f ^ p ^ k\n⊢ f ^ p ^ (k + 1) = (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1))) f)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 68
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case inl\nn m : ℕ\nhnm : n ≤ m\nt s : Fin n → ℕ\nhe : (accumulate n m) t = (accumulate n m) s\ni : Fin n\nh : ↑i + 1 < m\nthis : (accumulate n m) s ⟨↑i, ⋯⟩ = s ⟨↑i, ⋯⟩ + (accumulate n m) s ⟨↑i + 1, h⟩\n⊢ t i = s i",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Nat.instIsOrd... | [] | rwa [← he, accumulate_rec i.2 h t, add_right_cancel_iff] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 258,
"column": 20
} | {
"line": 258,
"column": 77
} | {
"line": 258,
"column": 77
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : LinearOrder σ\np : MvPolynomial σ R\nhp : p.IsSymmetric\nh0 : p ≠ 0\ni j : σ\nhle : i ≤ j\nhlt : (ofLex (supDegree (⇑toLex) p)) i < (ofLex (supDegree (⇑toLex) p)) j\nk : σ\nhk : (fun x1 x2 ↦ x1 < x2) k i\n| (Finsupp.mapDomain (⇑(Equiv.swap i ... | [
"σ : Type u_1\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : LinearOrder σ\np : MvPolynomial σ R\nhp : p.IsSymmetric\nh0 : p ≠ 0\ni j : σ\nhle : i ≤ j\nhlt : (ofLex (supDegree (⇑toLex) p)) i < (ofLex (supDegree (⇑toLex) p)) j\nk : σ\nhk : (fun x1 x2 ↦ x1 < x2) k i\n| (Finsupp.mapDomain (⇑(Equiv.swap i j)) (ofLex (... | ← Equiv.swap_apply_of_ne_of_ne hk.ne (hk.trans_le hle).ne | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 233,
"column": 58
} | {
"line": 233,
"column": 92
} | {
"line": 234,
"column": 4
} | [
{
"pp": "σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\n⊢ (-1) ^ (k + 1) * ∑ x ∈ NewtonIdentities.pairs σ k, NewtonIdentities.weight σ R k x = 0",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZero... | [
"σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\n⊢ (-1) ^ (k + 1) * 0 = 0"
] | NewtonIdentities.weight_sum σ R k, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 935,
"column": 8
} | {
"line": 939,
"column": 40
} | {
"line": 940,
"column": 8
} | [
{
"pp": "case mpr.refine_1\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def... | [
"case h\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := ⋯\nX_def : X = {f | Valued.v f < embedding ↑d}\nx : K⟮X⟯\nhx ... | use Units.mk0 (Valued.v.restrict (x : K⸨X⸩)) (by
simp only [ne_eq, map_eq_zero]
intro h
simp only [h, map_zero] at hx
exact Units.ne_zero _ hx.symm) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 8
} | {
"line": 253,
"column": 4
} | [
{
"pp": "case succ\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : ExpChar R p\nf : MvPowerSeries σ R\nk : ℕ\nn_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k) ⋯) f) = f ^ p ^ k\n⊢ (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1)) ⋯) f) = f ^ p ^ (k + 1)",
"ppTerm... | [
"case succ\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : ExpChar R p\nf : MvPowerSeries σ R\nk : ℕ\nn_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k) ⋯) f) = f ^ p ^ k\n⊢ f ^ p ^ (k + 1) = (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1)) ⋯) f)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.LaurentSeries | {
"line": 946,
"column": 10
} | {
"line": 946,
"column": 27
} | {
"line": 946,
"column": 28
} | [
{
"pp": "case mpr.refine_2\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := ⋯\nX_def : X = {f | Valued.v f < embed... | [
"case mpr.refine_2\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def : X = {f | ... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Morita.Matrix | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 63
} | {
"line": 171,
"column": 4
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Submodule",
"Matrix.scalar",
"instHSMul"... | [] | simpa [-SetLike.coe_mem] using (v i).2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Morita.Matrix | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 63
} | {
"line": 171,
"column": 4
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Submodule",
"Matrix.scalar",
"instHSMul"... | [] | simpa [-SetLike.coe_mem] using (v i).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Morita.Matrix | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 63
} | {
"line": 171,
"column": 4
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Submodule",
"Matrix.scalar",
"instHSMul"... | [] | simpa [-SetLike.coe_mem] using (v i).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.NoetherNormalization | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 51
} | {
"line": 119,
"column": 52
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\nhv : v ∈ f.support\nhw : w ∈ f.support\nne : v ≠ w\n⊢ degreeOf 0 ((T f) ((MvPolynomial.monomial v) (MvPolynomial.coeff v f))) ≠\n degreeOf 0 ((T f) ((MvPolynomial.monomial w) (MvPolynomial.coeff w f)))",
... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\nhv : v ∈ f.support\nhw : w ∈ f.support\nne : v ≠ w\n⊢ ∑ i, r i * v i ≠ degreeOf 0 ((T f) ((MvPolynomial.monomial w) (MvPolynomial.coeff w f)))"
] | degreeOf_zero_t _ _ <| mem_support_iff.mp hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 175,
"column": 50
} | {
"line": 175,
"column": 75
} | {
"line": 175,
"column": 75
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : NeZero p\nn : ℕ\n⊢ ((map (Int.castRingHom R)) (W_ ℤ n)).vars ⊆ range (n + 1)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"wittPolynomial",
"Nat.instMulZeroClass",
... | [
"p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : NeZero p\nn : ℕ\n⊢ ((map (Int.castRingHom R)) (W_ ℤ n)).vars ⊆ (W_ ℤ n).vars"
] | ← wittPolynomial_vars p ℤ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Teichmuller | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 6
} | {
"line": 40,
"column": 2
} | [
{
"pp": "case succ\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : CharP (R ⧸ I) p\nx : Perfection (R ⧸ I) p\nm : ℕ\n⊢ x.teichmullerAux (m + 1) ≡ x.teichmullerAux (m + 1 + 1) [SMOD I ^ (m + 1)]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [
"case succ\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : CharP (R ⧸ I) p\nx : Perfection (R ⧸ I) p\nm : ℕ\n⊢ x.teichmullerAux (m + 1 + 1) ≡ x.teichmullerAux (m + 1) [SMOD I ^ (m + 1)]"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.WittVector.Defs | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 51
} | {
"line": 292,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ\nn : ℕ\n⊢ constantCoeff (z • X 0) = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Finsupp.instAddZeroClass",
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
"instHSMul",
"Non... | [] | simp only [smul_zero, map_zsmul, constantCoeff_X] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 10
} | {
"line": 304,
"column": 2
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nφ : ℕ → MvPolynomial (idx × ℕ) ℤ\nh : ∀ (n : ℕ), (bind₁ φ) (W_ ℤ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℤ n)) Φ\n⊢ φ = wittStructureInt p Φ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Int",
... | [
"p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nφ : ℕ → MvPolynomial (idx × ℕ) ℤ\nh : ∀ (n : ℕ), (bind₁ φ) (W_ ℤ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℤ n)) Φ\nk : ℕ\n⊢ φ k = wittStructureInt p Φ k"
] | funext k | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 181,
"column": 7
} | {
"line": 181,
"column": 15
} | {
"line": 182,
"column": 7
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (fun i ↦ (MvPolynomial.eval x) (verschiebungPoly i)) = (verschiebung (mk p x)).coeff",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroClass",
"WittVe... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\nk : ℕ\n⊢ (MvPolynomial.eval x) (verschiebungPoly k) = (verschiebung (mk p x)).coeff k"
] | funext k | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.RingTheory.WittVector.MulP | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 45
} | {
"line": 65,
"column": 45
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR : Type u_2\n_Rcr : CommRing R\nx : 𝕎 R\n⊢ (x * ↑n).coeff = fun n_1 ↦ (aeval x.coeff) (wittMulN p n n_1)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"HMul.hMul",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR : Type u_2\n_Rcr : CommRing R\nx : 𝕎 R\nk : ℕ\n⊢ (x * ↑n).coeff k = (aeval x.coeff) (wittMulN p n k)"
] | funext k | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 124,
"column": 54
} | {
"line": 124,
"column": 79
} | {
"line": 124,
"column": 79
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx y : 𝕎 R\nh : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0\nP : ℕ → Prop := fun n ↦ y.coeff n = 0\nthis : DecidablePred P\nz : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n\nhx : select P z = x\nhy : select (fun i ↦ ¬P i) z =... | [
"p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx y : 𝕎 R\nh : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0\nP : ℕ → Prop := fun n ↦ y.coeff n = 0\nthis : DecidablePred P\nz : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n\nhx : select P z = x\nhy : select (fun i ↦ ¬P i) z = y\n⊢ z.coef... | select_add_select_not P z | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 334,
"column": 2
} | {
"line": 354,
"column": 57
} | {
"line": 357,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\nhg : IsPoly₂ p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R), f x y =... | [] | obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq' p φ ψ]
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x (0, i)⟩) (mk p fun i => ⟨x (1, i)⟩) k
simp only [ghostComponent_apply, aeval_eq_eval₂H... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 334,
"column": 2
} | {
"line": 354,
"column": 57
} | {
"line": 357,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\nhg : IsPoly₂ p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R), f x y =... | [] | obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq' p φ ψ]
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x (0, i)⟩) (mk p fun i => ⟨x (1, i)⟩) k
simp only [ghostComponent_apply, aeval_eq_eval₂H... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 35
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : 𝕎 (R ⧸ 𝔭)\nh : 𝔭 ^ (0 + 1) = 𝔭\ny : 𝕎 R\nhy : (map (Ideal.Quotient.mk 𝔭)) y = x\n⊢ (quotEquivOfEq h) ((ghostComponentModPPow 0) x) = (ghostComponent 0) x",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
... | [] | simp [← hy, ghostComponent_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 62
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nx : 𝕎 R\nn✝ i : ℕ\nhi : i < n✝ + 1\nn : ℕ\nx✝¹ x✝ :\n ↑{r | r ∈ Finset.Iic n✝ ∧ ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ r) (x.coeff r)) * ↑p ^ r).coeff n ≠ 0}\na : ℕ\n... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nx : 𝕎 R\nn✝ i : ℕ\nhi : i < n✝ + 1\nn : ℕ\nx✝¹ x✝ :\n ↑{r | r ∈ Finset.Iic n✝ ∧ ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ r) (x.coeff r)) * ↑p ^ r).coeff n ≠ 0}\na : ℕ\nleft✝¹ : a ∈... | rw [← Not.imp_symm (teichmuller_mul_pow_coeff_of_ne _) ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 10
} | {
"line": 185,
"column": 6
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nn : ℕ\nhn : p.natDegree < n\nd : ℕ\nhd :\n (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) =\n eval (↑n)\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i)\nh_le : ∀ (i ... | [
"case succ\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nn : ℕ\nhn : p.natDegree < n\nd : ℕ\nhd :\n (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) =\n eval (↑n)\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i)\nh_le : ∀ (i : ↥p.support... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 41
} | {
"line": 61,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi : Irreducible (P... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi : Irreducible (Polynomial.ma... | ← add_sub_cancel_left 1 (-(_ * _)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 6
} | {
"line": 50,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ ↑n ! * shiftedLegendre n = (⇑derivative)^[n] (X ^ n * (1 - X) ^ n)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.instOne",
"Semiring.toModule",
"HMul.hMul",
"LinearMap.instFunLike",
"HSub.hSub",
... | [
"n : ℕ\n⊢ (⇑derivative)^[n] (X ^ n * (1 - X) ^ n) = ↑n ! * shiftedLegendre n"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 54,
"column": 4
} | {
"line": 60,
"column": 12
} | {
"line": 61,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (⇑derivative)^[n] ((X - X ^ 2) ^ n) = (⇑derivative)^[n] (∑ m ∈ range (n + 1), n.choose m • (-1) ^ m * X ^ (n + m))",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Polynomial.derivative",
"NonUnitalNonAssocCommRi... | [] | congr
rw [sub_eq_add_neg, add_comm, add_pow]
congr! 1 with m hm
rw [neg_pow, pow_two, mul_pow, ← mul_assoc, mul_comm, mul_assoc, pow_mul_pow_sub, mul_assoc,
← pow_add, ← mul_assoc, nsmul_eq_mul, add_comm]
rw [Finset.mem_range] at hm
linarith | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 54,
"column": 4
} | {
"line": 60,
"column": 12
} | {
"line": 61,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (⇑derivative)^[n] ((X - X ^ 2) ^ n) = (⇑derivative)^[n] (∑ m ∈ range (n + 1), n.choose m • (-1) ^ m * X ^ (n + m))",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Polynomial.derivative",
"NonUnitalNonAssocCommRi... | [] | congr
rw [sub_eq_add_neg, add_comm, add_pow]
congr! 1 with m hm
rw [neg_pow, pow_two, mul_pow, ← mul_assoc, mul_comm, mul_assoc, pow_mul_pow_sub, mul_assoc,
← pow_add, ← mul_assoc, nsmul_eq_mul, add_comm]
rw [Finset.mem_range] at hm
linarith | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 51,
"column": 63
} | {
"line": 51,
"column": 71
} | {
"line": 51,
"column": 71
} | [
{
"pp": "case pos\nn : ℕ\nhn1 : n ≠ 1\nhn0 : n = 0\n⊢ Irreducible (0 - X)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddZeroClass.toAddZero",
"Irredu... | [
"case pos\nn : ℕ\nhn1 : n ≠ 1\nhn0 : n = 0\n⊢ Irreducible (-X)"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 71,
"column": 63
} | {
"line": 71,
"column": 71
} | {
"line": 71,
"column": 71
} | [
{
"pp": "case pos\nn : ℕ\nhn1 : n ≠ 1\nhn0 : n = 0\n⊢ Irreducible (0 - X)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"Rat",
"HSub.hSub",
"AddZeroClass.toAddZero",
... | [
"case pos\nn : ℕ\nhn1 : n ≠ 1\nhn0 : n = 0\n⊢ Irreducible (-X)"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 168,
"column": 21
} | {
"line": 168,
"column": 93
} | {
"line": 169,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g f : M →ₚₗ[R] N\n⊢ 0 • f = 0",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"PolynomialLaw.instSMul"... | [] | simp_rw [HSMul.hSMul, SMul.smul]; simp only [Nat.cast_zero, zero_smul f] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 168,
"column": 21
} | {
"line": 168,
"column": 93
} | {
"line": 169,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g f : M →ₚₗ[R] N\n⊢ 0 • f = 0",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"PolynomialLaw.instSMul"... | [] | simp_rw [HSMul.hSMul, SMul.smul]; simp only [Nat.cast_zero, zero_smul f] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 372,
"column": 39
} | {
"line": 372,
"column": 56
} | {
"line": 373,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹² : CommSemiring R\nM : Type u_1\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type u_2\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nS : Type v\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nf : M →ₚₗ[R] N\nA : Type u\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\nφ : A →... | [
"R : Type u\ninst✝¹² : CommSemiring R\nM : Type u_1\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type u_2\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nS : Type v\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nf : M →ₚₗ[R] N\nA : Type u\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\nφ : A →ₐ[R] S\np : ... | comp_toLinearMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 76
} | {
"line": 409,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_2\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : M →ₚₗ[R] N\ns : Finset S\np : MvPolynomial (Fin s.card) R ⊗[R] M\ns' : Finset S\np... | [
"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_2\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : M →ₚₗ[R] N\ns : Finset S\np : MvPolynomial (Fin s.card) R ⊗[R] M\ns' : Finset S\np' : MvPolyno... | have hA'B' : (φ R s').range ≤ (φ R t).range := le_trans hA'B (le_of_eq hB) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 176,
"column": 2
} | {
"line": 177,
"column": 32
} | {
"line": 179,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R⟦X⟧\ninst✝ : P.IsPrime\nhfg : P.FG\nhP : X ∉ P\n⊢ spanFinrank P ≤ spanFinrank (Ideal.map constantCoeff P) + 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"... | [] | · exact le_trans (spanFinrank_eq_spanFinrank_map_constantCoeff_of_X_notMem_of_fg_of_isPrime
hP hfg).le (Nat.le_succ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 89,
"column": 8
} | {
"line": 89,
"column": 29
} | {
"line": 90,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nr : R\nh : ¬r = 0\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ε\nε : ℝ\na✝ : 0 < ε\nn : ℕ\nhn : ∀ (n_1 : ℕ), n ≤ n_1 → ‖(coeff n_1) f‖ * |c| ^ n_1 < ε / ‖r‖\nN : ℕ\nhN : n ≤ N\n⊢ ‖r‖ * ‖(coeff N) f‖ * |c| ^ N < ‖r‖ * (ε... | [] | rw [mul_assoc]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 89,
"column": 8
} | {
"line": 89,
"column": 29
} | {
"line": 90,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nr : R\nh : ¬r = 0\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ε\nε : ℝ\na✝ : 0 < ε\nn : ℕ\nhn : ∀ (n_1 : ℕ), n ≤ n_1 → ‖(coeff n_1) f‖ * |c| ^ n_1 < ε / ‖r‖\nN : ℕ\nhN : n ≤ N\n⊢ ‖r‖ * ‖(coeff N) f‖ * |c| ^ N < ‖r‖ * (ε... | [] | rw [mul_assoc]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 436,
"column": 50
} | {
"line": 436,
"column": 67
} | {
"line": 436,
"column": 68
} | [
{
"pp": "case h\nR : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type v\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nT : Type u_3\ninst✝¹ : CommSemiring T\ninst✝ : Algebra R T\nA : Subalgebra R T\nφ : S →ₐ[R] T\nhφ : A ≤ φ.range\nt : T ⊗[R] M\nu : ↥A ⊗[R]... | [
"case h\nR : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type v\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nT : Type u_3\ninst✝¹ : CommSemiring T\ninst✝ : Algebra R T\nA : Subalgebra R T\nφ : S →ₐ[R] T\nhφ : A ≤ φ.range\nt : T ⊗[R] M\nu : ↥A ⊗[R] M\nhu : (rT... | comp_toLinearMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 50
} | {
"line": 521,
"column": 0
} | [
{
"pp": "case hpq\nR : Type u\ninst✝⁸ : CommSemiring R\nM : Type u_1\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_2\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nS : Type v\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nf : M →ₚₗ[R] N\nT : Type w\ninst✝¹ : CommSemiring T\ninst✝ : Algebra R T\nh... | [] | simp only [φ, aeval_X, Equiv.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 159,
"column": 33
} | {
"line": 159,
"column": 60
} | {
"line": 159,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ... | [
"R : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ε\nhg : ∀ (ε... | mul_le_iff_le_one_right ‹_› | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 57
} | {
"line": 419,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\na✝ : A\nf✝ f'✝ : A⟦X⟧\ninst✝ : IsAdicComplete I A\nf f' : A⟦X⟧\nhf : f ≈ f'\na : A⟦X⟧\nha : g * (a + H.div f') + ↑(H.mod f') = g * H.div f + ↑(H.mod f)\nhf1 : (H.mod f).degree < ↑((map (Ideal.Quotient.mk I)) g).or... | [] | exact (H.eq_of_mul_add_eq_mul_add hf'1 hf1 ha).2.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 74,
"column": 45
} | {
"line": 74,
"column": 51
} | {
"line": 74,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortCo... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 74,
"column": 45
} | {
"line": 74,
"column": 51
} | {
"line": 74,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortCo... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 74,
"column": 45
} | {
"line": 74,
"column": 51
} | {
"line": 74,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortCo... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 91
} | {
"line": 125,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\nn✝ n : ℕ\nS : ShortComplex (M... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\nn✝ n : ℕ\nS : ShortComplex (ModuleCat R) ... | simp only [S, this, AddCommGrpCat.epi_iff_surjective, AddCommGrpCat.hom_ofHom] at epi | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 148,
"column": 41
} | {
"line": 165,
"column": 68
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nM : ModuleCat R\ninst✝¹ : Nontrivial ↑M\ninst✝ : Module.Finite R ↑M\nrs : List R\nreg : IsWeaklyRegular (↑M) rs\nmem : ∀ r ∈ rs, r ∈ maximalIdeal R\n⊢ projectiveDimension (of R (↑M ⧸ Ideal.ofL... | [] | by
generalize len : rs.length = n
induction n generalizing M rs with
| zero =>
rw [List.length_eq_zero_iff.mp len, Ideal.ofList_nil, Submodule.bot_smul]
simpa using projectiveDimension_eq_of_iso (Submodule.quotEquivOfEqBot ⊥ rfl).toModuleIso
| succ n hn =>
match rs with
| [] => simp at len
|... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 88,
"column": 55
} | {
"line": 88,
"column": 59
} | {
"line": 88,
"column": 60
} | [
{
"pp": "case mem_mul_mem\nR : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nx y : R\nhy : y ∈ I\nz : R\nhz : z ∈ J\nw : ↥J\nhw : w ∈ Submodule.comap (Submodule.subtype J... | [
"case mem_mul_mem\nR : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nx y : R\nhy : y ∈ I\nz : R\nhz : z ∈ J\nw : ↥J\nhw : w ∈ Submodule.comap (Submodule.subtype J) (I * J)\ne... | eq0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 46
} | {
"line": 124,
"column": 4
} | [
{
"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 ... | rw [← MvPolynomial.aeval_algebraMap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 86
} | {
"line": 196,
"column": 2
} | [
{
"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... | [] | exact ((σ.restrictScalars P.Ring).comp mapTen).congr_arg (eTen.symm_apply_apply x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Cardinal.UnivLE | {
"line": 30,
"column": 38
} | {
"line": 30,
"column": 43
} | {
"line": 31,
"column": 4
} | [
{
"pp": "case refine_2.mk\nh✝ : univ.{v, u + 1} < univ.{u, v + 1}\nw✝ : Cardinal.{u}\nα : Type u\nh : univ.{v, u + 1} = lift.{max (u + 1) (v + 1), u} (Quot.mk (⇑isEquivalent) α)\n⊢ ∃ α, univ.{v, max u (v + 1)} ≤ lift.{v + 1, u} #α",
"ppTerm": "?refine_2.mk",
"assigned": true,
"usedConstants": [
... | [
"case h\nh✝ : univ.{v, u + 1} < univ.{u, v + 1}\nw✝ : Cardinal.{u}\nα : Type u\nh : univ.{v, u + 1} = lift.{max (u + 1) (v + 1), u} (Quot.mk (⇑isEquivalent) α)\n⊢ univ.{v, max u (v + 1)} ≤ lift.{v + 1, u} #α"
] | use α | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 31
} | {
"line": 226,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type v\ninst✝ : LinearOrder α\ns : Set (Set α)\nhα : cof α ≤ 1\nf : (x : Set α) → x ∈ s → Set α\nhf : ∀ (x : Set α) (_ : x ∈ s), IsClub (f x _)\nhxf : ∀ (x : Set α) (_ : x ∈ s), Disjoint x (f x _)\n⊢ Disjoint (⋃₀ s) (⋂ x, f ↑x ⋯)",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nα : Type v\ninst✝ : LinearOrder α\ns : Set (Set α)\nhα : cof α ≤ 1\nf : (x : Set α) → x ∈ s → Set α\nhf : ∀ (x : Set α) (_ : x ∈ s), IsClub (f x _)\nhxf : ∀ (x : Set α) (_ : x ∈ s), Disjoint x (f x _)\n⊢ ∀ s_1 ∈ s, Disjoint s_1 (⋂ x, f ↑x ⋯)"
] | rw [disjoint_sUnion_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 31
} | {
"line": 257,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nf : (x : Set α) → x ∈ s → Set α\nhf : ∀ (x : Set α) (_ : x ∈ s), IsClub (f x _)\nhxf : ∀ (x : Set α) (_ : x ∈ s), Disjoint x (f x _)\n⊢ Disjoint (⋃₀ s) (⋂ x, f ↑x ⋯)",
"p... | [
"case refine_2\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nf : (x : Set α) → x ∈ s → Set α\nhf : ∀ (x : Set α) (_ : x ∈ s), IsClub (f x _)\nhxf : ∀ (x : Set α) (_ : x ∈ s), Disjoint x (f x _)\n⊢ ∀ s_1 ∈ s, Disjoint s_1 (⋂ x, f ↑x ⋯)"
] | rw [disjoint_sUnion_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Lists | {
"line": 322,
"column": 2
} | {
"line": 323,
"column": 18
} | {
"line": 325,
"column": 0
} | [
{
"pp": "α : Type u_1\nb : Bool\na : Lists' α b\nl : Lists' α true\n⊢ sizeOf ⟨b, a⟩ < sizeOf (a.cons' l)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instIsOrderedAddMonoid",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"AddMonoid.toA... | [] | simp only [Sigma.mk.sizeOf_spec, Lists'.cons'.sizeOf_spec, lt_add_iff_pos_right]
apply sizeof_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Lists | {
"line": 322,
"column": 2
} | {
"line": 323,
"column": 18
} | {
"line": 325,
"column": 0
} | [
{
"pp": "α : Type u_1\nb : Bool\na : Lists' α b\nl : Lists' α true\n⊢ sizeOf ⟨b, a⟩ < sizeOf (a.cons' l)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instIsOrderedAddMonoid",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"AddMonoid.toA... | [] | simp only [Sigma.mk.sizeOf_spec, Lists'.cons'.sizeOf_spec, lt_add_iff_pos_right]
apply sizeof_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 172,
"column": 15
} | {
"line": 172,
"column": 95
} | {
"line": 174,
"column": 0
} | [
{
"pp": "case inr\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\na : Ordinal.{u}\nha : a < (succ #α).ord\nb : Ordinal.{u}\nhb : b < (succ #α).ord\nhne : a ≠ b\nhf : lfpApprox f x a = lfpApprox f x b\nhba : b ≤ a\n⊢ lfpApprox f x (succ #α).ord ∈ fixedPoints ⇑f",
"ppTerm": "?inr",
... | [] | exact lfpApprox_mem_fixedPoints_of_eq f hx (hne.symm.lt_of_le hba) hb.le hf.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 172,
"column": 15
} | {
"line": 172,
"column": 95
} | {
"line": 174,
"column": 0
} | [
{
"pp": "case inr\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\na : Ordinal.{u}\nha : a < (succ #α).ord\nb : Ordinal.{u}\nhb : b < (succ #α).ord\nhne : a ≠ b\nhf : lfpApprox f x a = lfpApprox f x b\nhba : b ≤ a\n⊢ lfpApprox f x (succ #α).ord ∈ fixedPoints ⇑f",
"ppTerm": "?inr",
... | [] | exact lfpApprox_mem_fixedPoints_of_eq f hx (hne.symm.lt_of_le hba) hb.le hf.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 172,
"column": 15
} | {
"line": 172,
"column": 95
} | {
"line": 174,
"column": 0
} | [
{
"pp": "case inr\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\na : Ordinal.{u}\nha : a < (succ #α).ord\nb : Ordinal.{u}\nhb : b < (succ #α).ord\nhne : a ≠ b\nhf : lfpApprox f x a = lfpApprox f x b\nhba : b ≤ a\n⊢ lfpApprox f x (succ #α).ord ∈ fixedPoints ⇑f",
"ppTerm": "?inr",
... | [] | exact lfpApprox_mem_fixedPoints_of_eq f hx (hne.symm.lt_of_le hba) hb.le hf.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 472,
"column": 19
} | {
"line": 472,
"column": 65
} | {
"line": 473,
"column": 2
} | [
{
"pp": "o : ONote\nx✝ : NF 0\nh₂ : o.NF\n⊢ (0 - o).repr = repr 0 - o.repr",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ONote.NF",
"ONote.oadd",
"Ordinal.zero_sub",
"ONote.instZero",
"HSub.hSub",
"ONote.zero",
"ONote.instSub",
"instHSub",... | [] | by cases o <;> exact (Ordinal.zero_sub _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.ZFC.VonNeumann | {
"line": 137,
"column": 85
} | {
"line": 138,
"column": 51
} | {
"line": 140,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.card ≤ (V_ o).card",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Cardinal",
"congrArg",
"Eq.mp",
"Ordinal.card_toZFSet",
"ZFSet.vonNeumann",
"LE.le",
"Cardinal.instLE",
"Ordinal.card",
"Ordinal.toZFSet_... | [] | by
simpa using card_mono o.toZFSet_subset_vonNeumann | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 642,
"column": 6
} | {
"line": 642,
"column": 14
} | {
"line": 643,
"column": 6
} | [
{
"pp": "case neg\ne : ONote\nn : ℕ+\na o' : ONote\nm : ℕ\nh : (e.oadd n a).NF\ne0 : ¬e = 0\np : (e - 1).oadd n a.split'.1 = o' ∧ a.split'.2 = m\n⊢ e.oadd n a.split.1 = scale 1 o' ∧ a.split.2 = m",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"p"
],
"use... | [
"case neg\ne : ONote\nn : ℕ+\na o' : ONote\nm : ℕ\nh : (e.oadd n a).NF\ne0 : ¬e = 0\n⊢ (e - 1).oadd n a.split'.1 = o' ∧ a.split'.2 = m → e.oadd n a.split.1 = scale 1 o' ∧ a.split.2 = m"
] | revert p | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 663,
"column": 6
} | {
"line": 663,
"column": 14
} | {
"line": 664,
"column": 6
} | [
{
"pp": "case neg\ne : ONote\nn : ℕ+\na o' : ONote\nm : ℕ\nh : (e.oadd n a).NF\ne0 : ¬e = 0\np : (e - 1).oadd n a.split'.1 = o' ∧ a.split'.2 = m\n⊢ o'.NF ∧ ω ^ e.repr * ↑↑n + a.repr = ω * o'.repr + ↑m",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"p"
],
... | [
"case neg\ne : ONote\nn : ℕ+\na o' : ONote\nm : ℕ\nh : (e.oadd n a).NF\ne0 : ¬e = 0\n⊢ (e - 1).oadd n a.split'.1 = o' ∧ a.split'.2 = m → o'.NF ∧ ω ^ e.repr * ↑↑n + a.repr = ω * o'.repr + ↑m"
] | revert p | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 742,
"column": 2
} | {
"line": 742,
"column": 33
} | {
"line": 743,
"column": 2
} | [
{
"pp": "o₁ o₂ : ONote\ninst✝¹ : o₁.NF\ninst✝ : o₂.NF\na : ONote\nm : ℕ\ne₁ : o₁.split = (a, m)\n⊢ (o₁ ^ o₂).NF",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"ONote.NF",
"AddMonoidWithOne.toNatCast",
"Nat.cast",
"Ordinal.addMonoidWithOne",
"instHAdd",
... | [
"o₁ o₂ : ONote\ninst✝¹ : o₁.NF\ninst✝ : o₂.NF\na : ONote\nm : ℕ\ne₁ : o₁.split = (a, m)\nna : a.NF\n⊢ (o₁ ^ o₂).NF"
] | have na := (nf_repr_split e₁).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 765,
"column": 8
} | {
"line": 765,
"column": 14
} | {
"line": 766,
"column": 6
} | [
{
"pp": "case pos\no₁ : ONote\ninst✝¹ : o₁.NF\nb' : ONote\nk : ℕ\nthis : b'.NF\nna : zero.NF\ne₁ : o₁.split = (zero, 0)\ninst✝ : NF 0\ne₂ : split' 0 = (b', k)\n⊢ (if True then zero.oadd ⟨1, ⋯⟩ zero else zero).NF",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"ONote.NF",
"instD... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 772,
"column": 8
} | {
"line": 772,
"column": 14
} | {
"line": 773,
"column": 6
} | [
{
"pp": "case pos\no₁ o₂ : ONote\ninst✝¹ : o₁.NF\ninst✝ : o₂.NF\nb' : ONote\nk : ℕ\ne₂ : o₂.split' = (b', k)\nthis : b'.NF\nna : zero.NF\nm : ℕ\ne₁ : o₁.split = (zero, m + 1)\nh : m = 0\n⊢ (zero.oadd ⟨1, ⋯⟩ zero).NF",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"ONote.NF",
"o... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 885,
"column": 10
} | {
"line": 885,
"column": 19
} | {
"line": 885,
"column": 20
} | [
{
"pp": "case h₁\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\nω0 : O... | [
"case h₁\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\nω0 : Ordinal.{0} :... | opow_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.