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.PowerSeries.WellKnown | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 25
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring A'\ninst✝¹ : Algebra ℚ A\ninst✝ : Algebra ℚ A'\nf : A →+* A'\n⊢ (map f) (sin A) = sin A'",
"usedConstants": [
"Rat.instOfNat",
"RingHom.instRingHomClass",
"instHDiv",
"Semiring.toModule",
"Algebra.algebraMap",... | ext
simp [sin, apply_ite f] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WellKnown | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 25
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring A'\ninst✝¹ : Algebra ℚ A\ninst✝ : Algebra ℚ A'\nf : A →+* A'\n⊢ (map f) (sin A) = sin A'",
"usedConstants": [
"Rat.instOfNat",
"RingHom.instRingHomClass",
"instHDiv",
"Semiring.toModule",
"Algebra.algebraMap",... | ext
simp [sin, apply_ite f] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Binomial | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 13
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : BinomialRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\n⊢ binomialSeries A 0 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Grassmannian | {
"line": 133,
"column": 10
} | {
"line": 133,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ ((single a) r * x).cardSupp ≤ x.cardSupp",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ (x * (single a) r).cardSupp ≤ x.cardSupp",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Grassmannian | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 26
} | [
{
"pp": "case succ\nΓ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\nIH : (x ^ n).cardSupp ≤ x.cardSupp ^ n\n⊢ (x ^ (n + 1)).cardSupp ≤ x.cardSupp ^ (n + 1)",
"usedConstants": [
"Eq.mpr",
"Hah... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 159,
"column": 18
} | {
"line": 159,
"column": 29
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nhκ : Fact (ℵ₀ ≤ κ)\n⊢ 0 ∈ {x | x.cardSupp < κ}",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 12
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝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 : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | intro ab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 12
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝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 : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | intro ab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | [
{
"pp": "case h₁\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | [
{
"pp": "case h₂\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module ℚ M\ninst✝ : IsOrderedModule ℚ M\nstrata : HahnEmbedding.ArchimedeanStrata ℚ M\nf : (c : FiniteArchimedeanClass M) → ↥(strata.stratum c) →+o ℝ\nhf : ∀ (c : FiniteArchimedeanClass M), F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 98,
"column": 31
} | {
"line": 98,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nf₁ : M →+o DivisibleHull M := DivisibleHull.coeOrderAddMonoidHom M\nhf₁ : Function.Injective ⇑f₁\nhf₁class :\n ∀ (a : M), ArchimedeanClass.mk a = (DivisibleHull.archimedeanClassOrderIso M).symm (ArchimedeanCla... | hf₃class, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 17
} | [
{
"pp": "case coeff.h.refine_1.left\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 : SummableFamil... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 53
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 53
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ a * (antipode R) a = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Henselian | {
"line": 83,
"column": 38
} | {
"line": 83,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n⊢ IsUnit a ∧ IsUnit y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 30
} | [
{
"pp": "case H.h\nR : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ((LinearMap.mul' R A[G] ∘ₗ LinearMap.rTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a) b =\n ((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 30
} | [
{
"pp": "case H.h\nR : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\na : G\nb : A\n⊢ ((LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a) b =\n ((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Henselian | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 87
} | [
{
"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\n⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ I",
"usedConstants": [
"P... | let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * (f'.eval b)⁻¹ʳ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 900,
"column": 2
} | {
"line": 900,
"column": 13
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nm : ℕ\ne : Bool\ns : ℕ\n⊢ (single 0) (OfScientific.ofScientific m e s) = OfScientific.ofScientific m e s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Henselian | {
"line": 208,
"column": 18
} | {
"line": 208,
"column": 68
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Pol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 65,
"column": 56
} | {
"line": 65,
"column": 67
} | [
{
"pp": "R : 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 : IsLocalizedModu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 68,
"column": 41
} | {
"line": 68,
"column": 52
} | [
{
"pp": "R : 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 : IsLocalizedModu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 55
} | [
{
"pp": "case ass\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np : Ideal R\ninst✝ : p.IsPrime\nass : p ∈ associatedPrimes R M\n⊢ Ideal.comap (algebraMap R (Localization.AtPrime p)) (maximalIdeal (Localization.AtPrime p)) ∈ associatedPrimes R M",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 41
} | [
{
"pp": "R : 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 : IsLocalizedModu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\np : Ideal R\nhp : p ∈ (annihilator R M).minimalPrimes\nprime : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\n⊢ Nontrivial (LocalizedModule p.prime... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 146,
"column": 4
} | {
"line": 147,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\np : Ideal R\nhp : p ∈ (annihilator R M).minimalPrimes\nprime : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\nthis✝ : Nontrivial (LocalizedModule p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\np : Ideal R\nhp : p ∈ (annihilator R M).minimalPrimes\nprime : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\nthis : Nontrivial (LocalizedModule p.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot (L ⊓ K) K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon (L ⊓ K) {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon L {k}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot L K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon L {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon (L ⊓ K) {k}",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 149,
"column": 2
} | {
"line": 157,
"column": 36
} | [
{
"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",
"Order.PFilter.nonempty",
"Submodule.completeLattice",
"Partia... | refine Order.IsPFilter.of_def ?nonempty ?directed ?mem_of_le
· obtain ⟨J, hJ⟩ := G.nonempty
exact ⟨J, J, hJ, isTorsionQuot_self F J⟩
· rintro I ⟨K, hK, hIK⟩ J ⟨L, hL, hJL⟩
refine ⟨I ⊓ J, ?_, inf_le_left, inf_le_right⟩
exact ⟨K ⊓ L, G.inf_mem hK hL,
(hIK.anti_right inf_le_left).inf (hJL.anti_right ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 149,
"column": 2
} | {
"line": 157,
"column": 36
} | [
{
"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",
"Order.PFilter.nonempty",
"Submodule.completeLattice",
"Partia... | refine Order.IsPFilter.of_def ?nonempty ?directed ?mem_of_le
· obtain ⟨J, hJ⟩ := G.nonempty
exact ⟨J, J, hJ, isTorsionQuot_self F J⟩
· rintro I ⟨K, hK, hIK⟩ J ⟨L, hL, hJL⟩
refine ⟨I ⊓ J, ?_, inf_le_left, inf_le_right⟩
exact ⟨K ⊓ L, G.inf_mem hK hL,
(hIK.anti_right inf_le_left).inf (hJL.anti_right ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 81,
"column": 47
} | {
"line": 81,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nx✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}\nB : RingFilterBasis A\nhB : B.sets = {x | ∃ I ∈ F, ↑I = x}\nI : Ideal A\nhI : I ∈ F\na : A\nJ : Ideal A\nhJ : J ∈ F\nhbasis : ↑J ∈ B\nhsub : ↑J ⊆ (fun x ↦ x * a) ⁻¹' ↑I\nx : A\nhx : x ∈ J\n⊢ x ∈ Submodule.colon I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 25
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\ns : Set (WithIdealFilter F)\n⊢ s ∈ 𝓝 0 ↔ ∃ I ∈ F, idealSet I ⊆ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 57
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 55,
"column": 19
} | {
"line": 55,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nσ : Type u_3\ninst✝ : Finite σ\n⊢ ringKrullDim R + ↑(Nat.card σ) ≤ ringKrullDim (MvPolynomial σ R)",
"usedConstants": [
"CharP.cast_eq_zero",
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"ringKrullDim_succ_le_ringKrullDim_polynomial"... | induction σ using Finite.induction_empty_option with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 203,
"column": 78
} | {
"line": 203,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nthis✝ : p.IsPrime\nhp : maximalIdeal (Localization p.primeCompl) ∈ (span (⇑(algebraMap R (Localization p.primeCompl)) '' ↑s)).minimalPrimes\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 227,
"column": 10
} | {
"line": 227,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nn : ℕ\nH :\n ∀ m < n + 1,\n ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),\n p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m\nw✝ : IsLocalRing R\nthis✝ : (maximalIdeal R).IsPrime\nq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 28,
"column": 19
} | {
"line": 28,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis : IsPrincipalIdealRing (R ⧸ P)\n⊢ RingHom.ker (Ideal.Quotient.mk P) ≤ I",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis✝¹ : IsPrincipalIdealRing (R ⧸ P)\nthis✝ : (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal\nthis : (Ideal.comap (Ideal.Quotient.mk P) (Ideal.map (Ideal.Quotient.mk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ m.height = 1",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
"WithBot",
... | refine le_antisymm ?_ ?_
· suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
· apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ m.height = 1",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
"WithBot",
... | refine le_antisymm ?_ ?_
· suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
· apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nhI : p ∈ (span ↑s).minimalPrimes\n⊢ Cardinal.toENat (Submodule.spanRank (span ↑s)) ≤ ↑s.card",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"ENat.instNatCast",
"Cardinal",
"C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Module | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 22
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : CommRing R\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsLocalRing R\ndim : Module.supportDim R N = 0\nx✝ : Nontrivial N := ⋯\np : PrimeSpectrum R\nhp : p ∈ Module.support R N\nnmem : ¬p = { asIdeal := maximalIdeal R, isPrime := ⋯ }\nthis : p < { ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Module | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 15
} | [
{
"pp": "case a\nR : Type u_1\ninst✝³ : CommRing R\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsLocalRing R\ndim : Module.supportDim R N = 0\nx✝ : Nontrivial N := ⋯\n⊢ {{ asIdeal := maximalIdeal R, isPrime := ⋯ }} ≤ Module.support R N",
"usedConstants": [
"Eq.mpr",
"Pri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 310,
"column": 10
} | {
"line": 310,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ∞\nI : Ideal R\nhp : p ∈ I.minimalPrimes\nhI : Submodule.spanRank I ≤ ↑n\n⊢ Cardinal.toENat (Submodule.spanRank I) ≤ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 60
} | [
{
"pp": "case h.e'_4\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ Cardinal.toENat (Submodule.spanRank I) = ↑(Submodule.spanFinrank I)",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 340,
"column": 17
} | {
"line": 340,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R\nhx : (Quotient.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 49,
"column": 71
} | {
"line": 49,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 24
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 98,
"column": 73
} | {
"line": 98,
"column": 84
} | [
{
"pp": "R : 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 ∧ RelSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 20
} | [
{
"pp": "R : 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 ∧ RelSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 102,
"column": 27
} | {
"line": 102,
"column": 45
} | [
{
"pp": "R : 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 ∧ RelSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 102,
"column": 75
} | {
"line": 102,
"column": 89
} | [
{
"pp": "R : 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 ∧ RelSe... | by simp [← hh] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 68,
"column": 31
} | {
"line": 68,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 69,
"column": 30
} | {
"line": 69,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 15
} | [
{
"pp": "case nil\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nreg : Sequence.IsRegular M []\n⊢ supportDim R (M ⧸ ⊥) + ↑[].length = supportDim R M",
"usedConstants": [
"Ch... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nx : R\nrs' : List R\nih :\n ∀ {M : Type u_2} [inst : AddCommGroup M] [inst_1 : Module R M] [Module.Finite R M],\n Sequence.IsRegular M rs' → supportDim R (M ⧸ ofList rs' • ⊤) + ↑rs'.length = supportDim R M\nM : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Lasker | {
"line": 66,
"column": 2
} | {
"line": 90,
"column": 70
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\n⊢ ∃ t,\n t.inf id = N ∧\n (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧\n ... | classical
refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
fun t ↦ t.inf id, ?_, ?_, ?_⟩
· ext
grind [Finset.inf_image, Submodule.mem_finsetInf]
· simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Lasker | {
"line": 66,
"column": 2
} | {
"line": 90,
"column": 70
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\n⊢ ∃ t,\n t.inf id = N ∧\n (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧\n ... | classical
refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
fun t ↦ t.inf id, ?_, ?_, ?_⟩
· ext
grind [Finset.inf_image, Submodule.mem_finsetInf]
· simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Lasker | {
"line": 66,
"column": 2
} | {
"line": 90,
"column": 70
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\n⊢ ∃ t,\n t.inf id = N ∧\n (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧\n ... | classical
refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
fun t ↦ t.inf id, ?_, ?_, ?_⟩
· ext
grind [Finset.inf_image, Submodule.mem_finsetInf]
· simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Lasker | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 43
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : InfIrred N\na : R\nb : M\nhab : a • b ∈ N\nf : ℕ → Submodule R M := fun n ↦ { carrier := {x | a ^ n • x ∈ N}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Lasker | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 37
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\na : R\nb : M\nf : ℕ → Submodule R M := fun n ↦ { carrier := {x | a ^ n • x ∈ N}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhf : Monotone f\nn : ℕ\nhab : a •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalIso | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : ⨆ i, PrimeSpectrum.basicOpen (f i) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LittleWedderburn | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 15
} | [
{
"pp": "n : ℕ\nIH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nval✝ : Fintype D\nhn : card D = n\nR : Subring D\nhR : R < ⊤\nx y : D\nhx : x ∈ R\nhy : y ∈ R\nthis : ∀ (g : ↥R), g * ⟨... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 72,
"column": 6
} | {
"line": 73,
"column": 35
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nX : ModuleCat R\ninj : Module.Injective R ↑X\nx✝ : Small.{v, u} (Localization S) := small_of_surjective Localization.mkHom_surjective\n⊢ Module.Injective (Localization S) ↑((localizedModuleFunctor S).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n → injectiveDimension (M.localizedModule S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n → injectiveDimension (M.localizedModule S... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\nh : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)\np : Ideal R\nhp : p.IsMaximal\nthis : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 156,
"column": 15
} | {
"line": 156,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nX : ModuleCat R\nproj : Module.Projective R ↑X\nthis : Small.{v, u} (Localization S)\n⊢ Module.Projective (Localization S) ↑((localizedModuleFunctor S).1 X)",
"usedConstants": [
"AddCommGroup.toAddCommMonoid",
"Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 80,
"column": 15
} | {
"line": 80,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n → projectiveDimension (M.localizedModule S) ≤ ↑↑n",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n → projectiveDimension (M.localizedModule S) ≤ ↑↑n",
"usedConsta... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 49
} | [
{
"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\nthis : Small.{v,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 13
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nH : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)\nI : Ideal R\nx✝¹ : FinitePresentation R ↥I := finit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 12
} | [
{
"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 ... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 147,
"column": 15
} | {
"line": 147,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 169,
"column": 15
} | {
"line": 169,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 68
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\nG : Type u\ninst✝³ : Group G\ninst✝² : MulSemiringAction G B\ninst✝¹ : SMulCommClass G A B\ninst✝ : TopologicalSpace G\nQ : Ideal B\nN N' : OpenNormalSubgroup G\ne : N ≤ N'\nh : FixedPoints.subalgebra A B ↥↑N'.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 65,
"column": 19
} | {
"line": 65,
"column": 48
} | [
{
"pp": "case pos.succ\nA : 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\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 35
} | [
{
"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\nh : length B M ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 116,
"column": 19
} | {
"line": 116,
"column": 48
} | [
{
"pp": "case pos.succ\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\ns : Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 53
} | [
{
"pp": "A : 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 : length A M = ⊤\nthis : length B (B ⊗[A] M) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 56
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nh : IsUnit fun i ↦ if i = i₁ then 0 else 1\n⊢ IsUnit 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"NeZero.one"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 53,
"column": 45
} | {
"line": 53,
"column": 66
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nha : ¬IsUnit fun i ↦ if i = i₁ then 0 else 1\nh : IsUnit fun i ↦ if i = i₁ then 1 else 0\n⊢ IsUnit 0",
"usedConstants": [
"Eq.mpr",
"Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 60,
"column": 43
} | {
"line": 60,
"column": 54
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nh : IsUnit (1, 0)\n⊢ IsUnit 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"NeZero.one",
"IsUnit",
"id",
"MulZeroOneClass.toM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 54
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nha : ¬IsUnit (1, 0)\nh : IsUnit (0, 1)\n⊢ IsUnit 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"NeZero.one",
"IsUnit",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Rat | {
"line": 28,
"column": 8
} | {
"line": 28,
"column": 77
} | [
{
"pp": "q : ℚ\n⊢ IsRelPrime q.num ↑⟨↑q.den, ⋯⟩",
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"False",
"Int.instIsStrictOrderedRing",
"Rat.num",
"congrArg",
"CommSemiring.toSemiring",
"Int.instLinearOrder",
"Int.euclideanDomain",
"Rat.den_ne_zero._... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Rat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 41
} | [
{
"pp": "q : ℚ\n⊢ (↑(IsFractionRing.den ℤ q)).natAbs = q.den",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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