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.Algebra.ContinuedFractions.Computation.ApproximationCorollaries | {
"line": 113,
"column": 22
} | {
"line": 113,
"column": 87
} | [
{
"pp": "K : Type u_1\nv : K\ninst✝⁴ : Field K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Archimedean K\nε : K\nε_pos : ε > 0\nN' : ℕ\none_div_ε_lt_N' : 1 / ε < ↑N'\nN : ℕ := max N' 5\nn : ℕ\nn_ge_N : n ≥ N\ng : GenContFract K := of v\nnot_terminatedAt_n : ¬g.Terminat... | exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Idempotents | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\nI : Type u_3\ne : I → R\ninst✝ : Fintype I\nhe : OrthogonalIdempotents e\n⊢ ∑ i, i.elim (1 - ∑ i, e i) e = 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Finset.univ",
"Ring.toNonAssocRing",
"AddGroupWithOne.... | rw [Fintype.sum_option] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Trace | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\n⊢ trace R (Dual R M) ∘ₗ Dual.transpose = trace R M",
"usedConstants": [
"dualTensorHomEquiv",
"Algebra.to_smulCommClass",
"Semiring.toModule... | let e := dualTensorHomEquiv R M M | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Field.GeomSum | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 53
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx : K\nh : x ≠ 1\nn : ℕ\n⊢ ∑ i ∈ range n, x ^ i = (x ^ n - 1) / (x - 1)",
"usedConstants": [
"False",
"eq_false",
"AddGroupWithOne.toAddGroup",
"congrArg",
"_private.Mathlib.Algebra.Field.GeomSum.0.geom_sum_eq._simp_1_1",
"Ad... | have : x - 1 ≠ 0 := by simp_all [sub_eq_iff_eq_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Field.TransferInstance | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : DivisionRing β\nadd_group_with_one : AddGroupWithOne α := e.addGroupWithOne\ninv : Inv α := e.Inv\ndiv : Div α := e.div\nmul : Mul α := e.mul\nnpow : Pow α ℕ := Equiv.pow ℕ e\nzpow : Pow α ℤ := Equiv.pow ℤ e\nnnratCast : NNRatCast α := e.nnratCast\n⊢ Divis... | let ratCast := e.ratCast | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Field.TransferInstance | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : Field β\nadd_group_with_one : AddGroupWithOne α := e.addGroupWithOne\nneg : Neg α := e.Neg\ninv : Inv α := e.Inv\ndiv : Div α := e.div\nmul : Mul α := e.mul\nnpow : Pow α ℕ := Equiv.pow ℕ e\nzpow : Pow α ℤ := Equiv.pow ℤ e\nnnratCast : NNRatCast α := e.nnr... | let ratCast := e.ratCast | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n⊢ Function.Injective toWord",
"usedConstants": [
"FreeGroup.Red.Step",
"FreeGroup.toWord",
"FreeGroup.reduce.exact",
"Quot.ind",
"List",
"Bool",
"Prod",
"Eq",
"FreeGroup",
"Quot.mk"
]
}
] | rintro ⟨L₁⟩ ⟨L₂⟩; exact reduce.exact | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n⊢ Function.Injective toWord",
"usedConstants": [
"FreeGroup.Red.Step",
"FreeGroup.toWord",
"FreeGroup.reduce.exact",
"Quot.ind",
"List",
"Bool",
"Prod",
"Eq",
"FreeGroup",
"Quot.mk"
]
}
] | rintro ⟨L₁⟩ ⟨L₂⟩; exact reduce.exact | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 109,
"column": 2
} | {
"line": 114,
"column": 84
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\nhx : x ^ (Polynomial.map (algebraMap R S) f).natDegree = -∑ i, (Polynomial.map (algebraMap R S) f).coeff ↑i * x ^ ↑i\nφ : ℕ → R\nhφ : ∀ n < ... | conv_rhs at hx =>
congr
congr
· skip
ext i
rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 natDegree_map_le), map_mul, mul_assoc] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 146,
"column": 48
} | {
"line": 146,
"column": 65
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : f.IsWeaklyEisensteinAt 𝓟\nx : R\nhroot : f.coeff f.natDegree * x ^ f.natDegree + ∑ x_1 ∈ range f.natDegree, f.coeff x_1 * x ^ x_1 = 0\nhmo : f.Monic\ni : ℕ\nhi : f.natDegree ≤ i\nk : ℕ\nhk : i = f.natDegree + k\n⊢ x ^ f.natDegree ∈ 𝓟",
... | Finset.sum_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ⇑f\ninj : Function.Injective ⇑(algebraMap R B)\ncl : ∀ {x : B}, IsIntegral R x ↔ ∃ y, (algebraMap R B... | convert! inj | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain R\nM : Submonoid R\nhM : M ≤ R⁰\ninst✝ : IsLocalization M S\nK : Type u_1 := FractionRing R\ng : S →+* K := IsLocalization.map K (RingHom.id R) hM\n⊢ IsIntegrally... | letI := g.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 78
} | [
{
"pp": "case h.e'_1.h.e'_4.h.e'_5.hf\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nP : Ideal R\nhP : P.IsPrime\nhfl : f.leadingCoeff ∉ P\nhfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P\nhfd0 : 0 < f.degree\nh0 : f.coeff 0 ∉ P ^ 2\nhu : f.IsPrimitive\n⊢ Function.Injective ⇑(algebraMap (R ⧸ P... | · exact FaithfulSMul.algebraMap_injective (R ⧸ P) (FractionRing (R ⧸ P)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Quaternion | {
"line": 613,
"column": 83
} | {
"line": 613,
"column": 97
} | [
{
"pp": "R : Type u_3\nc₁ c₂ c₃ : R\na : ℍ[R,c₁,c₂,c₃]\ninst✝ : CommRing R\n⊢ a + star a = ↑(2 * a.re + c₂ * a.imI)",
"usedConstants": [
"Eq.mpr",
"QuaternionAlgebra.imI",
"HMul.hMul",
"Ring.toNonAssocRing",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwo... | self_add_star' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Quaternion | {
"line": 1221,
"column": 59
} | {
"line": 1221,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na b : ℍ[R]\nx✝¹ : ℚ\nx✝ : ℍ[R]\n⊢ x✝¹ • x✝ = ↑x✝¹ • x✝",
"usedConstants": [
"NegZeroClass.toNeg",
"instHSMul",
"QuaternionAlgebra.imI",
"instSMulOfMul",
"DivisionRing.t... | ext <;> exact Rat.smul_def .. | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Group.NatPowAssoc | {
"line": 134,
"column": 42
} | {
"line": 134,
"column": 62
} | [
{
"pp": "R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn : ℤ\nm : ℕ\n⊢ ↑(n ^ m) * ↑n = ↑n ^ (m + 1)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring",
"inst... | Int.cast_npow R n m, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 263,
"column": 60
} | {
"line": 264,
"column": 84
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nn : ℕ\nIH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !\nthis : (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i\ni : ℕ\nhi : i ∈ range n\n⊢ (↑((n + 1).choose i) * Δ_[1]^[n] fun r ↦ r ^ i) = 0",
"usedConstants": [
"Eq.mpr",
"NonA... | by
rw [fwdDiff_iter_pow_eq_zero_of_lt (by have := mem_range.1 hi; lia), mul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Translate | {
"line": 59,
"column": 23
} | {
"line": 59,
"column": 37
} | [
{
"pp": "α : Type u_2\nG : Type u_5\ninst✝ : AddCommGroup G\na b : G\nf : G → α\n⊢ τ (a + b) f = τ b (τ a f)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"id",
"instHAdd",
"HAdd.hAdd",
"translate",
"AddCommSemigroup.toAddCommMagma... | translate_add' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Category.NonemptyFinLinOrd | {
"line": 196,
"column": 4
} | {
"line": 206,
"column": 29
} | [
{
"pp": "case refine_1\nX Y : NonemptyFinLinOrd\nf : X ⟶ Y\nhf : Epi f\nH : ∀ (y : ↑Y.toLinOrd), Nonempty ↑(⇑(ConcreteCategory.hom f) ⁻¹' {y})\nφ : ↑Y.toLinOrd → ↑X.toLinOrd := fun y ↦ ↑⋯.some\nhφ : ∀ (y : ↑Y.toLinOrd), (ConcreteCategory.hom f) (φ y) = y\n⊢ Monotone φ",
"usedConstants": [
"Eq.mpr",
... | · intro a b
contrapose
intro h
simp only [not_le] at h ⊢
suffices b ≤ a by
apply lt_of_le_of_ne this
rintro rfl
exfalso
simp at h
have H : f (φ b) ≤ f (φ a) := f.hom.hom.monotone (le_of_lt h)
simpa only [hφ] using H | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 18
} | [
{
"pp": "case h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf g : P ⟶ Q\n⊢ (f + g).f = (g + f).f",
"usedConstants": [
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommGroup.toAddComm... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 55
} | [
{
"pp": "case i_surj.refine_3\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nP : Type := Fin (n + 2) × Fin (n + 3)\nS : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}\nφ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)\ni' : Fin (n + 2)\nj' : Fin (n + 3)... | · simp only [φ, Fin.castLT_castSucc, Fin.succ_pred] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 659,
"column": 42
} | {
"line": 667,
"column": 46
} | [
{
"pp": "A✝ B✝ : SimplexCategory\nf : A✝ ⟶ B✝\nhf : IsIso ((forget SimplexCategory).map f)\ny₁ y₂ : Fin (B✝.len + 1)\nh : y₁ ≤ y₂\n⊢ (ConcreteCategory.hom (inv ((forget SimplexCategory).map f))) y₁ ≤\n (ConcreteCategory.hom (inv ((forget SimplexCategory).map f))) y₂",
"usedConstants": [
"not_le",
... | by
by_cases h' : y₁ < y₂
· by_contra h''
apply not_le.mpr h'
convert! f.toOrderHom.monotone (le_of_not_ge h'')
all_goals
exact (ConcreteCategory.congr_hom (Iso.inv_hom_id
(asIso ((forget Simpl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 685,
"column": 35
} | {
"line": 685,
"column": 74
} | [
{
"pp": "n : SimplexCategory\nf : n ⟶ n\nhf : Function.Injective ⇑(Hom.toOrderHom f)\nh : n.len = n.len\n⊢ Function.Surjective (Hom.toOrderHom f).toFun",
"usedConstants": [
"Eq.mpr",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"congrArg",
"PartialOrder.toPreorder",... | rwa [← Finite.injective_iff_surjective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 685,
"column": 35
} | {
"line": 685,
"column": 74
} | [
{
"pp": "n : SimplexCategory\nf : n ⟶ n\nhf : Function.Injective ⇑(Hom.toOrderHom f)\nh : n.len = n.len\n⊢ Function.Surjective (Hom.toOrderHom f).toFun",
"usedConstants": [
"Eq.mpr",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"congrArg",
"PartialOrder.toPreorder",... | rwa [← Finite.injective_iff_surjective] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 685,
"column": 35
} | {
"line": 685,
"column": 74
} | [
{
"pp": "n : SimplexCategory\nf : n ⟶ n\nhf : Function.Injective ⇑(Hom.toOrderHom f)\nh : n.len = n.len\n⊢ Function.Surjective (Hom.toOrderHom f).toFun",
"usedConstants": [
"Eq.mpr",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"congrArg",
"PartialOrder.toPreorder",... | rwa [← Finite.injective_iff_surjective] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 682,
"column": 2
} | {
"line": 685,
"column": 75
} | [
{
"pp": "n m : SimplexCategory\nf : n ⟶ m\nhf : Mono f\n⊢ IsIso f ↔ n.len = m.len",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.Mono",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | refine ⟨fun _ ↦ len_eq_of_isIso f, fun h ↦ ?_⟩
obtain rfl : n = m := by aesop
rw [mono_iff_injective] at hf
exact isIso_of_bijective ⟨hf, by rwa [← Finite.injective_iff_surjective]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 682,
"column": 2
} | {
"line": 685,
"column": 75
} | [
{
"pp": "n m : SimplexCategory\nf : n ⟶ m\nhf : Mono f\n⊢ IsIso f ↔ n.len = m.len",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.Mono",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | refine ⟨fun _ ↦ len_eq_of_isIso f, fun h ↦ ?_⟩
obtain rfl : n = m := by aesop
rw [mono_iff_injective] at hf
exact isIso_of_bijective ⟨hf, by rwa [← Finite.injective_iff_surjective]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 803,
"column": 2
} | {
"line": 805,
"column": 25
} | [
{
"pp": "n : ℕ\nθ : ⦋n + 1⦌ ⟶ ⦋n⦌\ninst✝ : Epi θ\n⊢ ∃ i, θ = σ i",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Mono",
"congrArg",
"PartialOrder.toPreorder",
"SimplexCategory.eq_σ_comp_of_not_injective",
"id",
"instOfNatNat",
"SimplexCategory.mono_iff_injectiv... | obtain ⟨i, θ', h⟩ := eq_σ_comp_of_not_injective θ (by
rw [← mono_iff_injective]
grind [→ le_of_mono]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 470,
"column": 28
} | {
"line": 470,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C ⥤ D\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { app := fun A ↦ { left := whiskerLeft (drop.obj A) (x✝¹ ≫ x✝), right := (x✝¹ ≫ x✝).app (point.obj A), w := ⋯ },\n naturality := ⋯ } =\n { a... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 470,
"column": 28
} | {
"line": 470,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C ⥤ D\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { app := fun A ↦ { left := whiskerLeft (drop.obj A) (x✝¹ ≫ x✝), right := (x✝¹ ≫ x✝).app (point.obj A), w := ⋯ },\n naturality := ⋯ } =\n { a... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 470,
"column": 28
} | {
"line": 470,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C ⥤ D\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { app := fun A ↦ { left := whiskerLeft (drop.obj A) (x✝¹ ≫ x✝), right := (x✝¹ ≫ x✝).app (point.obj A), w := ⋯ },\n naturality := ⋯ } =\n { a... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 131,
"column": 46
} | {
"line": 133,
"column": 22
} | [
{
"pp": "i n m : ℕ\nj : Fin (m + 1)\nhi : n + i = m\nhj : i ≤ ↑j\n⊢ ↑((ConcreteCategory.hom (σ₀Iter i hi)) j) = ↑j - i",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"SimplexCategory.σ₀Iter",
"SimplexCategory.instConcreteCategoryOrderHomFinHAd... | by
dsimp [σ₀Iter, Hom.mk, ConcreteCategory.hom, Hom.toOrderHom]
rw [if_neg (by lia)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 40
} | [
{
"pp": "X : SSet\nx y : X.N\nf : ⦋x.dim⦌ ⟶ ⦋y.dim⦌\nhf : Mono f\nh✝ : (ConcreteCategory.hom (X.map f.op)) y.simplex = x.simplex\n⊢ x.dim ≤ y.dim",
"usedConstants": [
"SimplexCategory.len_le_of_mono",
"SSet.N.toS",
"SimplexCategory.mk",
"SSet.S.dim"
]
}
] | exact SimplexCategory.len_le_of_mono f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 51,
"column": 67
} | {
"line": 51,
"column": 81
} | [
{
"pp": "X : SSet\nd : ℕ\ninst✝ : X.HasDimensionLT d\nh : ∀ i < d, Finite ↑(X.nonDegenerate i)\nthis✝ : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)\nx : X.N\nhj : ¬x.dim < d\nthis : x.simplex ∈ X.nonDegenerate x.dim\n⊢ d ≤ x.dim",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 51,
"column": 67
} | {
"line": 51,
"column": 81
} | [
{
"pp": "X : SSet\nd : ℕ\ninst✝ : X.HasDimensionLT d\nh : ∀ i < d, Finite ↑(X.nonDegenerate i)\nthis✝ : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)\nx : X.N\nhj : ¬x.dim < d\nthis : x.simplex ∈ X.nonDegenerate x.dim\n⊢ d ≤ x.dim",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 51,
"column": 67
} | {
"line": 51,
"column": 81
} | [
{
"pp": "X : SSet\nd : ℕ\ninst✝ : X.HasDimensionLT d\nh : ∀ i < d, Finite ↑(X.nonDegenerate i)\nthis✝ : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)\nx : X.N\nhj : ¬x.dim < d\nthis : x.simplex ∈ X.nonDegenerate x.dim\n⊢ d ≤ x.dim",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 179,
"column": 4
} | {
"line": 180,
"column": 68
} | [
{
"pp": "case mp\nX : SSet\nx y : X.op.N\n⊢ (∃ f,\n (ConcreteCategory.hom (X.map f.op)) (mk (opObjEquiv y.simplex) ⋯).simplex =\n (mk (opObjEquiv x.simplex) ⋯).simplex) →\n ∃ f, (ConcreteCategory.hom (X.op.map f.op)) y.simplex = x.simplex",
"usedConstants": [
"SSet.S.simplex",
"SS... | · rintro ⟨f, hf⟩
exact ⟨SimplexCategory.rev.map f, by simp [op_map, dsimp% hf]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 170,
"column": 75
} | {
"line": 171,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nx y : ComposableArrows C 0\ne : Edge x y\n⊢ ComposableArrows.mk₁ (homEquiv e) = ComposableArrows.mk₁ (ComposableArrows.hom e.edge)",
"usedConstants": [
"CategoryTheory.nerve.homEquiv._proof_2",
"Equiv.instEquivLike",
"CategoryTheory.Composabl... | by
simp [homEquiv, ComposableArrows.mk₁_eqToHom_comp, ComposableArrows.mk₁_comp_eqToHom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 80
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | exact h₁₂ (by simpa only [← h, ← h₁] using ComplexShape.rel_π₂ c₁ c₁₂ i₁ h₂) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 207,
"column": 4
} | {
"line": 219,
"column": 49
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | · ext ⟨i₁, i₂⟩ h
simp only [totalAux.ιMapObj_D₂_assoc, comp_zero]
by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
· rw [totalAux.d₂_eq K c₁₂ i₁ h₃ i₁₂']; swap
· rw [← ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃, ← c₁₂.next_eq' h₁, h]
simp only [Linear.units_smul_comp, assoc, totalAux.ιMapObj_D₂]
by... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 239,
"column": 8
} | {
"line": 244,
"column": 40
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | ext X₃ j
dsimp
ext i₁ i₂ i₃ h
simp only [ι_mapTrifunctorMapMap_assoc, categoryOfGradedObjects_id, Functor.map_id,
NatTrans.id_app, ι_mapTrifunctorMapMap, id_comp,
NatTrans.naturality_app_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 239,
"column": 8
} | {
"line": 244,
"column": 40
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | ext X₃ j
dsimp
ext i₁ i₂ i₃ h
simp only [ι_mapTrifunctorMapMap_assoc, categoryOfGradedObjects_id, Functor.map_id,
NatTrans.id_app, ι_mapTrifunctorMapMap, id_comp,
NatTrans.naturality_app_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.TotalComplexSymmetry | {
"line": 98,
"column": 6
} | {
"line": 99,
"column": 65
} | [
{
"pp": "case pos\nC : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁴ : TotalComplexShape c₁ c₂ c\ninst✝³ : TotalComplexShape c₂ c₁ c\ninst✝² : T... | have h₃ : ComplexShape.π c₂ c₁ c (ComplexShape.next c₂ i₂, i₁) = j' := by
rw [← ComplexShape.next_π₁ c₁ c h₂ i₁, h₁, c.next_eq' h₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 128,
"column": 67
} | {
"line": 128,
"column": 89
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn n' : ℤ\nh : n + x = n'\np q : ℤ\nhpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n\n⊢ (up ℤ).π (up ℤ) (up ℤ) (p + x, q) = n'",
"usedConstants": [
... | by dsimp at hpq ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 132,
"column": 9
} | {
"line": 132,
"column": 31
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn n' : ℤ\nh : n + x = n'\np q : ℤ\nhpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n'\n⊢ (up ℤ).π (up ℤ) (up ℤ) (p - x, q) = n",
"usedConstants": [
... | by dsimp at hpq ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 236,
"column": 5
} | {
"line": 236,
"column": 27
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn n' : ℤ\nh : n + y = n'\np q : ℤ\nhpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n\n⊢ (up ℤ).π (up ℤ) (up ℤ) (p, q + y) = n'",
"usedConstants": [
... | by dsimp at hpq ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 239,
"column": 61
} | {
"line": 239,
"column": 83
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn n' : ℤ\nh : n + y = n'\np q : ℤ\nhpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n'\n⊢ (up ℤ).π (up ℤ) (up ℤ) (p, q - y) = n",
"usedConstants": [
... | by dsimp at hpq ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 345,
"column": 24
} | {
"line": 345,
"column": 73
} | [
{
"pp": "case h.h\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\ny : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (((shiftFunctor₂ C y).map f).f i₁).f i₂ ≫\n ((shiftFunctor₂ C y).obj L... | L.ι_totalShift₂Iso_hom_f y i₁ i₂ n h _ rfl _ rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 101,
"column": 34
} | {
"line": 108,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na n n' : ℤ\nhn' : a + n = n'\n⊢ (t.ge n).shift a = t.ge n'",
"usedConstants": [
... | by
ext X
constructor
· intro hX
exact ((t.ge n').prop_iff_of_iso ((shiftEquiv C a).unitIso.symm.app X)).1
(t.ge_shift n (-a) n' (by lia) _ hX)
· intro hX
exact t.ge_shift _ _ _ hn' X hX | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.BifunctorShift | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 80
} | [
{
"pp": "case h.h\nC₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝⁹ : Category.{v_1, u_1} C₁\ninst✝⁸ : Category.{v_2, u_2} C₂\ninst✝⁷ : Category.{v_3, u_3} D\ninst✝⁶ : HasZeroMorphisms C₁\ninst✝⁵ : Preadditive C₂\ninst✝⁴ : Preadditive D\nK₁ : CochainComplex C₁ ℤ\nK₂ L₂ : CochainComplex C₂ ℤ\nf₂ : K₂ ⟶ L₂\nF :... | simp [ι_mapBifunctorShift₂Iso_hom_f _ _ _ _ _ _ _ _ (q + y) (n + y) rfl rfl,
ι_mapBifunctorShift₂Iso_hom_f_assoc _ _ _ _ _ _ _ _ (q + y) (n + y) rfl rfl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 82,
"column": 12
} | {
"line": 82,
"column": 14
} | [
{
"pp": "case h.a.refine_2\nC : Type u\ninst✝⁶ : Category.{v, u} C\nP : ObjectProperty C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : P.IsTriangulated\nY : C\nhY : P.rightOrthogonal Y\nX₁ X₂ : C\nf ... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 100,
"column": 14
} | {
"line": 100,
"column": 16
} | [
{
"pp": "case h.a.refine_2.refine_1\nC : Type u\ninst✝⁶ : Category.{v, u} C\nP : ObjectProperty C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : P.IsTriangulated\nX : C\nhX : P.leftOrthogonal X\nY₂ Y₃... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 705,
"column": 8
} | {
"line": 706,
"column": 48
} | [
{
"pp": "case e_a.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : Has... | simpa only [← ComplexShape.next_π₁ c₃ c₂₃ h₂ i₃]
using ComplexShape.rel_π₁ c₃ c₂₃ h₂ i₃ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.Double | {
"line": 211,
"column": 4
} | {
"line": 212,
"column": 51
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nι : Type u_2\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nX : C\nj : ι\nh : ∃ k, c.Rel j k\n⊢ (eval C c j ⋙ coyoneda.obj (op X)).CorepresentableBy (double (𝟙 X) ⋯)",
"usedConstants": [
... | exact evalCompCoyonedaCorepresentableByDoubleId _
(fun hj ↦ c.not_rel_of_eq hj h.choose_spec) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Double | {
"line": 211,
"column": 4
} | {
"line": 212,
"column": 51
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nι : Type u_2\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nX : C\nj : ι\nh : ∃ k, c.Rel j k\n⊢ (eval C c j ⋙ coyoneda.obj (op X)).CorepresentableBy (double (𝟙 X) ⋯)",
"usedConstants": [
... | exact evalCompCoyonedaCorepresentableByDoubleId _
(fun hj ↦ c.not_rel_of_eq hj h.choose_spec) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Double | {
"line": 211,
"column": 4
} | {
"line": 212,
"column": 51
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nι : Type u_2\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nX : C\nj : ι\nh : ∃ k, c.Rel j k\n⊢ (eval C c j ⋙ coyoneda.obj (op X)).CorepresentableBy (double (𝟙 X) ⋯)",
"usedConstants": [
... | exact evalCompCoyonedaCorepresentableByDoubleId _
(fun hj ↦ c.not_rel_of_eq hj h.choose_spec) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 93
} | [
{
"pp": "case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\nX' : C\ng : X' ⟶ X\np q : ℤ\nf : X ⟶ K.X q\nn : ℤ\nh : p + n = q\n⊢ (fromSingleEquiv h) (fromSingleMk (g ≫ f) h) =\n (fromSingleEquiv h) ((ofHom ((singleFunctor C p).map g)... | simp [fromSingleEquiv, singleFunctor, singleFunctors, HomologicalComplex.single_map_f_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 219,
"column": 31
} | {
"line": 219,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\nL : CochainComplex C ℤ\ng : L ⟶ K\n⊢ (toSingleEquiv h) (toSingleMk (g.f p ≫ f) h) = (toSingleEquiv h) ((ofHom g).comp (toSingleMk f h) ⋯)... | by simp [toSingleEquiv, singleFunctor, singleFunctors] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 142,
"column": 49
} | {
"line": 142,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | hα | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 209,
"column": 83
} | {
"line": 211,
"column": 5
} | [
{
"pp": "I : Type u\ninst✝⁴ : AddMonoid I\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : MonoidalCategory C\nX₁ X₂ X₃ : GradedObject I C\ninst✝¹ : X₂.HasTensor X₃\ninst✝ : X₁.HasTensor (tensorObj X₂ X₃)\ni₁ i₂ i₃ j : I\nh : i₁ + i₂ + i₃ = j\ni₂₃ : I\nh' : i₂ + i₃ = i₂₃\n⊢ ιTensorObj₃ X₁ X₂ X₃ i₁ i₂ i₃ ... | by
subst h'
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 253,
"column": 70
} | {
"line": 255,
"column": 5
} | [
{
"pp": "I : Type u\ninst✝⁴ : AddMonoid I\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : MonoidalCategory C\nX₁ X₂ X₃ : GradedObject I C\ninst✝¹ : X₁.HasTensor X₂\ninst✝ : (tensorObj X₁ X₂).HasTensor X₃\ni₁ i₂ i₃ j : I\nh : i₁ + i₂ + i₃ = j\ni₁₂ : I\nh' : i₁ + i₂ = i₁₂\n⊢ ιTensorObj₃' X₁ X₂ X₃ i₁ i₂ i₃... | by
subst h'
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 317,
"column": 8
} | {
"line": 317,
"column": 12
} | [
{
"pp": "case e_a.h₁\nC : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k : ι\nf : i ⟶ j\ng : j ⟶ k\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\nfg : i ⟶ k\nh : f ≫ g = fg\nfg' : i' ⟶ k'\nh' : f' ≫ g' = fg'\nα : mk₂ f g ⟶ mk₂... | hβ₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Square | {
"line": 334,
"column": 9
} | {
"line": 334,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nsq : Square C\nF : C ⥤ D\n⊢ F.map sq.f₁₂ ≫ F.map sq.f₂₄ = F.map sq.f₁₃ ≫ F.map sq.f₃₄",
"usedConstants": [
"CategoryTheory.Square.f₂₄",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"... | by simpa using F.congr_map sq.fac | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Artinian.Module | {
"line": 647,
"column": 4
} | {
"line": 647,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\n⊢ IsNilpotent (Ring.jacobson R)",
"usedConstants"... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 273,
"column": 17
} | {
"line": 273,
"column": 88
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nn : ℕ\nih : ∀ (y : ↥N), ↑(((toEnd R L ↥N) x ^ n) y) = ((toEn... | simp only [pow_succ', Module.End.mul_apply, ih, LieSubmodule.coe_toEnd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 273,
"column": 17
} | {
"line": 273,
"column": 88
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nn : ℕ\nih : ∀ (y : ↥N), ↑(((toEnd R L ↥N) x ^ n) y) = ((toEn... | simp only [pow_succ', Module.End.mul_apply, ih, LieSubmodule.coe_toEnd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 273,
"column": 17
} | {
"line": 273,
"column": 88
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nn : ℕ\nih : ∀ (y : ↥N), ↑(((toEnd R L ↥N) x ^ n) y) = ((toEn... | simp only [pow_succ', Module.End.mul_apply, ih, LieSubmodule.coe_toEnd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 40
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\np : Submodule R L\n⊢ (lieSpan R L ↑p).toSubmodule = p ↔ ∃ K, K.toSubmodule = p",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Submodule",
"LieRing.toAddCommGroup",
"congrAr... | rw [p.exists_lieSubalgebra_coe_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Basic | {
"line": 417,
"column": 89
} | {
"line": 419,
"column": 5
} | [
{
"pp": "R : Type u\nL₁ : Type v\nL₂ : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L₁\ninst✝² : LieAlgebra R L₁\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L₁ →ₗ⁅R⁆ L₂\nh₁ : ∀ (x y : L₁), f (x + y) = f x + f y\nh₂ :\n ∀ (m : R) (x : L₁),\n { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 736,
"column": 21
} | {
"line": 736,
"column": 30
} | [
{
"pp": "case smul\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nx : L\nt : R\nu : L\nhx✝ : u ∈ lieSpan R L s\nhu : ⟨u, hx✝⟩ ∈ lieSpan R (↥(lieSpan R L s)) (Subtype.val ⁻¹' s)\n⊢ ⟨t • u, ⋯⟩ ∈ lieSpan R (↥(lieSpan R L s)) (Subtype.val ⁻¹' s)",
"usedConst... | revert hu | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 752,
"column": 23
} | {
"line": 752,
"column": 32
} | [
{
"pp": "case a.smul\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nι : Type u_1\nf : ι → ↥K\nx : L\nt : R\nu : L\nhx✝ : u ∈ lieSpan R L (range (Subtype.val ∘ f))\nhu : u ∈ map K.incl (lieSpan R (↥K) (range f))\n⊢ t • u ∈ map K.incl (lieSpan R (↥... | revert hu | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 67,
"column": 66
} | {
"line": 69,
"column": 48
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\n⊢ map N.incl (comap N.incl N') = N ⊓ N'",
"usedConstants": [
"LieSubmodule.map",
"LieSubmodule.instSetLike",
... | by
rw [← toSubmodule_inj]
exact (N : Submodule R M).map_comap_subtype N' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 30
} | [
{
"pp": "case a\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : LieModule R L M\ns : Set M := {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x}\naux : ∀ ... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\n⊢ ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'",
"usedConstants": [
"LieAl... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\n⊢ ⁅I, J⁆ ≤ ⁅J, I⁆",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"Eq.mpr",
"LieRing.toAddCommGroup",
"congrArg",
"PartialOrder.to... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\n⊢ ⁅I, N⁆ ≤ N",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\n⊢ ⁅⊥, N⁆ ≤ ⊥",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\n⊢ ⁅I, N⁆ = ⊥ ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0",
"usedConstants": [
"LieAlgebra... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : m ∈ ⁅I, N⁆\n⊢ m ∈ ⁅J, N'⁆",
"usedConstant... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 52
} | {
"line": 163,
"column": 74
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : ∀ (N_1 : LieSubmodule R L M), {x | ∃ x_1 n, ⁅... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nh : ⁅I, N⁆ ⊔ ⁅I, N'⁆ ≤ ⁅I, N ⊔ N'⁆\n⊢ ⁅I, N ⊔ N'⁆ ≤ ⁅I, N⁆ ⊔ ⁅I, N'⁆",
"usedCon... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh : ⁅I, N⁆ ⊔ ⁅J, N⁆ ≤ ⁅I ⊔ J, N⁆\n⊢ ⁅I ⊔ J, N⁆ ≤ ⁅I, N⁆ ⊔ ⁅J, N⁆",
"usedConstant... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 17
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\n⊢ (if hx : IsIntegral A x then ⋯.min (fun x_1 ↦ x_1.Monic ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0).Monic",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
... | rw [dif_pos hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Abelian | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 17
} | [
{
"pp": "case refine_2.mem\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0\nx✝¹ x✝ : ↥(lieSpan R L s)\nx y : L\nhy : y ∈ lieSpan R L s\nw : L\nhw : w ∈ s\n⊢ ⁅⟨w, ⋯⟩, ⟨y, hy⟩⁆ = 0",
"usedConstants": [
"LieAlgebra.... | | mem w hw => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 267,
"column": 23
} | {
"line": 267,
"column": 38
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Algebra A B\nx : B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : f.Monic\nhg : g.Monic\nhe : f * g = minpoly A x\n⊢ IsUnit f ∨ g = 1",
"usedConstants": [
"Eq.mpr",
"Polynomial.in... | ← hg.isUnit_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\nI : LieIdeal R L\ninst✝ : LieModule.IsTrivial L M\n⊢ ⁅I, N⁆ ≤ ⊥",
"usedConstants": [
"LieAlge... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 67
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\n⊢ p * C p.leadingCoeff⁻¹ = minpoly A x",
"usedConstants": [
"Iff.mpr",
"Polynomial.leadingCoeff",
"Ne"... | have : p.leadingCoeff ≠ 0 := leadingCoeff_ne_zero.mpr hp1.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerBasis | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\npb : PowerBasis R S\ny : S\na✝ : Nontrivial S\n⊢ ∃ f, y = (aeval pb.gen) f",
"usedConstants": [
"PowerBasis.exists_eq_aeval"
]
}
] | obtain ⟨f, _, hf⟩ := exists_eq_aeval pb y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PowerBasis | {
"line": 191,
"column": 36
} | {
"line": 191,
"column": 53
} | [
{
"pp": "S : Type u_2\ninst✝² : Ring S\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\nroot : ∑ i ∈ Finset.range pb.dim, p.coeff i • pb.gen ^ i = 0\n⊢ (monomial ↑i) (p.coeff ↑i) = 0",
"usedConstants": [
... | Finset.sum_range, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PowerBasis | {
"line": 254,
"column": 90
} | {
"line": 269,
"column": 69
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : Ring S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\npb : PowerBasis A S\ny : S'\nhy : (aeval y) (minpoly A pb.gen) = 0\nf : A[X]\n⊢ ((pb.basis.constr A) fun i ↦ y ^ ↑i) ((aeval pb.gen) f) = (aeval y) f",
"use... | by
cases subsingleton_or_nontrivial A
· rw [(Subsingleton.elim _ _ : f = 0), aeval_zero, map_zero, aeval_zero]
rw [← aeval_modByMonic_eq_self_of_root (minpoly.aeval _ _), ← aeval_modByMonic_eq_self_of_root hy]
by_cases hf : f %ₘ minpoly A pb.gen = 0
· simp only [hf, map_zero]
have : (f %ₘ minpoly A pb.gen).... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 85
} | [
{
"pp": "case mul_X\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\nf p : MvPolynomial σ (R ⧸ I)\ni : σ\nhp :\n (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n (fun i ... | simp only [hp, coe_eval₂Hom, Ideal.Quotient.lift_mk, eval₂_mul, map_mul, eval₂_X] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 251,
"column": 4
} | {
"line": 253,
"column": 25
} | [
{
"pp": "case C\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\nf : MvPolynomial σ R\n⊢ ∀ (a : R),\n eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n (fun i ↦ (Ideal.Quotient.mk (Ideal.map C I)) (X i))\n ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom... | intro r
rw [Ideal.Quotient.lift_mk, eval₂Hom_C, RingHom.comp_apply, eval₂_C, Ideal.Quotient.lift_mk,
RingHom.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 251,
"column": 4
} | {
"line": 253,
"column": 25
} | [
{
"pp": "case C\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\nf : MvPolynomial σ R\n⊢ ∀ (a : R),\n eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n (fun i ↦ (Ideal.Quotient.mk (Ideal.map C I)) (X i))\n ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom... | intro r
rw [Ideal.Quotient.lift_mk, eval₂Hom_C, RingHom.comp_apply, eval₂_C, Ideal.Quotient.lift_mk,
RingHom.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 181,
"column": 79
} | {
"line": 184,
"column": 99
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : R[X]\nhsep : p.Separable\n⊢ Squarefree p",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instAddMonoidWithOneENat",
"Classical.or_iff_not_imp_right",
"congrArg",
"CommSemiring.toSemiring",
"IsUnit",
"id",
"LE.l... | by
classical
rw [squarefree_iff_emultiplicity_le_one p]
exact fun f => or_iff_not_imp_right.mpr fun hunit => emultiplicity_le_one_of_separable hunit hsep | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 239,
"column": 37
} | {
"line": 254,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nn : ℕ\nu : Rˣ\nhn : IsUnit ↑n\n⊢ (X ^ n - C ↑u).Separable",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Nontrivial",
"Iff.mpr",
"Polynomial.derivative",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Units.val",
"E... | by
nontriviality R
rcases n.eq_zero_or_pos with (rfl | hpos)
· simp at hn
apply (separable_def' (X ^ n - C (u : R))).2
obtain ⟨n', hn'⟩ := hn.exists_left_inv
refine ⟨-C ↑u⁻¹, C (↑u⁻¹ : R) * C n' * X, ?_⟩
rw [derivative_sub, derivative_C, sub_zero, derivative_pow X n, derivative_X, mul_one]
calc
-C ↑... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 326,
"column": 74
} | {
"line": 333,
"column": 36
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nι : Type u_1\nf : ι → F\ns : Finset ι\nH : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y\n⊢ (∏ i ∈ s, (X - C (f i))).Separable",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Finset",
"HSub.hSub",
"RingHom",
"Membership.mem",
... | by
rw [← prod_attach]
exact
separable_prod'
(fun x _hx y _hy hxy =>
@pairwise_coprime_X_sub_C _ _ { x // x ∈ s } (fun x => f x)
(fun x y hxy => Subtype.ext <| H x.1 x.2 y.1 y.2 hxy) _ _ hxy)
fun _ _ => separable_X_sub_C | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 337,
"column": 75
} | {
"line": 337,
"column": 93
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → F\n⊢ (∀ (x y : ι), f x = f y → x = y) ↔ Function.Injective f",
"usedConstants": [
"iff_self",
"Iff",
"of_eq_true",
"Eq"
]
}
] | Function.Injective | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Perfect | {
"line": 375,
"column": 93
} | {
"line": 378,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\nf : R[X]\ninst✝ : DecidableEq R\n⊢ Finset.image (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots.toFinset ⊆ f.roots.toFinset",
"usedConstants": [
"Multiset.toFinset",
"Multiset.subset_of_le",
... | by
rw [Finset.image_toFinset, ← (roots f).toFinset_nsmul _ (expChar_pow_pos R p n).ne',
toFinset_subset]
exact subset_of_le (roots_expand_pow_map_iterateFrobenius_le p n f) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Perfect | {
"line": 405,
"column": 7
} | {
"line": 405,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\ninst✝ : PerfectRing R p\ny : R\nH : ((expand R (p ^ n)) (X - C y)).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) (X - C y).roots\n⊢ (X ^ p ^ n - C y).roots = p ^ n • {(iterateFrobeniusEquiv R p n... | roots_X_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : Free R M\nf : M →ₗ[R] M\nhf : LinearMap.det f = 0\nthis : Module.Finite R M\nb : Basis (Fin (finrank R M)) R M := finBasis R M\n⊢ ⊥ < f.ker",
"usedConstants": [
"Eq.mpr"... | suffices ∃ x, f x = 0 ∧ x ≠ 0 by simpa [bot_lt_iff_ne_bot, ker_eq_bot'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
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