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.HahnSeries.Summable | {
"line": 445,
"column": 6
} | {
"line": 445,
"column": 14
} | [
{
"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✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : SummableFamily Γ R α... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 459,
"column": 59
} | {
"line": 459,
"column": 67
} | [
{
"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 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 459,
"column": 59
} | {
"line": 459,
"column": 67
} | [
{
"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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 459,
"column": 59
} | {
"line": 459,
"column": 67
} | [
{
"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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 32
} | [
{
"pp": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nσ : Type u_3\ninst✝ : Finite σ\nf g : R⟦σ →₀ ℕ⟧\nn : σ →₀ ℕ\n⊢ (MvPowerSeries.coeff n) (f * g).coeff = (MvPowerSeries.coeff n) (f.coeff * g.coeff)",
"usedConstants"... | change (f * g).coeff n = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 204,
"column": 26
} | {
"line": 206,
"column": 71
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nr : R\n⊢ toPowerSeries.toFun ((algebraMap R A⟦ℕ⟧) r) = (algebraMap R (PowerSeries A)) r",
"usedConstants": [
"ZeroHom.funLike",
"False",
"Nat.instMulZeroClass",
"Ring... | by
ext n
cases n <;> simp [algebraMap_apply, PowerSeries.algebraMap_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 479,
"column": 6
} | {
"line": 479,
"column": 14
} | [
{
"pp": "case coeff.h.refine_1.right\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFami... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 483,
"column": 6
} | {
"line": 483,
"column": 14
} | [
{
"pp": "case neg\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α\nt : Summa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 483,
"column": 6
} | {
"line": 483,
"column": 14
} | [
{
"pp": "case neg\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α\nt : Summa... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 483,
"column": 6
} | {
"line": 483,
"column": 14
} | [
{
"pp": "case neg\nΓ : Type u_1\nΓ' : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ns : SummableFamily Γ R α\nt : Summa... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Henselian | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : ∃ b, (Ideal.Quotient.mk I) a * b = 1\n⊢ ∃ b, (Ideal.Quotient.mk ⊥.jacobson) a * b = 1",
"usedConstants": []
}
] | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Henselian | {
"line": 157,
"column": 8
} | {
"line": 157,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nhR : HenselianLocalRing R\n⊢ maximalIdeal R ≤ ⊥.jacobson",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"HenselianLocalRing.toIsLocalRing",
"setOf",
... | Ideal.jacobson, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Henselian | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 49
} | [
{
"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... | apply IsUnit.of_map (Ideal.Quotient.mk I) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 63
} | [
{
"pp": "case h.h\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 : IsLoc... | mem_colon_singleton, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 67,
"column": 12
} | {
"line": 67,
"column": 32
} | [
{
"pp": "case h\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 : IsLocal... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Henselian | {
"line": 222,
"column": 10
} | {
"line": 222,
"column": 39
} | [
{
"pp": "case succ.refine_2\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 ... | refine Submodule.sum_mem _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 110,
"column": 24
} | {
"line": 110,
"column": 44
} | [
{
"pp": "case h\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 : IsLocal... | mem_colon_singleton, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 44
} | [
{
"pp": "case nonempty\nA : Type u_1\ninst✝ : Ring A\nF G : IdealFilter A\nJ : Ideal A\nhJ : J ∈ ↑G\n⊢ {L | ∃ K ∈ G, F.IsTorsionQuot L K}.Nonempty",
"usedConstants": [
"Semiring.toModule",
"IdealFilter.IsTorsionQuot",
"PartialOrder.toPreorder",
"setOf",
"Membership.mem",
... | exact ⟨J, J, hJ, isTorsionQuot_self F J⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Henselian | {
"line": 256,
"column": 6
} | {
"line": 262,
"column": 40
} | [
{
"pp": "case refine_2\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 -... | · show a - a₀ ∈ I
specialize ha (0 + 1)
rw [hc, pow_one, ← Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one, sub_eq_add_neg] at ha
rw [← SModEq.sub_mem, ← add_zero a₀]
refine ha.symm.trans (SModEq.rfl.add ?_)
rw [SModEq.zero, Ideal.neg_mem_iff]
exact Ideal.mul_mem_right _ _ h... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 28
} | [
{
"pp": "case mpr\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nh₁ : F.IsUniform\nh₂ : F • F = F\n⊢ ∀ (I : Ideal A), (∃ J ∈ F, ∀ x ∈ J, Submodule.colon I {x} ∈ F) → I ∈ F",
"usedConstants": [
"Ideal",
"Ring.toSemiring"
]
}
] | rintro I ⟨J, hJ, hcolon⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 94
} | [
{
"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 : IsArtinianRing (R ⧸ I)\nf : R →+* Localizatio... | apply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (q.map f) _ (IsNoetherian.noetherian _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.Pure | {
"line": 123,
"column": 26
} | {
"line": 123,
"column": 39
} | [
{
"pp": "case refine_2.h\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 p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"Algebra.algeb... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.LocalRing | {
"line": 30,
"column": 28
} | {
"line": 30,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nh : ringKrullDim R = 1\nx : R\nhx : x ≠ 0\nJ : Ideal R\nx✝ : J ∈ {J | Ideal.span {x} ≤ J ∧ J.IsPrime}\nhJ1 : Ideal.span {x} ≤ J\nhJ2 : J.IsPrime\nhJ3 : J = ⊥\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.KrullDimension.LocalRing | {
"line": 30,
"column": 28
} | {
"line": 30,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nh : ringKrullDim R = 1\nx : R\nhx : x ≠ 0\nJ : Ideal R\nx✝ : J ∈ {J | Ideal.span {x} ≤ J ∧ J.IsPrime}\nhJ1 : Ideal.span {x} ≤ J\nhJ2 : J.IsPrime\nhJ3 : J = ⊥\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 104,
"column": 52
} | {
"line": 104,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\nthis : ringKrullDim (MvPolynomial σ R) = ⊤\n⊢ ringKrullDim R + ⊤ ≤ ringKrullDim (MvPolynomial σ R)",
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
"WithBot",
"instTopENat... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 104,
"column": 52
} | {
"line": 104,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\nthis : ringKrullDim (MvPolynomial σ R) = ⊤\n⊢ ringKrullDim R + ⊤ ≤ ringKrullDim (MvPolynomial σ R)",
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
"WithBot",
"instTopENat... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 104,
"column": 52
} | {
"line": 104,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\nthis : ringKrullDim (MvPolynomial σ R) = ⊤\n⊢ ringKrullDim R + ⊤ ≤ ringKrullDim (MvPolynomial σ R)",
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
"WithBot",
"instTopENat... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.KrullDimension.LocalRing | {
"line": 30,
"column": 28
} | {
"line": 30,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nh : ringKrullDim R = 1\nx : R\nhx : x ≠ 0\nJ : Ideal R\nx✝ : J ∈ {J | Ideal.span {x} ≤ J ∧ J.IsPrime}\nhJ1 : Ideal.span {x} ≤ J\nhJ2 : J.IsPrime\nhJ3 : J = ⊥\n⊢ x = 0",
"usedConstants": [
"Submodule",
"False"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 14
} | [
{
"pp": "case h.zero\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nthis : p.IsPrime\nH :\n ∀ m < 0,\n ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),\n p ∈ (span ↑s).minimalPrimes → s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 286,
"column": 60
} | {
"line": 296,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height",
"usedConstants": [
"Iff.mpr",
"Ideal.height_le_spanRank",
"Eq.mpr",
"False",
"instCompleteLinearOrderENat",
... | by
obtain ⟨J, hJ₁, hJ₂, hJ₃⟩ := exists_spanRank_le_and_le_height_of_le_height I _
(ENat.coe_toNat_le_self I.height)
rw [ENat.coe_toNat_eq_self.mpr (Ideal.height_ne_top hI)] at hJ₃
refine ⟨J, hJ₁, le_antisymm ?_ (le_trans ?_ (J.height_le_spanRank ?_)),
le_antisymm (Ideal.height_mono hJ₁) hJ₃⟩
· convert! ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nS : Finset R\nhS : ↑S ⊆ ↑(maximalIdeal R)\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ span ↑S) + ↑S.card",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Finset",
"IsLocalR... | apply ringKrullDim_le_ringKrullDim_quotient_add_card
rwa [IsLocalRing.ringJacobson_eq_maximalIdeal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nS : Finset R\nhS : ↑S ⊆ ↑(maximalIdeal R)\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ span ↑S) + ↑S.card",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Finset",
"IsLocalR... | apply ringKrullDim_le_ringKrullDim_quotient_add_card
rwa [IsLocalRing.ringJacobson_eq_maximalIdeal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 560,
"column": 14
} | {
"line": 560,
"column": 51
} | [
{
"pp": "R : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² ... | exact quot_hom_ext _ _ _ fun _ => rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Lasker | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 31
} | [
{
"pp": "case pos\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\nhq : (q.colon Set.univ).radical = ↑p\nS : Submonoid R := ⨅ q ... | obtain ⟨b, hb, a, ha⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 82
} | [
{
"pp": "case zero\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nS : Submonoid R\nthis : Small.{v, u} (Localization S)\nM : ModuleCat R\ninst✝ : HasInjectiveDimensionLE M 0\ninjle : Module.Injective R ↑M\n⊢ Module.Injective (Localization S) ↑(M.localizedModule S)",
... | exact Module.injective_of_isLocalizedModule S (M.localizedModuleMkLinearMap S) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 48
} | [
{
"pp": "case succ\nR : 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\nP : Type v\nw✝³ : AddCommGroup P\nw✝² : Module R P\nw✝¹ : Module.Free R P\nw✝ : Module.Finite R P\nf : P →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : Sh... | simp only [HasProjectiveDimensionLE] at ih ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.LocalRing.Length | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 37
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : IsLocalRing A\ninst✝⁶ : IsLocalRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsLocalHom (algebraMap A B)\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\ninst✝¹ : Module B M\ninst✝ : IsScalarTower A B M\np q : Submodule... | let f : p →ₗ[B] q := inclusion h.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 95,
"column": 2
} | {
"line": 100,
"column": 44
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\n⊢ ∃ β, R[β] = ⊤",
"usedConstants": [
"Algebra.instIsSeparableResid... | obtain ⟨β₀, hβ₀⟩ := Field.exists_primitive_element (ResidueField R) (ResidueField S)
obtain ⟨β, hβ⟩ := residue_surjective (R := S) β₀
refine ⟨β, adjoin_residue_eq_top_iff_adjoin_eq_top β |>.mp ?_⟩
rw [hβ,
← IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite _ _),
hβ₀, Inter... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 95,
"column": 2
} | {
"line": 100,
"column": 44
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\n⊢ ∃ β, R[β] = ⊤",
"usedConstants": [
"Algebra.instIsSeparableResid... | obtain ⟨β₀, hβ₀⟩ := Field.exists_primitive_element (ResidueField R) (ResidueField S)
obtain ⟨β, hβ⟩ := residue_surjective (R := S) β₀
refine ⟨β, adjoin_residue_eq_top_iff_adjoin_eq_top β |>.mp ?_⟩
rw [hβ,
← IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite _ _),
hβ₀, Inter... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 106,
"column": 58
} | {
"line": 116,
"column": 48
} | [
{
"pp": "R : 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\n⊢ Module.finrank R S = Module.finrank (ResidueField R) (ResidueField S)",
"usedConsta... | by
have : Module.Free R S := Module.free_of_flat_of_isLocalRing
have e := AddEquiv.toLinearEquiv (R := R ⧸ maximalIdeal R) (Ideal.quotEquivOfEq <|
Algebra.FormallyUnramified.map_maximalIdeal (R := R) (S := S)).toAddEquiv
?_
· rw [← finrank_quotient_map (R := R) (S := S)]
exact e.finrank_eq -- again ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 75,
"column": 2
} | {
"line": 81,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\n⊢ [¬IsLocalRing R, Nontrivial (MaximalSpectrum R), ∃ m₁ m₂, m₁.IsMaximal ∧ m₂.IsMaximal ∧ m₁ ≠ m₂].TFAE",
"usedConstants": [
"Nontrivial",
"_private.Mathlib.RingTheory.LocalRing.NonLocalRing.0.IsLocalRing.not_isLocalRing_tfae.... | tfae_have 1 → 2
| h => not_subsingleton_iff_nontrivial.mp fun _ ↦ h of_singleton_maximalSpectrum
tfae_have 2 → 3
| ⟨⟨m₁, hm₁⟩, ⟨m₂, hm₂⟩, h⟩ => ⟨m₁, m₂, ⟨hm₁, hm₂, fun _ ↦ h (by congr)⟩⟩
tfae_have 3 → 1
| ⟨m₁, m₂, ⟨hm₁, hm₂, h⟩⟩ => fun _ ↦ h <| (eq_maximalIdeal hm₁).trans (eq_maximalIdeal hm₂).symm
tfae_fin... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 75,
"column": 2
} | {
"line": 81,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\n⊢ [¬IsLocalRing R, Nontrivial (MaximalSpectrum R), ∃ m₁ m₂, m₁.IsMaximal ∧ m₂.IsMaximal ∧ m₁ ≠ m₂].TFAE",
"usedConstants": [
"Nontrivial",
"_private.Mathlib.RingTheory.LocalRing.NonLocalRing.0.IsLocalRing.not_isLocalRing_tfae.... | tfae_have 1 → 2
| h => not_subsingleton_iff_nontrivial.mp fun _ ↦ h of_singleton_maximalSpectrum
tfae_have 2 → 3
| ⟨⟨m₁, hm₁⟩, ⟨m₂, hm₂⟩, h⟩ => ⟨m₁, m₂, ⟨hm₁, hm₂, fun _ ↦ h (by congr)⟩⟩
tfae_have 3 → 1
| ⟨m₁, m₂, ⟨hm₁, hm₂, h⟩⟩ => fun _ ↦ h <| (eq_maximalIdeal hm₁).trans (eq_maximalIdeal hm₂).symm
tfae_fin... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.EulerIdentity | {
"line": 46,
"column": 8
} | {
"line": 46,
"column": 27
} | [
{
"pp": "case refine_4\nR : Type u_1\nσ : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nφ : MvPolynomial σ R\ninst✝ : AddCancelCommMonoid M\nw : σ → M\nn n' : M\ni : σ\nh✝ : φ ∈ Submodule.span R ((fun i ↦ single i 1) '' {d | (weight w) d = n})\nh' : n' + w i = n\nr : R\np : MvPolynomial σ R\nx✝ : p ∈ Submodu... | (pderiv i).map_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 30
} | [
{
"pp": "case refine_2\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\nt : Finset (Fin m)\nht : t ∈ powersetCard (i + 1) univ\nhne : ∃ x ∈ Iic ⟨i, him⟩, x ∉ t\nht' : #t = #(Iic ⟨i, him⟩)\n⊢ toLex (Finsupp.indicator t fun x x_1 ↦ 1) < toLex (Finsupp.indicator (Iic ⟨i, him... | simp_rw [← mem_sdiff] at hne | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 145,
"column": 40
} | {
"line": 145,
"column": 48
} | [
{
"pp": "case inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\nb : MvPolynomial n R\nhab : p = 0 * b\nhle : totalDegree 0 ≤ b.totalDegree\n⊢ IsUnit 0 ∨ IsUnit b",
"usedConstant... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 145,
"column": 40
} | {
"line": 145,
"column": 48
} | [
{
"pp": "case inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\nb : MvPolynomial n R\nhab : p = 0 * b\nhle : totalDegree 0 ≤ b.totalDegree\n⊢ IsUnit 0 ∨ IsUnit b",
"usedConstant... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 145,
"column": 40
} | {
"line": 145,
"column": 48
} | [
{
"pp": "case inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\nb : MvPolynomial n R\nhab : p = 0 * b\nhle : totalDegree 0 ≤ b.totalDegree\n⊢ IsUnit 0 ∨ IsUnit b",
"usedConstant... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 146,
"column": 40
} | {
"line": 146,
"column": 48
} | [
{
"pp": "case inr.inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na : MvPolynomial n R\nha₀ : a ≠ 0\nhab : p = a * 0\nhle : a.totalDegree ≤ totalDegree 0\n⊢ IsUnit a ∨ IsUnit 0",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 146,
"column": 40
} | {
"line": 146,
"column": 48
} | [
{
"pp": "case inr.inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na : MvPolynomial n R\nha₀ : a ≠ 0\nhab : p = a * 0\nhle : a.totalDegree ≤ totalDegree 0\n⊢ IsUnit a ∨ IsUnit 0",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 146,
"column": 40
} | {
"line": 146,
"column": 48
} | [
{
"pp": "case inr.inl\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na : MvPolynomial n R\nha₀ : a ≠ 0\nhab : p = a * 0\nhle : a.totalDegree ≤ totalDegree 0\n⊢ IsUnit a ∨ IsUnit 0",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 38
} | [
{
"pp": "case nontrivial\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc : c.support.Nontrivial\nh_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nι : n ↪ n ⊕ n →₀ ℕ := ⋯\naux : sumSMulXSMulY c = Finsupp.embDomain ι c\nhcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c ... | · rwa [hsupp, Finset.map_nontrivial] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 13
} | [
{
"pp": "case inr\nσ : 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\n⊢ p.support.sup ⇑toLex < toLex ?m.112",
"usedConstants": []
... | pick_goal 3 | Batteries.Tactic.«_aux_Batteries_Tactic_PermuteGoals___elabRules_Batteries_Tactic_tacticPick_goal-__1» | Batteries.Tactic.«tacticPick_goal-_» |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 70,
"column": 43
} | {
"line": 70,
"column": 51
} | [
{
"pp": "case pos\nσ : Type u_1\ninst✝ : DecidableEq σ\nt : Finset σ × σ\nh1 : (t.1.erase t.2, t.2).2 ∈ (t.1.erase t.2, t.2).1\nht : (t.1.erase t.2, t.2) = t\n⊢ False",
"usedConstants": [
"False",
"congrArg",
"Finset",
"False.elim",
"false_and",
"Membership.mem",
"E... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 70,
"column": 43
} | {
"line": 70,
"column": 51
} | [
{
"pp": "case neg\nσ : Type u_1\ninst✝ : DecidableEq σ\nt : Finset σ × σ\nh1✝ : t.2 ∉ t.1\nh1 : (cons t.2 t.1 h1✝, t.2).2 ∉ (cons t.2 t.1 h1✝, t.2).1\nht : (cons t.2 t.1 h1✝, t.2) = t\n⊢ False",
"usedConstants": [
"False",
"eq_false",
"Finset.cons",
"congrArg",
"Finset",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 12
} | [
{
"pp": "case pos\nσ : Type u_1\ninst✝ : DecidableEq σ\nt : Finset σ × σ\nh1 : t.2 ∉ t.1\nh3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1\n⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t",
"usedConstants": [
"False",
"eq_false",
"Finset.co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 12
} | [
{
"pp": "case pos\nσ : Type u_1\ninst✝ : DecidableEq σ\nt : Finset σ × σ\nh1 : t.2 ∉ t.1\nh3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1\n⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t",
"usedConstants": [
"False",
"eq_false",
"Finset.co... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 12
} | [
{
"pp": "case pos\nσ : Type u_1\ninst✝ : DecidableEq σ\nt : Finset σ × σ\nh1 : t.2 ∉ t.1\nh3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1\n⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t",
"usedConstants": [
"False",
"eq_false",
"Finset.co... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 115,
"column": 2
} | {
"line": 118,
"column": 58
} | [
{
"pp": "case inr\nσ : Type u_1\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nt : Finset σ × σ\nh : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)\nh1 : t.2 ∉ t.1\n⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)",
"usedConstants": [
"le_iff_eq_or_lt",
"Eq.mpr",
... | · rw [pairMap_of_snd_notMem_fst σ h1]
simp only [h1] at h
simp only [card_cons, mem_cons, true_or, implies_true, and_true]
exact (le_iff_eq_or_lt.mp h.left).resolve_left h.right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 214,
"column": 2
} | {
"line": 218,
"column": 32
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ∃ i, ¬p ∣ d i\n⊢ MvPolynomial.coeff d ((trunc' R (p • n)) ((expand p hp) φ)) =\n MvPolynomial.coeff d ((MvPolynomial.expand p) ((trunc' R n) φ))",
... | · obtain ⟨i, hi⟩ := h
rw [MvPolynomial.coeff_expand_of_not_dvd _ hi]
by_cases hd : d ≤ p • n
· rw [coeff_trunc', if_pos hd, coeff_expand_of_not_dvd _ hp _ hi]
rw [coeff_trunc', if_neg hd] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 161,
"column": 42
} | {
"line": 161,
"column": 50
} | [
{
"pp": "case hab\nσ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\na : ℕ × ℕ\nha : a ∈ {a ∈ antidiagonal k | a.1 < k}\nf : Finset σ × σ → MvPolynomial σ R\np : Finset σ × σ\nhp : #p.1 = a.1\n⊢ #p.1 < k",
"usedConstants": [
"Finset.mem_filter._simp_1",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.LaurentSeries | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 12
} | [
{
"pp": "case zero\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nhD : 0 ≠ 0\n⊢ Valued.v f ≤ 0 ↔ ∀ n < -log 0, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Multiplicative.linearOrde... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.LaurentSeries | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 12
} | [
{
"pp": "case zero\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nhD : 0 ≠ 0\n⊢ Valued.v f ≤ 0 ↔ ∀ n < -log 0, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Multiplicative.linearOrde... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LaurentSeries | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 12
} | [
{
"pp": "case zero\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nhD : 0 ≠ 0\n⊢ Valued.v f ≤ 0 ↔ ∀ n < -log 0, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Multiplicative.linearOrde... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Noetherian.OfPrime | {
"line": 43,
"column": 20
} | {
"line": 43,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nhsup : (I ⊔ span {a}).FG\nhcolon : (Submodule.colon I ↑(span {a})).FG\nw✝¹ : ℕ\nf : Fin w✝¹ → R\nhf : span (Set.range f) = I ⊔ span {a}\nw✝ : ℕ\ni : Fin w✝ → R\nhi : Submodule.span R (Set.range i) = Submodule.colon I ↑(span {a})\nk : Fin w✝¹\n⊢ f k ... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 826,
"column": 6
} | {
"line": 827,
"column": 33
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0\nd : ℕ\nhd : Multiplicative.toAdd (unzero ⋯) = -↑d\n⊢ ∀ n < d + 1, (PowerSeries.coeff n) (F - ↑((trunc (d + 1)) F)) = 0",
"usedConstants": [
"Eq.mpr",
"PowerSeries.co... | simpa only [map_sub, sub_eq_zero, Polynomial.coeff_coe, coeff_trunc] using
fun _ h ↦ (if_pos h).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.LaurentSeries | {
"line": 908,
"column": 64
} | {
"line": 947,
"column": 16
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\n⊢ IsUniformInducing ⇑(algebraMap K⟮X⟯ K⸨X⸩)",
"usedConstants": [
"Int.instAddCommGroup",
"RatFunc.instFaithfulSMulPolynomialLaurentSeries",
"WithZero.instNontrivial",
"Filter.instMembership",
"Multiplicative.group",
"subset_refl._si... | by
rw [isUniformInducing_iff, Filter.comap]
ext S
simp only [Filter.mem_mk, Set.mem_setOf_eq, uniformity_eq_comap_nhds_zero,
Filter.mem_comap]
constructor
· rintro ⟨T, ⟨⟨R, ⟨hR, pre_R⟩⟩, pre_T⟩⟩
obtain ⟨d, hd⟩ := Valued.mem_nhds.mp hR
use {P : K⟮X⟯ | Valued.v P < embedding d.1}
simp only [Valu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.NoetherNormalization | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 25
} | [
{
"pp": "case hf.a\nk : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nc : k\ni : Fin (n + 1)\nv : Fin (n + 1) →₀ ℕ\n⊢ MvPolynomial.coeff v\n ((MvPolynomial.aeval fun i ↦\n (T1 f c) (if i = 0 then MvPolynomial.X 0 else MvPolynomial.X i + -c • MvPolynomial.X 0 ^ r i))\n ... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.LaurentSeries | {
"line": 1080,
"column": 6
} | {
"line": 1080,
"column": 14
} | [
{
"pp": "case h.right\nK : Type u_2\ninst✝ : Field K\na✝ : adicCompletion K⟮X⟯ (idealX K)\nthis : ∀ (s : Set (adicCompletion K⟮X⟯ (idealX K))), s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v.restrict x < ↑γ} ⊆ s\nha : a✝ = 0\nS : Set (WithZero (Multiplicative ℤ))\nγ : WithZero (Multiplicative ℤ)\nγ_ne_zero : γ ≠ 0\nγ_le : Set.I... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.OrderOfVanishing.Noetherian | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : K\nha : a = 0\n⊢ (ordFrac R) a =\n (MonoidWithZero.inverse.comp\n (IsDedekindDomain.HeightOneSpectrum.val... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.OrderOfVanishing.Noetherian | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : K\nha : a = 0\n⊢ (ordFrac R) a =\n (MonoidWithZero.inverse.comp\n (IsDedekindDomain.HeightOneSpectrum.val... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.OrderOfVanishing.Noetherian | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : K\nha : a = 0\n⊢ (ordFrac R) a =\n (MonoidWithZero.inverse.comp\n (IsDedekindDomain.HeightOneSpectrum.val... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Defs | {
"line": 231,
"column": 2
} | {
"line": 232,
"column": 75
} | [
{
"pp": "case a.h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nIH : ∀ m < n, 0 < m → (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ m) = 0\nhn : 0 < n\n⊢ C ⅟↑p ^ n - C ↑p ^ 0 * (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ 0) ^ p ^ (n - 0) * C ⅟↑p ^ n = 0",
"usedConstants": [
"one_pow",
"Finsupp.instAddZeroClass",
"E... | · simp only [one_mul, pow_zero]
simp only [one_pow, one_mul, xInTermsOfW_zero, sub_self, bind₁_X_right] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.Defs | {
"line": 251,
"column": 60
} | {
"line": 254,
"column": 49
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ wittMul p 0 = X (0, 0) * X (1, 0)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.instRingHomClass",
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgHom.algHomClass",
"HMul.hMul",
... | by
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
simp only [wittMul, wittStructureRat, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero,
map_mul, bind₁_X_right, map_wittStructureInt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Defs | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 31
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx y : 𝕎 R\nn : ℕ\n⊢ (x * y).coeff n = peval (wittMul p n) ![x.coeff, y.coeff]",
"usedConstants": [
"HMul.hMul",
"WittVector.peval",
"congrArg",
"WittVector.mk",
"instOfNatNat",
"WittVector.wittMul... | simp [(· * ·), Mul.mul, eval] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.Defs | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 31
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx y : 𝕎 R\nn : ℕ\n⊢ (x * y).coeff n = peval (wittMul p n) ![x.coeff, y.coeff]",
"usedConstants": [
"HMul.hMul",
"WittVector.peval",
"congrArg",
"WittVector.mk",
"instOfNatNat",
"WittVector.wittMul... | simp [(· * ·), Mul.mul, eval] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Defs | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 31
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx y : 𝕎 R\nn : ℕ\n⊢ (x * y).coeff n = peval (wittMul p n) ![x.coeff, y.coeff]",
"usedConstants": [
"HMul.hMul",
"WittVector.peval",
"congrArg",
"WittVector.mk",
"instOfNatNat",
"WittVector.wittMul... | simp [(· * ·), Mul.mul, eval] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfection | {
"line": 723,
"column": 2
} | {
"line": 733,
"column": 89
} | [
{
"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\nf : PreTilt O p\nn : ℕ\nhfn : (coeff n) f ≠ 0\nh : ∃ n, (coeff n) f ≠ 0\n⊢ ModP.preVal K v O p ((coeff (Nat.find h)... | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le (Nat.find_min' h hfn)
induction k with
| zero => rfl
| succ k ih => ?_
obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff (Nat.find h + k + 1) f)
have h1 : (Ideal.Quotient.mk _ x : ModP O p) ≠ 0 := hx.symm ▸ hfn
have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP O p)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Perfection | {
"line": 723,
"column": 2
} | {
"line": 733,
"column": 89
} | [
{
"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\nf : PreTilt O p\nn : ℕ\nhfn : (coeff n) f ≠ 0\nh : ∃ n, (coeff n) f ≠ 0\n⊢ ModP.preVal K v O p ((coeff (Nat.find h)... | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le (Nat.find_min' h hfn)
induction k with
| zero => rfl
| succ k ih => ?_
obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff (Nat.find h + k + 1) f)
have h1 : (Ideal.Quotient.mk _ x : ModP O p) ≠ 0 := hx.symm ▸ hfn
have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP O p)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 12
} | [
{
"pp": "p : ℕ\nR S : Type u\nidx : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nh : ⦃R : Type u_2⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nf g : ⦃R : Type u_2⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_2⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff)... | ext ⟨i, n⟩ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 316,
"column": 33
} | {
"line": 323,
"column": 22
} | [
{
"pp": "p : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nhf : IsPoly p f\ng : R →+* S\nx : 𝕎 R\n⊢ (WittVector.map g) (f x) = f ((WittVector.map g) x)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mp... | by
-- this could be turned into a tactic “macro” (taking `hf` as parameter)
-- so that applications do not have to worry about the universe issue
-- see `IsPoly₂.map` for a slightly more general proof strategy
obtain ⟨φ, hf⟩ := hf
ext n
simp_rw [map_coeff, hf, map_aeval, funext (map_coeff g _), RingHom.ext_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Identities | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 28
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → ((⇑verschiebung)^[n] x).coeff m = 0\nm : ℕ\nh : m < n + 1\n⊢ ((⇑verschiebung)^[n + 1] x).coeff m = 0",
"usedConstants": [
"Function.iterate_succ_apply'",
"Eq.mpr",
... | rw [iterate_succ_apply'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.WittVector.Identities | {
"line": 170,
"column": 17
} | {
"line": 170,
"column": 80
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ k : ℕ\nih : ((⇑verschiebung)^[k] x).coeff (k✝ + k) = x.coeff k✝\n⊢ ((⇑verschiebung)^[k + 1] x).coeff (k✝ + (k + 1)) = x.coeff k✝",
"usedConstants": [
"Function.iterate_succ_apply'",
"Eq.mpr",
... | rw [iterate_succ_apply', Nat.add_succ, verschiebung_coeff_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.WittVector.Identities | {
"line": 217,
"column": 2
} | {
"line": 218,
"column": 17
} | [
{
"pp": "case calc_2\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx y : 𝕎 R\ni j : ℕ\n⊢ ((⇑verschiebung)^[i + j] ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y)).coeff (i + j) =\n ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y).coeff 0",
"usedConstants": [
"Eq.mpr",
... | · convert! iterate_verschiebung_coeff (p := p) (R := R) _ _ _ using 2
rw [zero_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 44
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nx : 𝕎 R\nn : ℕ\n⊢ ∀ m < n + 1,\n (x - ∑ i ∈ Finset.Iic n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i).coeff m = 0",
"usedConstants": [
"Eq.mpr... | ← le_coeff_eq_iff_le_sub_coeff_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 24
} | [
{
"pp": "case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type\nw✝¹ : Field K\nw✝ : CharP K p\nH : Set.univ.Infinite\nh : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite\nx : K\nhx : ¬x = 0\n⊢ ∃ i, (∃ y, i = y + y⁻¹ ∧ ¬y = 0) ∧ (i = x + x⁻¹ ∨ x = 0)",
"usedConstants": [
"Eq.mpr",
"False",
"DivisionC... | · simp only [hx, or_false, exists_eq_right]
exact ⟨_, rfl, hx⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 14
} | [
{
"pp": "n : ℕ\n⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"id",
"Polynomial.hermite.eq_2",
... | rw [hermite] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 14
} | [
{
"pp": "n : ℕ\n⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"id",
"Polynomial.hermite.eq_2",
... | rw [hermite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 14
} | [
{
"pp": "n : ℕ\n⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"id",
"Polynomial.hermite.eq_2",
... | rw [hermite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 185,
"column": 82
} | {
"line": 191,
"column": 66
} | [
{
"pp": "n k : ℕ\nhnk : Even (n + k)\n⊢ (hermite n).coeff k = (-1) ^ ((n - k) / 2) * ↑(n - k - 1)‼ * ↑(n.choose k)",
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"Preorder.toLT",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"congrArg",
"Nat.mul_div_cancel_left... | by
rcases le_or_gt k n with h_le | h_lt
· rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
obtain ⟨m, hm⟩ := hnk
rw [(by lia : n = 2 * m + k),
Nat.add_sub_cancel, Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]
· simp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 180,
"column": 10
} | {
"line": 180,
"column": 80
} | [
{
"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 ... | Finset.sum_coe_sort _ (fun x => (p.coeff ↑x) * (_ + d - ↑x).choose _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 125,
"column": 24
} | {
"line": 125,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ (IsLeftCancelMulZero R ∧ IsRightCancelMulZero R) ∧ IsCancelAdd R ∧ IsCancelAdd R ↔\n (IsLeftCancelMulZero R ∧ IsRightCancelMulZero R) ∧ IsCancelAdd R",
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
"Distrib.toAdd",
... | and_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Morse | {
"line": 59,
"column": 43
} | {
"line": 59,
"column": 47
} | [
{
"pp": "case pos\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... | hfp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 55,
"column": 65
} | {
"line": 57,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nS : Subsemigroup R\nhS : 0 ∉ S\n⊢ ∃ P ∈ kaplanskySet S, ∀ I ∈ kaplanskySet S, P ≤ I → I = P",
"usedConstants": [
"Semiring.toModule",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"zorn_le₀",
"Subm... | by
obtain ⟨P, hP⟩ := zorn_le₀ (kaplanskySet S) (fun _ ↦ exists_mem_kaplanskySet_le hS)
exact ⟨P, hP.1, fun _ hI H ↦ le_antisymm (hP.2 hI H) H⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 73
} | [
{
"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... | have hAB' : (φ R s).range ≤ (φ R t).range := le_trans hAB (le_of_eq hB) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Radical.NatInt | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 25
} | [
{
"pp": "z : ℤ\nhz : z ≠ 0\np : ℕ\nx✝ : Prime ↑p ∧ 0 ≤ ↑p ∧ ↑p ∣ z\npp : Nat.Prime p\ndp : p ∣ z.natAbs\n⊢ p ∈ z.natAbs.primeFactors",
"usedConstants": [
"False",
"Nat.Prime",
"Dvd.dvd",
"eq_false",
"congrArg",
"and_self",
"Finset",
"Membership.mem",
"id... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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