module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 31
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nn : ℕ\nm : M\n⊢ dp R n (r • m) = r ^ n • dp R n m",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"DividedPowerAlgebra.Rel.smul",
... | rw [dp_def, dp_def, ← map_smul]
exact mkAlgHom_rel R Rel.smul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 31
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nn : ℕ\nm : M\n⊢ dp R n (r • m) = r ^ n • dp R n m",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"DividedPowerAlgebra.Rel.smul",
... | rw [dp_def, dp_def, ← map_smul]
exact mkAlgHom_rel R Rel.smul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 21
} | [
{
"pp": "case hc\nA : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn m : ℕ\nx : A\nhm : m ≠ 0\... | rw [dif_pos hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 29
} | [
{
"pp": "case dp.mem.mem\nR : Type u_3\nM : Type u_4\nι : Type u_5\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx✝ : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nx : DividedPowerAlgebra R M\nhx : x ∈ Set.range fun n ↦ n.prod fun i k ↦ dp... | obtain ⟨n, rfl⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 383,
"column": 2
} | {
"line": 385,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I",
"usedConstants": [
"DividedPowers.dpow_mem",
"Ideal.span_le",
"Eq.mpr",
"Semiring.toModule",
"con... | rw [span_le]
rintro y ⟨n, hn, x, hx, hxy⟩
exact hxy ▸ hI.dpow_mem hn (hS hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 383,
"column": 2
} | {
"line": 385,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I",
"usedConstants": [
"DividedPowers.dpow_mem",
"Ideal.span_le",
"Eq.mpr",
"Semiring.toModule",
"con... | rw [span_le]
rintro y ⟨n, hn, x, hx, hxy⟩
exact hxy ▸ hI.dpow_mem hn (hS hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 424,
"column": 4
} | {
"line": 428,
"column": 31
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}",
"usedConstants": [
"Eq.mpr",
"Ideal.su... | have h : J ∈ insert ⊤ {J : hI.SubDPIdeal | S ⊆ ↑J.carrier} :=
Set.mem_insert_of_mem _
(fun x hx ↦ subset_span ⟨1, one_ne_zero, x, hx, by rw [hI.dpow_one (hS hx)]⟩)
refine sInf_le_of_le ⟨J, ?_⟩ (le_refl _)
simp only [h, ciInf_pos, J] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 424,
"column": 4
} | {
"line": 428,
"column": 31
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier ≤ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}",
"usedConstants": [
"Eq.mpr",
"Ideal.su... | have h : J ∈ insert ⊤ {J : hI.SubDPIdeal | S ⊆ ↑J.carrier} :=
Set.mem_insert_of_mem _
(fun x hx ↦ subset_span ⟨1, one_ne_zero, x, hx, by rw [hI.dpow_one (hS hx)]⟩)
refine sInf_le_of_le ⟨J, ?_⟩ (le_refl _)
simp only [h, ciInf_pos, J] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Frobenius | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 78
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\n⊢ Ideal.under R (Ideal.map (MulSemiringAction.toRingEquiv G... | rw [← Ideal.comap_symm, ← Ideal.comap_coe, Ideal.under, Ideal.comap_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 77
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\ng : Γ\nhg : g ∈ s.hsum.support\n⊢ g ∈ ⋃ a, (s a).support",
"usedConstants": [
"HahnSeries.support",
"congrArg",
"AddMonoid.toAddZeroClass",
"Membership.mem",... | rw [mem_support, coeff_hsum, finsum_eq_sum _ (s.finite_co_support _)] at hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 726,
"column": 4
} | {
"line": 727,
"column": 87
} | [
{
"pp": "case neg\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\ng : Γ\nh : ¬0 < x.orderTop\n⊢ {a | ((if 0 < x.orderTop then x else 0) ^ a).coeff g ≠ 0}.Fin... | · simp only [h, ↓reduceIte]
exact pow_finite_co_support (orderTop_zero (R := R) (Γ := Γ) ▸ WithTop.top_pos) g | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 796,
"column": 6
} | {
"line": 796,
"column": 49
} | [
{
"pp": "Γ : 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⟧\nr : R\nhr : r * x.leadingCoeff = 1\noinv : Γ\nhxo : oinv + x.order = 0\ny : R⟦Γ⟧ := x - (single x.order) x.leadingCoeff\nhy : ¬y = 0\nhr' : IsRegular r\n⊢ 0 <... | rw [(order_single_mul_of_isRegular hr' hy)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 806,
"column": 6
} | {
"line": 806,
"column": 18
} | [
{
"pp": "Γ : 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 : IsUnit x.leadingCoeff\nhxo : IsAddUnit x.order\nu i : R\nui : u * i = 1\niu : i * u = 1\nh : ↑{ val := u, inv := i, val_inv := ui, inv_val := iu } = x.le... | Units.val_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 848,
"column": 8
} | {
"line": 848,
"column": 20
} | [
{
"pp": "case mpr\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommGroup Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedAddMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R⟦Γ⟧\nu i : R\nui : u * i = 1\niu : i * u = 1\nhx : ↑{ val := u, inv := i, val_inv := ui, inv_val := iu } = x.leadingCoeff\n⊢ IsUnit x",
... | Units.val_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 13
} | [
{
"pp": "case h.h.refine_2.a.inr\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'\... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 24
} | [
{
"pp": "case refine_2\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 : ... | obtain ⟨k, hk⟩ := hr | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF G : IdealFilter A\n⊢ Order.IsPFilter {L | ∃ K ∈ G, F.IsTorsionQuot L K}",
"usedConstants": [
"Order.IsPFilter.of_def",
"Semiring.toModule",
"IdealFilter.IsTorsionQuot",
"PartialOrder.toPreorder",
"setOf",
"Membership.mem",
"E... | refine Order.IsPFilter.of_def ?nonempty ?directed ?mem_of_le | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.Pure | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\n⊢ IsIdempotentElem I",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"inf_of_le_left",
"Submodule.comple... | simp [IsIdempotentElem, ← Ideal.inf_eq_mul_of_pure] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Pure | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\n⊢ IsIdempotentElem I",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"inf_of_le_left",
"Submodule.comple... | simp [IsIdempotentElem, ← Ideal.inf_eq_mul_of_pure] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Pure | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\n⊢ IsIdempotentElem I",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"inf_of_le_left",
"Submodule.comple... | simp [IsIdempotentElem, ← Ideal.inf_eq_mul_of_pure] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 64
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\np : Ideal R\nhp : p ∈ I.minimalPrimes\nthis : p.IsPrime\nf : R →+* Localization.AtPrime p := algebraMap R (Localization.AtPrime p)\n⊢ IsLocalRing.maximalIdeal (Localization.AtPri... | rwa [IsLocalization.minimalPrimes_map p.primeCompl (Localization.AtPrime p) I,
Set.mem_preimage, Localization.AtPrime.under_maximalIdeal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 64
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\np : Ideal R\nhp : p ∈ I.minimalPrimes\nthis : p.IsPrime\nf : R →+* Localization.AtPrime p := algebraMap R (Localization.AtPrime p)\n⊢ IsLocalRing.maximalIdeal (Localization.AtPri... | rwa [IsLocalization.minimalPrimes_map p.primeCompl (Localization.AtPrime p) I,
Set.mem_preimage, Localization.AtPrime.under_maximalIdeal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 64
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\np : Ideal R\nhp : p ∈ I.minimalPrimes\nthis : p.IsPrime\nf : R →+* Localization.AtPrime p := algebraMap R (Localization.AtPrime p)\n⊢ IsLocalRing.maximalIdeal (Localization.AtPri... | rwa [IsLocalization.minimalPrimes_map p.primeCompl (Localization.AtPrime p) I,
Set.mem_preimage, Localization.AtPrime.under_maximalIdeal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surje... | mk_ker, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 65,
"column": 8
} | {
"line": 65,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\np : Ideal R\ninst✝² : p.IsMaximal\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nx✝ : Field (R ⧸ p) := Quotient.field p\ne : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial... | mk_ker, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 27
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nx : R\nn : ℕ\nhn :\n ∀ (p : LTSeries (PrimeSpectrum R)),\n x ∈ (RelSeries.last p).asIdeal →\n p.length = n →\n ∃ q,\n x ∈ (q.toFun 1).asIdeal ∧\n n = q.length ∧ RelSeries.head p = RelSeries.head q... | simpa [h1, hp] using hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 98,
"column": 8
} | {
"line": 99,
"column": 59
} | [
{
"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\nh : ∀ (m : MaximalSpectrum R), Projective (M.localizedModule m.asIdeal.primeCompl)\nthis : Module.FinitePresentation R ↑M\np : Ideal R\nhp : p.IsMaximal\n⊢ Module.Projecti... | let : Small.{v} (Localization.AtPrime p) :=
small_of_surjective Localization.mkHom_surjective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 131,
"column": 4
} | {
"line": 132,
"column": 85
} | [
{
"pp": "case H\nR : Type u\ninst✝¹² : CommRing R\nM : Type v\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u'\ninst✝⁹ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁸ : ∀ (P : Ideal R) [inst : P.IsMaximal], Small.{v', u'} (Rₚ P)\ninst✝⁷ : (P : Ideal ... | apply ((Module.End.isUnit_iff _).mp
(IsLocalizedModule.map_units (LocalizedModule.mkLinearMap P.primeCompl M) s)).1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.LocalRing.Length | {
"line": 108,
"column": 4
} | {
"line": 116,
"column": 79
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\n⊢ length B (B... | obtain ⟨s, hs_bot, hs_top⟩ := isFiniteLength_iff_exists_compositionSeries.mp h
rw [← length_compositionSeries s hs_bot hs_top]
suffices ∀ k, length B ((s k).baseChange B) =
k * length B (B ⧸ (maximalIdeal A).map (algebraMap A B)) by
rw [← Fin.val_last s.length, ← this, ← RelSeries.last, hs_top, ba... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.Length | {
"line": 108,
"column": 4
} | {
"line": 116,
"column": 79
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\n⊢ length B (B... | obtain ⟨s, hs_bot, hs_top⟩ := isFiniteLength_iff_exists_compositionSeries.mp h
rw [← length_compositionSeries s hs_bot hs_top]
suffices ∀ k, length B ((s k).baseChange B) =
k * length B (B ⧸ (maximalIdeal A).map (algebraMap A B)) by
rw [← Fin.val_last s.length, ← this, ← RelSeries.last, hs_top, ba... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\np : Ideal R\ninst✝⁹ : p.IsPrime\nRₚ : Type u_3\ninst✝⁸ : CommRing Rₚ\ninst✝⁷ : Algebra R Rₚ\ninst✝⁶ : IsLocalization.AtPrime Rₚ p\ninst✝⁵ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁴ : CommRing Sₚ\ninst✝³ : Algebr... | mk_ker, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 62
} | [
{
"pp": "case h.e'_3.h.e'_6.h.h\ni n m : ℕ\nhin : i < n\nhim : i + 1 < m\nt : Fin n → ℕ\na✝ : Fin n\n⊢ a✝ ∈ {x | i + 1 ≤ ↑x} ↔ a✝ ≠ ⟨i, hin⟩ ∧ a✝ ∈ {x | i ≤ ↑x}",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",... | simp_rw [mem_filter_univ, i.succ_le_iff, lt_iff_le_and_ne] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.LaurentSeries | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx : R⟦X⟧\nn : ℕ\n⊢ ((ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff n) x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIsStrictOrderedRing",
"Semiring.toModule",
"HahnSeries.instNonAssocSemirin... | rw [ofPowerSeries_apply_coeff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx : R⟦X⟧\nn : ℕ\n⊢ ((ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff n) x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIsStrictOrderedRing",
"Semiring.toModule",
"HahnSeries.instNonAssocSemirin... | rw [ofPowerSeries_apply_coeff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LaurentSeries | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx : R⟦X⟧\nn : ℕ\n⊢ ((ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff n) x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIsStrictOrderedRing",
"Semiring.toModule",
"HahnSeries.instNonAssocSemirin... | rw [ofPowerSeries_apply_coeff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 173,
"column": 23
} | {
"line": 173,
"column": 35
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nf : Finset σ × σ → MvPolynomial σ R\n⊢ ∑ t ∈ (range k).disjiUnion (fun x ↦ powersetCard x univ) ⋯ ×ˢ univ, f t =\n ∑ a ∈ antidiagonal k with a.1 < k, ∑ A ∈ powersetCard a.1 univ, ∑ j, f (A, j)",
"u... | sum_product, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 395,
"column": 35
} | {
"line": 395,
"column": 59
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn✝ : ℤ\nn : ℕ\n⊢ (single (-↑(n + ↑1))) 1⁻¹ = (single 1) 1⁻¹ ^ (-↑(n + ↑1))",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"NegZeroClass.toNeg",
"ZeroHom.funLike",
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"DivisionCommM... | ← HahnSeries.inv_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 71
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nP : K⟮X⟯\nf g : K[X]\nh : g ≠ 0\n⊢ (polynomialValuationX K) (RatFunc.mk f g) =\n (valuation K⸨X⸩ (PowerSeries.idealX K)) ((algebraMap K⟮X⟯ K⸨X⸩) (RatFunc.mk f g))",
"usedConstants": [
"Int.instAddCommGroup",
"RatFunc.instFaithfulSMulPolynomialLaurentSer... | rw [Polynomial.valuation_of_mk K f h, RatFunc.mk_eq_mk' f h, Eq.comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 687,
"column": 2
} | {
"line": 710,
"column": 80
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nℱ : Filter K⸨X⸩\nhℱ : Cauchy ℱ\n⊢ ∃ N, ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"HahnSeries.support",
"WithZero.instNontrivial",
"Filter.instMembership",
"Iff.mpr",
"Int.sub_one_lt_of_l... | let entourage : Set (K⸨X⸩ × K⸨X⸩) := {P : K⸨X⸩ × K⸨X⸩ | Valued.v.restrict (P.snd - P.fst) < 1}
let ζ : (MonoidWithZeroHom.ValueGroup₀ (Valued.v (R := K⸨X⸩)))ˣ :=
Units.mk0 1 (zero_ne_one.symm)
obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := mem_prod_iff.mp <| Filter.le_def.mp hℱ.2 entourage
<| (Valued.hasBasis_uniformity... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LaurentSeries | {
"line": 687,
"column": 2
} | {
"line": 710,
"column": 80
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nℱ : Filter K⸨X⸩\nhℱ : Cauchy ℱ\n⊢ ∃ N, ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"HahnSeries.support",
"WithZero.instNontrivial",
"Filter.instMembership",
"Iff.mpr",
"Int.sub_one_lt_of_l... | let entourage : Set (K⸨X⸩ × K⸨X⸩) := {P : K⸨X⸩ × K⸨X⸩ | Valued.v.restrict (P.snd - P.fst) < 1}
let ζ : (MonoidWithZeroHom.ValueGroup₀ (Valued.v (R := K⸨X⸩)))ˣ :=
Units.mk0 1 (zero_ne_one.symm)
obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := mem_prod_iff.mp <| Filter.le_def.mp hℱ.2 entourage
<| (Valued.hasBasis_uniformity... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 162,
"column": 47
} | {
"line": 163,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R\n⊢ ord R (-x) = ord R x",
"usedConstants": [
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Ring.ord",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemir... | by
simp [ord_eq_of_associated (x := -x) (y := x) (by simp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OrderOfVanishing.Noetherian | {
"line": 118,
"column": 86
} | {
"line": 119,
"column": 99
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx y : R\n⊢ min (ord R x) (ord R y) ≤ ord R (x + y)",
"usedConstants": [
"Eq.mpr",
"le_refl",
"instCompleteLinearOrderENat",
"Ring.ord",
"congrArg",
"CommSemiring.toSemiring... | by
grw [ord_eq_addVal x, ord_eq_addVal y, ord_eq_addVal (x + y), IsDiscreteValuationRing.addVal_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Teichmuller | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 34
} | [
{
"pp": "case inr\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\ny : R\nN : ℕ\nh : ∀ n ≥ N, ∃ z, (Ideal.Quotient.mk I) z = (coeff (R ⧸ I) p n) x ∧ z ^ p ^ n ≡ y [SMOD I ^ (n + 1)]\nn : ℕ\nhn : N... | obtain ⟨z, hz₁, hz₂⟩ := h n hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.WittVector.Defs | {
"line": 273,
"column": 75
} | {
"line": 275,
"column": 48
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ constantCoeff (wittMul p n) = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
... | by
apply constantCoeff_wittStructureInt p _ _ n
simp only [mul_zero, map_mul, constantCoeff_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Perfection | {
"line": 162,
"column": 43
} | {
"line": 162,
"column": 79
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\nN : Type u_3\ninst✝ : CommMonoid N\nf : M →* N\nx y : M\nx✝ : ℕ\n⊢ (coeffMonoidHom N p x✝) ⟨fun n ↦ f ((powMulEquiv M (p ^ n)).symm (x * y)), ⋯⟩ =\n (coeffMonoidHom N p x✝)\n (⟨fun n ... | simp_rw [map_mul, coeffMonoidHom_mk] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Perfection | {
"line": 162,
"column": 43
} | {
"line": 162,
"column": 79
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\nN : Type u_3\ninst✝ : CommMonoid N\nf : M →* N\nx y : M\nx✝ : ℕ\n⊢ (coeffMonoidHom N p x✝) ⟨fun n ↦ f ((powMulEquiv M (p ^ n)).symm (x * y)), ⋯⟩ =\n (coeffMonoidHom N p x✝)\n (⟨fun n ... | simp_rw [map_mul, coeffMonoidHom_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Perfection | {
"line": 162,
"column": 43
} | {
"line": 162,
"column": 79
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\nN : Type u_3\ninst✝ : CommMonoid N\nf : M →* N\nx y : M\nx✝ : ℕ\n⊢ (coeffMonoidHom N p x✝) ⟨fun n ↦ f ((powMulEquiv M (p ^ n)).symm (x * y)), ⋯⟩ =\n (coeffMonoidHom N p x✝)\n (⟨fun n ... | simp_rw [map_mul, coeffMonoidHom_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 52
} | [
{
"pp": "case h.e'_2\np : ℕ\nR : Type u_1\nidx : Type u_2\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ (aeval fun i ↦ (map (Int.castRingHom R)) (wittStructureInt p Φ i)) (W_ ℤ n) =\n (eval₂Hom ((map (Int.castRingHom R)).comp C) fun i ↦ (map (Int.castRingHom R)) (wittStructur... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 52
} | [
{
"pp": "case h.e'_3\np : ℕ\nR : Type u_1\nidx : Type u_2\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ (aeval fun i ↦ (rename (Prod.mk i)) (W_ R n)) Φ =\n (eval₂Hom ((map (Int.castRingHom R)).comp C) fun i ↦ (map (Int.castRingHom R)) ((rename (Prod.mk i)) (W_ ℤ n))) Φ",
... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 341,
"column": 2
} | {
"line": 343,
"column": 21
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\n⊢ constantCoeff (wittStructureRat p Φ 0) = constantCoeff Φ",
"usedConstants": [
"Finsupp.instAddZeroClass",
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"MvPolynomial.aeva... | simp only [wittStructureRat, bind₁, map_aeval, xInTermsOfW_zero, constantCoeff_rename,
constantCoeff_wittPolynomial, aeval_X, constantCoeff_comp_algebraMap, eval₂Hom_zero'_apply,
RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 341,
"column": 2
} | {
"line": 343,
"column": 21
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\n⊢ constantCoeff (wittStructureRat p Φ 0) = constantCoeff Φ",
"usedConstants": [
"Finsupp.instAddZeroClass",
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"MvPolynomial.aeva... | simp only [wittStructureRat, bind₁, map_aeval, xInTermsOfW_zero, constantCoeff_rename,
constantCoeff_wittPolynomial, aeval_X, constantCoeff_comp_algebraMap, eval₂Hom_zero'_apply,
RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 341,
"column": 2
} | {
"line": 343,
"column": 21
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\n⊢ constantCoeff (wittStructureRat p Φ 0) = constantCoeff Φ",
"usedConstants": [
"Finsupp.instAddZeroClass",
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"MvPolynomial.aeva... | simp only [wittStructureRat, bind₁, map_aeval, xInTermsOfW_zero, constantCoeff_rename,
constantCoeff_wittPolynomial, aeval_X, constantCoeff_comp_algebraMap, eval₂Hom_zero'_apply,
RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfection | {
"line": 817,
"column": 2
} | {
"line": 817,
"column": 36
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nhp : Nat.Prime p\nthis✝ : Nontrivial (PreTilt O p)\nthis : NoZeroDivisors (PreTilt O p)\n⊢ IsDomain (PreTilt O p)",... | exact NoZeroDivisors.to_isDomain _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 180,
"column": 7
} | {
"line": 180,
"column": 55
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (eval₂Hom ((MvPolynomial.eval x).comp C) fun i ↦ (MvPolynomial.eval x) (verschiebungPoly i))\n (wittPolynomial p ℤ (n + 1)) =\n (ghostComponent (n + 1)) (verschiebung (mk p x))",
"usedConstants": [
"Finsupp.instAddZ... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 52
} | [
{
"pp": "case h.e'_2\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x)... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 52
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coef... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.MulP | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 52
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ (k : ℕ), (x * ↑n).coeff k = (aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ peval (wittAdd p k) ![(x * ↑n).coeff, x.coeff] =\n (aeval fun i ↦ (aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)) (witt... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 52
} | [
{
"pp": "case h.e'_3\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x)... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 52
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coef... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 52
} | [
{
"pp": "case h.e'_2\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n ↦ peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 52
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n ↦ peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 52
} | [
{
"pp": "case h.e'_3\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n ↦ peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 52
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n ↦ peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 50
} | [
{
"pp": "case h\np : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\ng : R →+* S\nx y : 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n ↦ peval (φ n)... | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.Truncated | {
"line": 149,
"column": 85
} | {
"line": 150,
"column": 57
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : TruncatedWittVector p n R\n⊢ WittVector.truncateFun n x.out = x",
"usedConstants": [
"TruncatedWittVector.coeff",
"congrArg",
"TruncatedWittVector.coeff_out",
"TruncatedWittVector.mk_coeff",
"Fin.val",
"funext",
... | by
simp only [WittVector.truncateFun, coeff_out, mk_coeff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 56
} | [
{
"pp": "case h\nR : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nthis : ↑p = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) ↑p\n⊢ ↑p ^ (n + 1) = 0",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Submodule.Qu... | rw [this, ← map_pow, Ideal.Quotient.eq_zero_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 47
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ∃ K x, ∃ (_ : CharP K p), Infinite K",
"usedConstants": [
"ZMod.commRing",
"FractionRing",
"Field.toSemifield",
"ZMod",
"Polynomial",
"ZMod.instField",
"Semifield.toDivisionSemiring",
"Polynomial.commRing",
"... | let K := FractionRing (Polynomial (ZMod p)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 11
} | [
{
"pp": "n k : ℕ\nhnk : n < n + k + 1\n⊢ (hermite n).coeff (n + k + 1) = 0",
"usedConstants": []
}
] | clear hnk | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 44,
"column": 68
} | {
"line": 45,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nr : R\n⊢ (opRingEquiv R) (op ((monomial n) r)) = (monomial n) (op r)",
"usedConstants": [
"RingEquiv.op",
"AddMonoidAlgebra.semiring",
"AddMonoidAlgebra.opRingEquiv",
"Semiring.toModule",
"Equiv.instEquivLike",
"Finsupp.eq... | by
simp [opRingEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ IsCancelMulZero R[X] ∧ Nontrivial R ↔ (IsCancelMulZero R ∧ Nontrivial R) ∧ IsCancelAdd R",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"congrArg",
"id",
"Distrib.toAdd",
"IsCancelMulZero",
"Polynomial",
"instDistribO... | Polynomial.isCancelMulZero_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 36
} | [
{
"pp": "case a\nn x : ℕ\na✝ : x ∈ range (n + 1)\n⊢ C (n.choose x • (-1) ^ x) * ↑((n + x).descFactorial n) • X ^ (n + x - n) =\n ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) * X ^ x",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"inst... | descFactorial_eq_div (by lia), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Morse | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 49
} | [
{
"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... | let π : S →ₐ[R] S ⧸ p := Ideal.Quotient.mkₐ R p | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 70
} | [
{
"pp": "case succ\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\nk i : ℕ\nq : A⟦X⟧ := H.seq f k\ns : A⟦X⟧ := f - g * q\nn : ℕ := ((map (Ideal.Quotient.mk I)) g).order.toNat\nhi : i ≥ n\nhq : ∀ {i : ℕ}, i ≥ n → (coeff i) s ∈ I ^ k\ns₀ : A[X] := (trunc n) s\ns... | refine coeff_mul_mem_ideal_of_coeff_left_mem_ideal' (fun i ↦ ?_) i | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\nk i : ℕ\n⊢ (coeff i) ((mk fun i ↦ (coeff (i + ((map (Ideal.Quotient.mk I)) g).order.toNat)) (f - g * H.seq f k)) * ↑⋯.unit⁻¹) ∈\n I ^ k",
"usedConstants": [
"Units.val",
"Semiring.toMod... | refine coeff_mul_mem_ideal_of_coeff_left_mem_ideal' (fun i ↦ ?_) i | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 12
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\ni : ℕ\n⊢ (coeff i) (H.div f) = ↑(H.divCoeff f i)",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"c... | simp [div] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 12
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\ni : ℕ\n⊢ (coeff i) (H.div f) = ↑(H.divCoeff f i)",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"c... | simp [div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 12
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\ni : ℕ\n⊢ (coeff i) (H.div f) = ↑(H.divCoeff f i)",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"c... | simp [div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 69
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ (coeff i) (H.div f - H.seq f k) ∈ I ^ k",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | simpa [coeff_div, SModEq.sub_mem] using ((H.divCoeff f i).2 k).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 69
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ (coeff i) (H.div f - H.seq f k) ∈ I ^ k",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | simpa [coeff_div, SModEq.sub_mem] using ((H.divCoeff f i).2 k).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 69
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ (coeff i) (H.div f - H.seq f k) ∈ I ^ k",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | simpa [coeff_div, SModEq.sub_mem] using ((H.divCoeff f i).2 k).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 23
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq q' : A⟦X⟧\nr r' : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nhr' : r'.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq : g * (q - q') = ↑(r' - r)\nh : ... | exact ⟨h.1, h.2.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RamificationInertia.Inertia | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 21
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\nq : Ideal S\nR : Type u_2\ninst✝³ : CommRing R\ninst✝² : Algebra R S\nhq : q.IsPrime\ninst✝¹ : Algebra (Localization.AtPrime (under R q)) (Localization.AtPrime q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (under R q) q\n⊢ q.inertiaDeg' R = Module.finrank (under ... | convert! dif_pos hq | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 132,
"column": 18
} | {
"line": 132,
"column": 55
} | [
{
"pp": "case h.e'_3.h.e'_2.h\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\na : 𝕎 k\nha : a ≠ 0\nm : ℕ\nb : 𝕎 k\nhc : b.coeff 0 ^ p ^ m ≠ 0\nhcm : a = (⇑verschiebung ∘ ⇑frobenius)^[m] b\nthis : (⇑verschiebung ∘ ⇑frobenius)^[m] b = (⇑verschiebu... | ← WittVector.verschiebung_frobenius x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 79,
"column": 4
} | {
"line": 82,
"column": 18
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range n\n⊢ (wittMul p x ^ p ^ (n - x)).vars ⊆ univ ×ˢ range n",
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Preorder.toLT",
"Finset... | apply Subset.trans (vars_pow _ _)
apply Subset.trans (wittMul_vars _ _)
apply product_subset_product (Subset.refl _)
simpa using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 79,
"column": 4
} | {
"line": 82,
"column": 18
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range n\n⊢ (wittMul p x ^ p ^ (n - x)).vars ⊆ univ ×ˢ range n",
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Preorder.toLT",
"Finset... | apply Subset.trans (vars_pow _ _)
apply Subset.trans (wittMul_vars _ _)
apply product_subset_product (Subset.refl _)
simpa using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 142,
"column": 2
} | {
"line": 164,
"column": 6
} | [
{
"pp": "p n : ℕ\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +\n remain... | have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast
rw [wittPolyProd, wittPolynomial, map_sum, map_sum]
conv_lhs =>
arg 1
rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
rename_C, rename_X, ← mvpz]
conv_lhs =>
arg 2
rw [su... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 142,
"column": 2
} | {
"line": 164,
"column": 6
} | [
{
"pp": "p n : ℕ\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +\n remain... | have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast
rw [wittPolyProd, wittPolynomial, map_sum, map_sum]
conv_lhs =>
arg 1
rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,
rename_C, rename_X, ← mvpz]
conv_lhs =>
arg 2
rw [su... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 64
} | [
{
"pp": "case inr.inr\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα✝ : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\nh✝ : Nonempty α\nhα : ℵ₀ < cof α\n⊢ IsClub (⋂₀ s)",
"usedConstants": [
"IsClub.dirSupClosed",
"IsClub.mk",
"PartialOrder.toPreor... | refine ⟨.sInter fun x hx ↦ (hs x hx).dirSupClosed, fun a ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 40
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\n⊢ ∃ f,\n ∀ (x y : 𝕎 k),\n (x * y).coeff (n + 1) =\n x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) + y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) +\n f (truncateFun (n + 1) x) (truncateFun (n +... | obtain ⟨f, hf⟩ := nth_mul_coeff' p k n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 28
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ count p n ≤ {k | p k}.ncard",
"usedConstants": [
"Eq.mpr",
"Set.encard_eq_top_iff._simp_1",
"Set.encard",
"Set.ncard_def",
"Nat.count_le_setENCard",
"ENat.instNatCast",
"instTopENat",
... | rw [Set.ncard_def, ← ENat.coe_le_coe, ENat.coe_toNat (by simpa)]
exact count_le_setENCard n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 28
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ count p n ≤ {k | p k}.ncard",
"usedConstants": [
"Eq.mpr",
"Set.encard_eq_top_iff._simp_1",
"Set.encard",
"Set.ncard_def",
"Nat.count_le_setENCard",
"ENat.instNatCast",
"instTopENat",
... | rw [Set.ncard_def, ← ENat.coe_le_coe, ENat.coe_toNat (by simpa)]
exact count_le_setENCard n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 44
} | [
{
"pp": "case refine_2\np : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha :... | rw [LinearMap.span_singleton_eq_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 298,
"column": 10
} | {
"line": 298,
"column": 92
} | [
{
"pp": "case inr\no₁ e₁ : ONote\nn₁ : ℕ+\na₁ : ONote\nh✝¹ : o₁ = e₁.oadd n₁ a₁\no₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF\nh✝ : o₂ = e₁.oadd n₂ a₂\nh₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF\nnh : ↑n₂ ≤ ↑n₁ ∧ (if ↑n₂ < ↑n₁ then Ordering.gt else Ordering.eq) = Ordering.lt\... | rw [ite_eq_iff] at nh; rcases nh.right with nh | nh <;> cases nh <;> contradiction | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 298,
"column": 10
} | {
"line": 298,
"column": 92
} | [
{
"pp": "case inr\no₁ e₁ : ONote\nn₁ : ℕ+\na₁ : ONote\nh✝¹ : o₁ = e₁.oadd n₁ a₁\no₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF\nh✝ : o₂ = e₁.oadd n₂ a₂\nh₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF\nnh : ↑n₂ ≤ ↑n₁ ∧ (if ↑n₂ < ↑n₁ then Ordering.gt else Ordering.eq) = Ordering.lt\... | rw [ite_eq_iff] at nh; rcases nh.right with nh | nh <;> cases nh <;> contradiction | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 363,
"column": 8
} | {
"line": 364,
"column": 20
} | [
{
"pp": "case neg\nb : Ordinal.{u_1}\nhb : 1 < b\ne x : Ordinal.{u_1}\nf : Ordinal.{u_1} →₀ Ordinal.{u_1}\nhf' : ∀ c ∈ f.support, c < e\nhx : x ≠ 0\nIH : (∀ (e : Ordinal.{u_1}), f e < b) → coeff b (eval b f) = f\nhf : ∀ (e_1 : Ordinal.{u_1}), (single e x + f) e_1 < b\ne' : Ordinal.{u_1}\nhe' : e' ∉ f.support\n⊢... | rw [notMem_support_iff.1 he']
exact hb.pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 363,
"column": 8
} | {
"line": 364,
"column": 20
} | [
{
"pp": "case neg\nb : Ordinal.{u_1}\nhb : 1 < b\ne x : Ordinal.{u_1}\nf : Ordinal.{u_1} →₀ Ordinal.{u_1}\nhf' : ∀ c ∈ f.support, c < e\nhx : x ≠ 0\nIH : (∀ (e : Ordinal.{u_1}), f e < b) → coeff b (eval b f) = f\nhf : ∀ (e_1 : Ordinal.{u_1}), (single e x + f) e_1 < b\ne' : Ordinal.{u_1}\nhe' : e' ∉ f.support\n⊢... | rw [notMem_support_iff.1 he']
exact hb.pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 14
} | [
{
"pp": "case eq\ne₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b\nh₂ : (e₂.oadd n₂ a₂).NF\nh' : (a₁.sub a₂).NFBelow e₁.repr\nh : e₁.cmp e₂ = Ordering.eq\nthis : e₁ = e₂\n⊢ (match Ordering.eq with\n | Ordering.lt => 0\n | Ordering.gt => e₁.oadd ... | subst e₂ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 483,
"column": 6
} | {
"line": 483,
"column": 14
} | [
{
"pp": "case eq\ne₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh₁ : (e₁.oadd n₁ a₁).NF\nh₂ : (e₂.oadd n₂ a₂).NF\nthis✝ : a₁.NF\nthis : a₂.NF\nh' : (a₁.sub a₂).repr = a₁.repr - a₂.repr\nh : e₁.cmp e₂ = Ordering.eq\nee : e₁ = e₂\n⊢ (match Ordering.eq with\n | Ordering.lt => 0\n | Ordering.gt... | subst e₂ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.SetTheory.ZFC.Basic | {
"line": 677,
"column": 70
} | {
"line": 679,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\ni : α\n⊢ f i ⊆ ⋃ (i : α), f i",
"usedConstants": [
"Eq.mpr",
"ZFSet",
"Membership.mem",
"Exists",
"id",
"ZFSet.mem_iUnion._simp_1",
"Exists.intro",
"ZFSet.iUnion",
"Eq",
"ZFSet.ins... | by
intro x hx
simpa using ⟨i, hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.ZFC.Rank | {
"line": 121,
"column": 42
} | {
"line": 121,
"column": 50
} | [
{
"pp": "case h\nx✝ x : PSet.{u}\nih : ∀ y ∈ x, lift.{u + 1, u} y.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) y\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x",
"usedConstants": []
}
] | | _ x ih
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
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