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.PolynomialLaw.Basic | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommSemiring R\nM : Type u_1\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_2\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M →ₚₗ[R] N\nS : Type u\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nS' : Type u\ninst✝¹ : CommSemiring S'\ninst✝ : Algebra R S'\nφ : S →ₐ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 49
} | [
{
"pp": "case h.e'_3.h.e'_13\nR : 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\nf : M →ₚₗ[R] N\nS : Type u\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nx : M\n⊢ 1 ⊗ₜ[R] x = (rTensor M (Algebra.algHom R... | · rw [rTensor_tmul, toLinearMap_apply, map_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 49
} | [
{
"pp": "case right\nR : Type u_1\nS : Type u_2\nN : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nA : Subalgebra R S\nhA : A.FG\nt : ↥A ⊗[R] N\nA' : Subalgebra R S\nhA' : A'.FG\nt' : ↥A' ⊗[R] N\nhj : (A ⊔ A').val.comp (Subalgebra.inc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\na : R\n⊢ IsRestricted c (PowerSeries.C a)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 73,
"column": 2
} | {
"line": 75,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf g : R⟦X⟧\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → |‖(coeff n) f‖| * |c| ^ n < ε\nhg : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → |‖(coeff n) g‖| * |c| ^ n < ε\nε : ℝ\nhε : 0 < ε\nfN gN : ℕ\nhfN : ∀ (n : ℕ), fN ≤ n → ‖(coeff n) f‖ * |c| ^ n < ε / 2\nh... | calc _ ≤ ‖(coeff n) f‖ * |c| ^ n + ‖(coeff n) g‖ * |c| ^ n := by grw [norm_add_le, add_mul]
_ < ε / 2 + ε / 2 := by gcongr <;> grind
_ = ε := by ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf : IsRestricted c f\n⊢ IsRestricted c (-f)",
"usedConstants": [
"MvPowerSeries.instAddCommGroup",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 81,
"column": 20
} | {
"line": 81,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf : IsRestricted c f\nr : R\nh : r = 0\n⊢ IsRestricted c (r • f)",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MvPowerSeries.instZero",
"Semiring.toModule",
"NormedRing.toRing",
"Ring.toNonAssocRing",
"co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 13
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ ∀ x ∈ range n,\n (coeff x) (X * largeSchroderSeries) * (n - x).largeSchroder =\n if 0 < x then (x - 1).largeSchroder * (n - x).largeSchroder else 0",
"usedConstants": [
"Nat"
]
}
] | intro x a | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 100,
"column": 23
} | {
"line": 100,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖‖(coeff n) f‖ * c ^ n‖ < ε\n⊢ ?m.12",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 112,
"column": 21
} | {
"line": 112,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf✝ : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖‖(coeff n) f‖ * c ^ n‖ < ε\nN : ℕ\nhf : ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < 1\ni : ℕ\nh : N ≤ i\n⊢ ‖(coeff i) f‖ * |c ^ i| ≤ 1",
"usedConstants": [
"Real.instIsOrderedRing",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf✝ : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖‖(coeff n) f‖ * c ^ n‖ < ε\nN : ℕ\nhf : ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < 1\ni : ℕ\nh : N ≤ i\n⊢ ‖(coeff i) f‖ * |c ^ i| ≤ 1",
"usedConstants": [
"Real.instIsOrderedRing",
"... | by simpa using (hf i h).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 15
} | [
{
"pp": "case pos.refine_1\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhI : X ∈ I\nhfg : (Ideal.map constantCoeff I).FG\n⊢ (insert X (⇑C '' generators (Ideal.map constantCoeff I))).Finite",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"CommSemiring.toSemiring",
"RingHom",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → R\nhT : ∀ g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 134,
"column": 21
} | {
"line": 134,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → ... | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → ... | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → ... | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 123,
"column": 11
} | {
"line": 123,
"column": 49
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhI : X ∉ I\nS : Set R\nhSI : span S = Ideal.map constantCoeff I\nhS : S.Finite\nr : R\nhr : r ∈ S\n⊢ r ∈ ⇑constantCoeff '' ↑I",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 27
} | [
{
"pp": "n : ℕ\n⊢ radical n ≤ 1 ↔ n ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 36
} | [
{
"pp": "n : ℕ\n⊢ n < radical n ↔ n = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 27
} | [
{
"pp": "z : ℤ\n⊢ radical z ≤ 1 ↔ z.natAbs ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 35
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\ni : ℕ\nh : i < ((map (Ideal.Quotient.mk I)) g).order.toNat\n⊢ (coeff i) g ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Category | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 62
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nr : R\n⊢ Epi (M.smulShortComplex r).g",
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"Submodule",
"instHSMul",
"ModuleCat.smulShortComplex._proof_1",
"CategoryTheory.Epi",
"ModuleCat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Category | {
"line": 54,
"column": 15
} | {
"line": 54,
"column": 85
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nr : R\nreg : IsSMulRegular (↑M) r\n⊢ Mono (M.smulShortComplex r).f",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"CategoryTheory.Mono",
"ModuleCat",
"congrArg",
"CommSemiring.toSemiring",
"CategoryTheor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ (coeff i) (H.div f - H.seq f k) ∈ I ^ k",
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Sem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Flat | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module S N\ninst✝⁵ : IsScalarTower R S N\np : Ideal R\ninst✝⁴ : p.IsPrime\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.div 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.mod 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 434,
"column": 71
} | {
"line": 434,
"column": 82
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 439,
"column": 71
} | {
"line": 439,
"column": 82
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nr : R\nreg : IsSMulRegular M r\nmem_ann : r ∈ annihilator R N\nf : N →ₗ[R] M\nx : N\nthis : r • f x = r • 0\n⊢ f x = 0 x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 80
} | [
{
"pp": "case inl\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Subsingleton M\n... | exact ⟨0, ⟨Submodule.zero_mem (Module.annihilator R N), IsSMulRegular.zero⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 57
} | [
{
"pp": "case neg.le\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsLocalRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nisf : ¬IsField R\nx : R\nhx : maximalIdeal R = R ∙ x\n⊢ ↑(Submodule.spanFinrank (maximalIdeal R)) ≤ ringKrullDim R",
"usedConstants": [
"WithBot.addMonoidWithOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 108,
"column": 2
} | {
"line": 112,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nR' : Type u_2\ninst✝¹ : CommRing R'\ne : R ≃+* R'\ninst✝ : IsRegularRing R\n⊢ IsRegularRing R'",
"usedConstants": [
"Eq.mpr",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",
"IsRegularRing.toIsNoetherian",
... | have := isNoetherianRing_of_ringEquiv R e
rw [isRegularRing_iff]
intro p hp
exact IsRegularLocalRing.of_ringEquiv <| IsLocalization.ringEquivOfRingEquiv
(Localization.AtPrime (p.comap e)) (Localization.AtPrime p) e (e.map_primeCompl_comap_eq p) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 108,
"column": 2
} | {
"line": 112,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nR' : Type u_2\ninst✝¹ : CommRing R'\ne : R ≃+* R'\ninst✝ : IsRegularRing R\n⊢ IsRegularRing R'",
"usedConstants": [
"Eq.mpr",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",
"IsRegularRing.toIsNoetherian",
... | have := isNoetherianRing_of_ringEquiv R e
rw [isRegularRing_iff]
intro p hp
exact IsRegularLocalRing.of_ringEquiv <| IsLocalization.ringEquivOfRingEquiv
(Localization.AtPrime (p.comap e)) (Localization.AtPrime p) e (e.map_primeCompl_comap_eq p) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 117,
"column": 69
} | {
"line": 117,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsRegularRing R\nx : R\n⊢ x ∈ (maximalIdeal R).primeCompl → x ∈ IsUnit.submonoid R",
"usedConstants": [
"Eq.mpr",
"mem_nonunits_iff._simp_1",
"Semiring.toModule",
"Classical.not_not._simp_1",
"congrArg"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
"HMul.hMul",
"congrArg",
"CommSem... | simp [H.eq_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
"HMul.hMul",
"congrArg",
"CommSem... | simp [H.eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
"HMul.hMul",
"congrArg",
"CommSem... | simp [H.eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 72
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 800,
"column": 12
} | {
"line": 800,
"column": 23
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 800,
"column": 12
} | {
"line": 800,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 800,
"column": 12
} | {
"line": 800,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 800,
"column": 12
} | {
"line": 800,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 889,
"column": 29
} | {
"line": 889,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorizati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 899,
"column": 29
} | {
"line": 899,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorizati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 24
} | [
{
"pp": "case h\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\nn✝ n : ℕ\nS : ShortCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 126,
"column": 79
} | {
"line": 130,
"column": 50
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ IsIsotypic R ↥N ↔ ∀ m ≤ N, ∀ [IsSimpleModule R ↥m], IsIsotypicOfType R ↥N ↥m",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.MapSubtype.orderIso",
"RingHomSurj... | by
rw [Subtype.forall', ← (Submodule.MapSubtype.orderIso N).forall_congr_right]
have e := Submodule.equivMapOfInjective _ N.subtype_injective
simp_rw [Submodule.MapSubtype.orderIso, Equiv.coe_fn_mk, ← (e _).isSimpleModule_iff,
← (e _).isIsotypicOfType_iff_type, IsIsotypic] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 83
} | [
{
"pp": "case bot\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ), pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 142,
"column": 15
} | {
"line": 142,
"column": 26
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 14
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ)... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 15
} | [
{
"pp": "case zero\nR : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nM : ModuleCat R\ninst✝¹ : Nontrivial ↑M\ninst✝ : Module.Finite R ↑M\nrs : List R\nreg : IsWeaklyRegular (↑M) rs\nmem : ∀ r ∈ rs, r ∈ maximalIdeal R\nlen : rs.length = 0\n⊢ projectiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 78
} | [
{
"pp": "R : Type u_5\ninst✝ : Semiring R\nI : Ideal R\n⊢ Submodule.IsFullyInvariant I ↔ I.IsTwoSided",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingEquiv.toEquiv",
"Semiring.toModule",
"Equiv.instEquivLike",
"HMul.hMul",
"RingEquiv.moduleEndSelf",
"MulOppos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 162,
"column": 6
} | {
"line": 165,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nn : ℕ\nhn :\n ∀ (M : ModuleCat R) [Nontrivial ↑M] [Module.Finite R ↑M] (rs : List R),\n IsWeaklyRegular (↑M) rs →\n (∀ r ∈ rs, r ∈ maximalIdeal R) →\n rs.length = n → projectiv... | rw [Nat.cast_add, Nat.cast_one, projectiveDimension_eq_of_iso
(Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner M x rs').toModuleIso, add_comm _ 1,
← add_assoc, ← projectiveDimension_quotSMulTop_eq_succ_of_isSMulRegular M x reg.1 mem.1,
← hn (ModuleCat.of R (QuotSMulTop x M)) rs' reg.2 mem.2... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 368,
"column": 25
} | {
"line": 368,
"column": 51
} | [
{
"pp": "R₀ : Type u_1\nR : Type u_2\nM : Type u\nN✝ : Type u_3\nS : Type u_4\ninst✝¹⁰ : CommSemiring R₀\ninst✝⁹ : Ring R\ninst✝⁸ : Algebra R₀ R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N✝\ninst✝² : Module R S\ninst✝¹ : IsSimpleModule R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ maximalIdeal R",
"usedConstants": [
"Submodule",
"instHSMul",
"Semiring.toModule",
"... | apply IsLocalRing.le_maximalIdeal reg.2.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 15
} | [
{
"pp": "case a\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ Ideal.ofList rs • ⊤",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Maximal.Topology | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : MaximalSpectrum R\n⊢ toPrimeSpectrum ⁻¹' {x.toPrimeSpectrum} = {x}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instSingletonSet",
"id",
"MaximalSpectrum",
"MaximalSpectrum.toPrimeSpectrum",
"CommRing.toCommSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 53,
"column": 78
} | {
"line": 64,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ imageOfDf f = PrimeSpectrum.comap C '' (zeroLocus {f})ᶜ",
"usedConstants": [
"Set.ext",
"Set.singleton_subset_iff",
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.mk",
"Ideal.subset_span",
"PrimeSpectrum.ext",
"... | by
ext x
refine ⟨fun hx => ⟨⟨map C x.asIdeal, isPrime_map_C_of_isPrime⟩, ⟨?_, ?_⟩⟩, ?_⟩
· rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]
obtain ⟨i, hi⟩ := hx
exact fun a => hi (mem_map_C_iff.mp a i)
· ext x
refine ⟨fun h => ?_, fun h => subset_span (mem_image_of_mem C.1 h)⟩
rw [← @coeff... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 70,
"column": 23
} | {
"line": 70,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (PrimeSpectrum.comap C '' Uᶜᶜ)",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.zeroLocus",
"congrArg",
"CommSemiring.toSemiring",
"Compl.com... | ← z, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | [
{
"pp": "case refine_1.a.h.h\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 15
} | [
{
"pp": "case refine_2.a.h.h\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 54
} | [
{
"pp": "case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"usedConstants": [
"Ring.toNonAssocRing",
"TwoSidedIdeal",
"List.map",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 54
} | [
{
"pp": "case cons.inr\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nhal : ∃ a ∈ l, f a ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"usedConstants": [
"Ring.toNonAssocRing",
"TwoSidedIdeal",
"L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | [
{
"pp": "case mk\nR : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\na✝ : List ι\nhs : ∃ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, f x ∈ I\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod ∈ I",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nval✝ : Multiset ι\nnodup✝ : val✝.Nodup\nhs : ∃ x ∈ { val := val✝, nodup := nodup✝ }, f x ∈ I\n⊢ { val := val✝, nodup := nodup✝ }.prod f ∈ I",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 43
} | [
{
"pp": "case refine_2.smul\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nr : R\nhr : r ∈ I\nm : M\nhm : m ∈ f.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TotallySplit | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 57
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | apply Module.nontrivial_of_rankAtStalk_pos (R := R) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nthis : ∀ {x : R} (h : x ∈ I * J), f (J.toCotangent ⟨x, ⋯⟩) = 0\nx : J.Cotangent\nhx : x ∈ Submodule.map J.toCot... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx y : R\n⊢ vA ((algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 28
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 36
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 45
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 24
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 189,
"column": 33
} | {
"line": 189,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 73
} | [
{
"pp": "p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : IsUnit ↑p\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 25
} | [
{
"pp": "p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : ¬IsUnit ↑p\na b : 𝕎 k\nhab : ↑p = a * b\n⊢ a ≠ 0 ∧ b ≠ 0",
"usedConstants": [
"WittVector.instZero",
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"MulZeroClass.toMul",
... | ← mul_ne_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 15
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range n\n⊢ range (x + 1) ⊆ range n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instOne",
"Finset",
"Order.add_one_le_iff._simp_1",
"id",
"HasSubset.Subset",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 17
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(0, x)} ⊆ univ ×ˢ range (n + 1)",
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ",
"Nat.instOne",
"SProd.sprod",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 17
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(1, x)} ⊆ univ ×ˢ range (n + 1)",
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ",
"Nat.instOne",
"SProd.sprod",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 26
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : solution p a₁ a₂ = 0\nthis : 0 = a₂.coeff 0 / a₁.coeff 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 62
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : frobeniusRotation p ha₁ ha₂ = 0\n⊢ solution p a₁ a₂ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 35
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\nkey : frobenius b * r' = q' * b\nhq''' : q' ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 31
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na : FractionRing (𝕎 k)\nm : ℕ\nr' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhr0 : ↑p ^ m * r' ≠ 0\nn : ℕ\nq' : 𝕎 k\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nhq0 : ↑p ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.Torsion | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 29
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : (ZMod p)ˣ\n⊢ x✝ ∈ rootsOfUnity (p - 1) (ZMod p) ↔ x✝ ∈ ⊤",
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"iff_true",
"M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 15
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα✝ : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\nh✝ : Nonempty α\nhα : ℵ₀ < cof α\na : α\nf : ↑s → α → α\nhf : ∀ (x : ↑s) (x_1 : α), f x x_1 ∈ ↑x ∧ x_1 ≤ f x x_1\ng : ℕ → α := fun t ↦ Nat.rec a (fun x IH ↦ sSup (S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 31
} | [
{
"pp": "case h\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\nhα : cof α ≠ ℵ₀\nhf : IsNormal f\nh✝ : Nonempty α\na : α\nval✝ : OrderTop α\n⊢ ⊤ ∈ Function.fixedPoints f ∧ a ≤ ⊤",
"usedConstants": [
"Eq.mpr",
"and_true",
"congrArg",
"le_top._simp_2",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 17
} | [
{
"pp": "case inr.refine_2.inr\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\nhα : cof α ≠ ℵ₀\nhf : IsNormal f\nh✝¹ : Nonempty α\na : α\nh✝ : NoMaxOrder α\nh : IsCofinal (Set.range fun n ↦ f^[n] a)\n⊢ #↑(Set.range fun n ↦ f^[n] a) ≤ ℵ₀",
"usedConstants": [
"Eq.mpr",
"Ca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 137,
"column": 2
} | {
"line": 215,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | have surjP : Function.Surjective (algebraMap P.Ring S) := fun x ↦ ⟨P.σ x, P.algebraMap_σ x⟩
apply (Algebra.FormallySmooth.iff_split_injection surjP).mpr
have surjP' : Function.Surjective (algebraMap P'.Ring S') := fun x ↦ ⟨P'.σ x, P'.algebraMap_σ x⟩
rcases (Algebra.FormallySmooth.iff_split_injection surjP').mp sm... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 137,
"column": 2
} | {
"line": 215,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | have surjP : Function.Surjective (algebraMap P.Ring S) := fun x ↦ ⟨P.σ x, P.algebraMap_σ x⟩
apply (Algebra.FormallySmooth.iff_split_injection surjP).mpr
have surjP' : Function.Surjective (algebraMap P'.Ring S') := fun x ↦ ⟨P'.σ x, P'.algebraMap_σ x⟩
rcases (Algebra.FormallySmooth.iff_split_injection surjP').mp sm... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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