module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 74
} | {
"line": 219,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nS : Type u\nA : Type v\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nhx : AlgebraicIndependent R x\nι' : Type u_4\ny : ι' → A\nhxS : range x ⊆ range ⇑(... | [] | simpa only [AlgHom.coe_toRingHom] using Subalgebra.inclusion_injective _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 118,
"column": 15
} | {
"line": 118,
"column": 26
} | {
"line": 118,
"column": 27
} | [
{
"pp": "F✝ : Type u_1\ninst✝² : Field F✝\nq✝ : ℕ\nF : Type u_1\ninst✝¹ : Field F\nq : ℕ\nhq : NeZero q\ninst✝ : CharP F q\ng g' : F[X]\nm : ℕ\nhg : g.Separable\nhg' : g'.Separable\ns : ℕ\nh_expand : (expand F (q ^ m)) g = (expand F (q ^ m * q ^ s)) g'\nhm : m ≤ m + s\n⊢ g.natDegree = g'.natDegree",
"ppTerm... | [
"F✝ : Type u_1\ninst✝² : Field F✝\nq✝ : ℕ\nF : Type u_1\ninst✝¹ : Field F\nq : ℕ\nhq : NeZero q\ninst✝ : CharP F q\ng g' : F[X]\nm : ℕ\nhg : g.Separable\nhg' : g'.Separable\ns : ℕ\nh_expand : (expand F (q ^ m)) g = (expand F (q ^ m)) ((expand F (q ^ s)) g')\nhm : m ≤ m + s\n⊢ g.natDegree = g'.natDegree"
] | expand_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 42
} | {
"line": 134,
"column": 42
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nq : ℕ\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime✝ : Nat.Prime q\nhchar✝ : CharP F q\nhg : g.Separable\nm : ℕ\nhm : (expand F (q ^ m)) g = f\ng' : F[X] := Classical.choose hf\nhg' : (Classical.choose hf).Separable\nm' : ℕ\nhm' : (expand F (q ^ m')) (Clas... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 39
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case right\nι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\ni : AlgebraicIndependent R x\nw : Set A\ni' : AlgebraicIndepOn R _root_.id w\nh : range x ⊆ w\np : Surjective fun i ↦ ⟨x i, ⋯⟩\nq : (Subtype.val '' range fu... | [
"case right\nι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\ni : AlgebraicIndependent R x\nw : Set A\ni' : AlgebraicIndepOn R _root_.id w\nh : range x ⊆ w\np : Surjective fun i ↦ ⟨x i, ⋯⟩\nq : (fun x_1 ↦ ↑⟨x x_1, ⋯⟩) '' univ = Su... | rw [← image_univ, image_image] at q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 46
} | {
"line": 415,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\n⊢ (X ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.natSepDegree_pow",
"congrArg",
"Polynomial.natSepDegree_X",
"instOfNatNat",
"Field.toSemifield",
"Pol... | [] | simp only [natSepDegree_pow, natSepDegree_X] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.SeparableDegree | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 46
} | {
"line": 415,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\n⊢ (X ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.natSepDegree_pow",
"congrArg",
"Polynomial.natSepDegree_X",
"instOfNatNat",
"Field.toSemifield",
"Pol... | [] | simp only [natSepDegree_pow, natSepDegree_X] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparableDegree | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 46
} | {
"line": 415,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\n⊢ (X ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.natSepDegree_pow",
"congrArg",
"Polynomial.natSepDegree_X",
"instOfNatNat",
"Field.toSemifield",
"Pol... | [] | simp only [natSepDegree_pow, natSepDegree_X] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 441,
"column": 6
} | {
"line": 443,
"column": 37
} | {
"line": 444,
"column": 4
} | [
{
"pp": "case refine_1\nF : Type u\ninst✝ : Field F\nf✝ f g : F[X]\nh✝ : f = 0 ∨ g = 0\nhf : f = 0\nx✝ : ∃ x, C x = g\nx : F\nh : C x = g\n⊢ 0 = 0 ∧ g = 0 ∨ ∃ r, IsUnit r ∧ C r = g",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.inst... | [] | by_cases hx : x = 0
· exact .inl ⟨rfl, by rw [← h, hx, map_zero]⟩
exact .inr ⟨x, Ne.isUnit hx, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparableDegree | {
"line": 441,
"column": 6
} | {
"line": 443,
"column": 37
} | {
"line": 444,
"column": 4
} | [
{
"pp": "case refine_1\nF : Type u\ninst✝ : Field F\nf✝ f g : F[X]\nh✝ : f = 0 ∨ g = 0\nhf : f = 0\nx✝ : ∃ x, C x = g\nx : F\nh : C x = g\n⊢ 0 = 0 ∧ g = 0 ∨ ∃ r, IsUnit r ∧ C r = g",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.inst... | [] | by_cases hx : x = 0
· exact .inl ⟨rfl, by rw [← h, hx, map_zero]⟩
exact .inr ⟨x, Ne.isUnit hx, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 910,
"column": 2
} | {
"line": 916,
"column": 53
} | {
"line": 917,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis : CompactSpace ↥c.p... | [
"I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis✝ : CompactSpace ↥c.pt\ni : ↥c.p... | have := ((Presheaf.isSheaf_iff_isSheaf_forget _ _ (forget _)).mp (D.obj k').IsSheaf).isSheafFor
(.ofArrows (fun x : σ ↦ D.map (fk'k ≫ fk (hiS x.2)) ⁻¹ᵁ U x.1) fun x ↦ homOfLE le_top)
(fun x _ ↦ by
obtain ⟨ix, hix, h⟩ : ∃ ix, ∃ (h : ix ∈ σ), (D.map (fk'k ≫ fk (hiS h))).base x ∈ U ix := by
simpa usi... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.SeparableDegree | {
"line": 811,
"column": 2
} | {
"line": 811,
"column": 15
} | {
"line": 812,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\nE' : I... | [
"F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\nE' : IntermediateF... | clear_value g | Lean.Elab.Tactic.evalClearValue | Lean.Parser.Tactic.clearValue |
Mathlib.FieldTheory.IsSepClosed | {
"line": 133,
"column": 2
} | {
"line": 135,
"column": 20
} | {
"line": 136,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\n⊢ ∃ z, z ^ n = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Field.toDivisionRing",
"... | [
"k : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\nhn' : 0 < n\n⊢ ∃ z, z ^ n = x"
] | have hn' : 0 < n := Nat.pos_of_ne_zero fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Galois.Basic | {
"line": 689,
"column": 2
} | {
"line": 694,
"column": 34
} | {
"line": 696,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\n⊢ IsCyclic Gal(K/F)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"isCyclic_of_prime_card",
"Nat.zero_le",
"False",
"Nat.rawCast",
... | [] | have := finrank_eq_two F K ▸ AlgEquiv.card_le
rw [← Nat.card_eq_fintype_card] at this
interval_cases h : Nat.card Gal(K/F)
· simp_all
· exact @isCyclic_of_subsingleton _ _ (Finite.card_le_one_iff_subsingleton.mp h.le)
· exact isCyclic_of_prime_card h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 689,
"column": 2
} | {
"line": 694,
"column": 34
} | {
"line": 696,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\n⊢ IsCyclic Gal(K/F)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"isCyclic_of_prime_card",
"Nat.zero_le",
"False",
"Nat.rawCast",
... | [] | have := finrank_eq_two F K ▸ AlgEquiv.card_le
rw [← Nat.card_eq_fintype_card] at this
interval_cases h : Nat.card Gal(K/F)
· simp_all
· exact @isCyclic_of_subsingleton _ _ (Finite.card_le_one_iff_subsingleton.mp h.le)
· exact isCyclic_of_prime_card h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsSepClosed | {
"line": 235,
"column": 3
} | {
"line": 235,
"column": 16
} | {
"line": 235,
"column": 16
} | [
{
"pp": "k : Type u\ninst✝² : Field k\nK : Type v\ninst✝¹ : Field K\ninst✝ : IsSepClosed k\n⊢ IsSepClosed k",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.PurelyInseparable | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 29
} | {
"line": 55,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Field K\nf : F →+* E\ng : E →+* K\nhf : f.IsPurelyInseparable\nhg : g.IsPurelyInseparable\n⊢ (g.comp f).IsPurelyInseparable",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraM... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Field K\nf : F →+* E\ng : E →+* K\nhf : f.IsPurelyInseparable\nhg : g.IsPurelyInseparable\nalgInst✝² : Algebra F E := f.toAlgebra\nalgInst✝¹ : Algebra E K := g.toAlgebra\nalgInst✝ : Algebra F K := (g.comp f).toAlgebra\nscalarTowe... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 146,
"column": 54
} | {
"line": 152,
"column": 68
} | {
"line": 153,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\ntfae_2_to_4 : (∀ (... | [] | by
simp only [g₁, g₂, Scheme.SpecToEquivOfField_symm_apply, AlgHom.toRingHom_eq_coe,
Category.assoc, ← f.SpecMap_residueFieldMap_fromSpecResidueField x, ← Spec.map_comp_assoc]
congr 2
ext a
simp only [CommRingCat.hom_comp, RingHom.comp_apply]
exact (AlgHom.commutes σ₁ a).trans (Alg... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 674,
"column": 2
} | {
"line": 689,
"column": 53
} | {
"line": 690,
"column": 2
} | [
{
"pp": "case inl\nk : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁶ : Field k\ninst✝⁵ : Field K\ninst✝⁴ : Algebra k K\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsPurelyInseparable k K\nq : ℕ\ninst✝ : ExpChar k q\nx : R ⊗[k] K\na✝ : Nontrivial (R ⊗[k] K)\nhq : Nat.Prime q\n⊢ ∃ n, x ^ q ^ n ∈ (algebraM... | [
"case inr\nk : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁶ : Field k\ninst✝⁵ : Field K\ninst✝⁴ : Algebra k K\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsPurelyInseparable k K\nq : ℕ\ninst✝ : ExpChar k q\nx : R ⊗[k] K\na✝ : Nontrivial (R ⊗[k] K)\nhq : q = 1\n⊢ ∃ n, x ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).r... | induction x with
| zero => exact ⟨0, 0, by simp⟩
| add x y h h' =>
have : ExpChar (R ⊗[k] K) q := expChar_of_injective_ringHom (algebraMap k _).injective q
simp_rw [RingHom.mem_range, ← RingHom.mem_rangeS, ← Subalgebra.mem_perfectClosure_iff] at h h' ⊢
exact add_mem h h'
| tmul x y =>
obtain ⟨n, a... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 69,
"column": 4
} | {
"line": 76,
"column": 11
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ... | [] | refine ⟨⟨s.sigma fun i ↦ if h : i ∈ s then V i h else ∅, isOpen_sigma_iff.mpr ?_⟩, ?_, ?_⟩
· intro i
by_cases h : i ∈ s
· simpa [h] using (V _ _).2
· simp [h]
· dsimp only
exact Set.isCompact_sigma hs fun i ↦ (by simp_all)
· aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 69,
"column": 4
} | {
"line": 76,
"column": 11
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ... | [] | refine ⟨⟨s.sigma fun i ↦ if h : i ∈ s then V i h else ∅, isOpen_sigma_iff.mpr ?_⟩, ?_, ?_⟩
· intro i
by_cases h : i ∈ s
· simpa [h] using (V _ _).2
· simp [h]
· dsimp only
exact Set.isCompact_sigma hs fun i ↦ (by simp_all)
· aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1130,
"column": 4
} | {
"line": 1130,
"column": 67
} | {
"line": 1131,
"column": 4
} | [
{
"pp": "case inr\nI : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ (i : I), IsAffine (D.obj i)\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\ninst✝² : IsAffine (Spec R)\nS : CommRingCat\ninst✝¹ : IsAffine (Spec S)\na : c.pt ⟶ Spe... | [
"case inr\nI : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ (i : I), IsAffine (D.obj i)\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\ninst✝² : IsAffine (Spec R)\nS : CommRingCat\ninst✝¹ : IsAffine (Spec S)\na : c.pt ⟶ Spec S\nφ : R ⟶... | have inst (i) : IsIso (e.app i) := by dsimp [e]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 25
} | {
"line": 138,
"column": 4
} | [
{
"pp": "S : Scheme\n𝒰✝¹ 𝒰✝ : PreZeroHypercover S\nK : Precoverage Scheme\nX : Scheme\n𝒰 : PreZeroHypercover X\ninst✝¹ : QuasiCompactCover 𝒰\nf : (x : 𝒰.I₀) → PreZeroHypercover (𝒰.X x)\ninst✝ : ∀ (x : 𝒰.I₀), QuasiCompactCover (f x)\nU : X.Opens\nhU✝ : IsAffineOpen U\ns : Set 𝒰.I₀\nhs : s.Finite\nV : (i ... | [
"S : Scheme\n𝒰✝¹ 𝒰✝ : PreZeroHypercover S\nK : Precoverage Scheme\nX : Scheme\n𝒰 : PreZeroHypercover X\ninst✝¹ : QuasiCompactCover 𝒰\nf : (x : 𝒰.I₀) → PreZeroHypercover (𝒰.X x)\ninst✝ : ∀ (x : 𝒰.I₀), QuasiCompactCover (f x)\nU : X.Opens\nhU✝ : IsAffineOpen U\ns : Set 𝒰.I₀\nhs : s.Finite\nV : (i : 𝒰.I₀) → i... | refine .of_finite (κ := Σ (i : s), t i.1 i.2) (fun p ↦ ⟨p.1, p.2⟩) (fun p ↦ W _ p.1.2 _ p.2.2)
(fun p ↦ hcW ..) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Category.TopCat.EffectiveEpi | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 25
} | {
"line": 56,
"column": 4
} | [
{
"pp": "B X : TopCat\nπ : X ⟶ B\nhπ : IsQuotientMap ⇑(ConcreteCategory.hom π)\nW✝ : TopCat\ne : X ⟶ W✝\nh : ∀ {Z : TopCat} (g₁ g₂ : Z ⟶ X), g₁ ≫ π = g₂ ≫ π → g₁ ≫ e = g₂ ≫ e\ng : B ⟶ W✝\nhm : π ≫ g = e\nthis : g = ofHom (hπ.liftEquiv ⟨Hom.hom e, ⋯⟩)\n⊢ g = ofHom (hπ.lift (Hom.hom e) ⋯)",
"ppTerm": "?m.232"... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 43,
"column": 4
} | {
"line": 47,
"column": 9
} | {
"line": 48,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nα : Type u_2\nB : C\nX Y : α → C\nf : (a : α) → X a ⟶ B\ng : (a : α) → Y a ⟶ X a\ni : (a : α) → X a ⟶ Y a\nhi : ∀ (a : α), i a ≫ g a = 𝟙 (X a)\ninst✝ : EffectiveEpiFamily X f\nW✝ : C\ne : (a : α) → Y a ⟶ W✝\nw : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ Y a₁) (g₂ : ... | [] | simp only [Category.assoc, EffectiveEpiFamily.fac]
rw [← Category.id_comp (e a), ← Category.assoc, ← Category.assoc]
apply w
simp only [Category.comp_id, Category.id_comp, ← Category.assoc]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 43,
"column": 4
} | {
"line": 47,
"column": 9
} | {
"line": 48,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nα : Type u_2\nB : C\nX Y : α → C\nf : (a : α) → X a ⟶ B\ng : (a : α) → Y a ⟶ X a\ni : (a : α) → X a ⟶ Y a\nhi : ∀ (a : α), i a ≫ g a = 𝟙 (X a)\ninst✝ : EffectiveEpiFamily X f\nW✝ : C\ne : (a : α) → Y a ⟶ W✝\nw : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ Y a₁) (g₂ : ... | [] | simp only [Category.assoc, EffectiveEpiFamily.fac]
rw [← Category.id_comp (e a), ← Category.assoc, ← Category.assoc]
apply w
simp only [Category.comp_id, Category.id_comp, ← Category.assoc]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EffectiveEpi | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 51
} | {
"line": 94,
"column": 2
} | [
{
"pp": "X Y S : Scheme\ninst✝³ : IsAffine X\ninst✝² : IsAffine Y\nπ : X ⟶ Y\ninst✝¹ : Surjective π\ninst✝ : Flat π\nf : X ⟶ S\nhf : fst π π ≫ f = snd π π ≫ f\nb : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace\nhfac : π.base ≫ b = f.base\n⊢ ∃ 𝒰, ∀ (i : 𝒰.I₀), ∃ u, fst π (𝒰.f i) ≫ f = snd π (𝒰.f i) ≫ u",
"... | [
"X Y S : Scheme\ninst✝³ : IsAffine X\ninst✝² : IsAffine Y\nπ : X ⟶ Y\ninst✝¹ : Surjective π\ninst✝ : Flat π\nf : X ⟶ S\nhf : fst π π ≫ f = snd π π ≫ f\nb : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace\nhfac : π.base ≫ b = f.base\n𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯\n⊢ ∃ 𝒰, ∀ (i : 𝒰.I₀),... | let 𝒰 := Y.openCoverOfIsOpenCover _ <| Y.isBasis_affineOpens.isOpenCover_mem_and_le
(S.isBasis_affineOpens.isOpenCover.comap b.hom) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.EffectiveEpi | {
"line": 131,
"column": 4
} | {
"line": 132,
"column": 71
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case refine_1\nX Y : Scheme\ninst✝³ : IsAffine X\ninst✝² : IsAffine Y\nπ : X ⟶ Y\ninst✝¹ : Surjective π\ninst✝ : Flat π\nthis✝ : Epi π\nZ : Scheme\nf : X ⟶ Z\nhf : pullback.fst π π ≫ f = pullback.snd π π ≫ f\n𝒰 : Y.OpenCover\nu : (i : 𝒰.I₀) → 𝒰.X i ⟶ Z\nhfac : ∀ (i : 𝒰.I₀), pullback.fst π (𝒰.f i) ... | [
"case refine_1\nX Y : Scheme\ninst✝³ : IsAffine X\ninst✝² : IsAffine Y\nπ : X ⟶ Y\ninst✝¹ : Surjective π\ninst✝ : Flat π\nthis✝ : Epi π\nZ : Scheme\nf : X ⟶ Z\nhf : pullback.fst π π ≫ f = pullback.snd π π ≫ f\n𝒰 : Y.OpenCover\nu : (i : 𝒰.I₀) → 𝒰.X i ⟶ Z\nhfac : ∀ (i : 𝒰.I₀), pullback.fst π (𝒰.f i) ≫ f = pullba... | rw [← cancel_epi (pullback.snd π (pullback.fst (𝒰.f i) (𝒰.f j) ≫ 𝒰.f i)),
← cancel_epi (pullback.congrHom rfl pullback.condition.symm).hom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 284,
"column": 8
} | {
"line": 284,
"column": 34
} | {
"line": 285,
"column": 8
} | [
{
"pp": "case h₀\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.499} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\ninst✝³ :... | [
"case h₀\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.499} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\ninst✝³ :\n ∀ {i j :... | rw [← congr($(hm a).left)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1256,
"column": 25
} | {
"line": 1256,
"column": 80
} | {
"line": 1256,
"column": 80
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | [] | simpa [← Scheme.Hom.resLE_eq_morphismRestrict] using! e | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1256,
"column": 25
} | {
"line": 1256,
"column": 80
} | {
"line": 1256,
"column": 80
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | [] | simpa [← Scheme.Hom.resLE_eq_morphismRestrict] using! e | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1256,
"column": 25
} | {
"line": 1256,
"column": 80
} | {
"line": 1256,
"column": 80
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | [] | simpa [← Scheme.Hom.resLE_eq_morphismRestrict] using! e | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 485,
"column": 25
} | {
"line": 485,
"column": 66
} | {
"line": 485,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nb c d : R\nk : ℤ\nP : ℕ → Sort u\nzero : P 0\none : P 1\neven : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))\nodd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)\nn : ℕ\n⊢ (n : ℕ) → ((k : ℕ) → k < 2 * n → P k) → P (2 * n)",
"p... | [] | rintro (_ | _) h; exacts [zero, even _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 485,
"column": 25
} | {
"line": 485,
"column": 66
} | {
"line": 485,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nb c d : R\nk : ℤ\nP : ℕ → Sort u\nzero : P 0\none : P 1\neven : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))\nodd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)\nn : ℕ\n⊢ (n : ℕ) → ((k : ℕ) → k < 2 * n → P k) → P (2 * n)",
"p... | [] | rintro (_ | _) h; exacts [zero, even _ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 508,
"column": 59
} | {
"line": 510,
"column": 54
} | {
"line": 512,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ W.b₆ = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",... | [] | by
rw [b₆_of_isCharTwoJNeZeroNF]
linear_combination 2 * W.a₆ * CharP.cast_eq_zero R 2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 526,
"column": 2
} | {
"line": 527,
"column": 79
} | {
"line": 528,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ W.Δ = W.a₆",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WeierstrassCurve.Δ",
"NegZeroClass.toNeg",
"HMul.hMul",
"AddGroupWithO... | [
"R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ -1 ^ 2 * W.a₆ - 8 * 0 ^ 3 - 27 * 0 ^ 2 + 9 * 1 * 0 * 0 = W.a₆"
] | rw [Δ, b₂_of_isCharTwoJNeZeroNF_of_char_two, b₄_of_isCharTwoJNeZeroNF,
b₆_of_isCharTwoJNeZeroNF_of_char_two, b₈_of_isCharTwoJNeZeroNF_of_char_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 38
} | {
"line": 402,
"column": 39
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Affine R\np q : R[X]\n⊢ (Algebra.leftMulMatrix (CoordinateRing.basis W')) (p • 1 + q • (mk W') Y) 0 0 *\n (Algebra.leftMulMatrix (CoordinateRing.basis W')) (p • 1 + q • (mk W') Y) 1 1 -\n (Algebra.leftMulMatrix (CoordinateRing.basis W')) (p • 1 + q • (mk... | [
"R : Type r\ninst✝ : CommRing R\nW' : Affine R\np q : R[X]\n⊢ ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 0)) 0 *\n ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 1)) 1 -\n ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W'... | Algebra.leftMulMatrix_eq_repr_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 255,
"column": 31
} | {
"line": 255,
"column": 91
} | {
"line": 255,
"column": 91
} | [
{
"pp": "case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ W'.dblX P + -8 * P x * P x ^ 3 - ((P x ^ 2) ^ 2 + -8 * P x * P y ^ 2) = 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr"... | [
"case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ (-3 * P x ^ 2) ^ 2 - W'.a₁ * (-3 * P x ^ 2) * 0 * (P y - -P y) - W'.a₂ * 0 ^ 2 * (P y - -P y) ^ 2 -\n 2 * P x * (P y - -P y) ^ 2 +\n -8 * P x * P x ^ 3 -\n ((P x ^ 2) ^ 2 + -8 * P x * P... | rw [dblX, dblU_of_Z_eq_zero hPz, negY_of_Z_eq_zero hPz, hPz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 261,
"column": 12
} | {
"line": 261,
"column": 43
} | {
"line": 261,
"column": 43
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ W'.dblU P ^ 2 - W'.a₁ * W'.dblU P * P z * (P y - W'.negY P) - W'.a₂ * P z ^ 2 ... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ W'.dblU P ^ 2 - W'.a₁ * W'.dblU P * P z * (W'.negY P - W'.negY P) - W'.a₂ * P z ^ 2 * (W'.... | Y_eq_negY_of_Y_eq hQz hx hy hy' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 299,
"column": 43
} | {
"line": 299,
"column": 74
} | {
"line": 299,
"column": 74
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ -W'.dblU P * (W'.dblU P ^ 2 - P x * (P y - W'.negY P) ^ 2) + P y * (P y - W'.n... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ -W'.dblU P * (W'.dblU P ^ 2 - P x * (W'.negY P - W'.negY P) ^ 2) + W'.negY P * (W'.negY P ... | Y_eq_negY_of_Y_eq hQz hx hy hy' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 90
} | {
"line": 390,
"column": 2
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.dblXYZ P =\n W.dblZ P •\n ![W.toAffine.addX (P x / P z ^ 2) ... | [
"F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\nhZ : ∀ {n : ℕ}, IsUnit (W.dblZ P ^ n)\n⊢ W.dblXYZ P =\n W.dblZ P •\n ![W.toA... | have hZ {n : ℕ} : IsUnit <| W.dblZ P ^ n := (isUnit_dblZ_of_Y_ne' hP hQ hPz hx hy).pow n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 525,
"column": 8
} | {
"line": 529,
"column": 97
} | {
"line": 531,
"column": 0
} | [
{
"pp": "case neg\nF : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : ¬P z = 0\nhQz : ¬Q z = 0\nhxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)\n⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q... | [] | have := toAffine_add_of_Z_ne_zero hP hQ hPz hQz hxy
by_cases hx : P x * Q z ^ 2 = Q x * P z ^ 2
· rwa [add_of_Y_ne' hP.left hQ.left hPz hQz hx <| not_and.mp hxy hx,
toAffine_smul _ <| isUnit_dblZ_of_Y_ne' hP.left hQ.left hPz hx <| not_and.mp hxy hx]
· rwa [add_of_X_ne hP.left hQ.left... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 525,
"column": 8
} | {
"line": 529,
"column": 97
} | {
"line": 531,
"column": 0
} | [
{
"pp": "case neg\nF : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : ¬P z = 0\nhQz : ¬Q z = 0\nhxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)\n⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q... | [] | have := toAffine_add_of_Z_ne_zero hP hQ hPz hQz hxy
by_cases hx : P x * Q z ^ 2 = Q x * P z ^ 2
· rwa [add_of_Y_ne' hP.left hQ.left hPz hQz hx <| not_and.mp hxy hx,
toAffine_smul _ <| isUnit_dblZ_of_Y_ne' hP.left hQ.left hPz hx <| not_and.mp hxy hx]
· rwa [add_of_X_ne hP.left hQ.left... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 212,
"column": 4
} | {
"line": 222,
"column": 60
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case hcard\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_1\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhxL : K[x] = ⊤\nr : ℕ\nhr : r < finrank K L\nthis : Fun... | [] | refine (Polynomial.natDegree_sub_le _ _).trans_lt
(max_lt ((Polynomial.natDegree_sum_le _ _).trans_lt ?_) ?_)
· simp only [Polynomial.map_smul,
map_div₀, map_pow, RingHom.coe_coe, Function.comp_apply,
Finset.mem_univ, forall_true_left, Finset.fold_max_lt, AlgHom.card]
refine ⟨finrank_pos... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 212,
"column": 4
} | {
"line": 222,
"column": 60
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case hcard\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_1\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhxL : K[x] = ⊤\nr : ℕ\nhr : r < finrank K L\nthis : Fun... | [] | refine (Polynomial.natDegree_sub_le _ _).trans_lt
(max_lt ((Polynomial.natDegree_sum_le _ _).trans_lt ?_) ?_)
· simp only [Polynomial.map_smul,
map_div₀, map_pow, RingHom.coe_coe, Function.comp_apply,
Finset.mem_univ, forall_true_left, Finset.fold_max_lt, AlgHom.card]
refine ⟨finrank_pos... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 16
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case hfg\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE : Type u_7\ninst✝¹ : Field E\ninst✝ : Algebra K E\npb : PowerBasis K L\nhE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits\nhfx : IsSeparable K pb.gen\nthis✝ : DecidableEq E := Classical.decE... | [] | intro x; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Basic | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 16
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case hfg\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE : Type u_7\ninst✝¹ : Field E\ninst✝ : Algebra K E\npb : PowerBasis K L\nhE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits\nhfx : IsSeparable K pb.gen\nthis✝ : DecidableEq E := Classical.decE... | [] | intro x; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 433,
"column": 83
} | {
"line": 435,
"column": 50
} | {
"line": 437,
"column": 0
} | [
{
"pp": "A : Type u\nB : Type v\nC : Type z\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\npb : PowerBasis A B\ne : Fin pb.dim ≃ (B →ₐ[A] C)\n⊢ embeddingsMatrixReindex A C (⇑pb.basis) e = (vandermonde fun i ↦ (e i) pb.gen)ᵀ",
"ppTerm": "?m.61",
... | [] | by
ext i j
simp [embeddingsMatrixReindex, embeddingsMatrix] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 69
} | {
"line": 499,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\nι : Type w\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n⊢ ... | [] | simpa only [traceMatrix_of_basis] using det_traceMatrix_ne_zero' pb | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.UniformRing | {
"line": 108,
"column": 21
} | {
"line": 108,
"column": 91
} | {
"line": 109,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : IsUniformAddGroup α\na✝ b✝ c✝ : Completion α\na b c : α\n⊢ ↑a * (↑b + ↑c) = ↑a * ↑b + ↑a * ↑c",
"ppTerm": "?m.899",
"assigned": true,
"usedConstants": [
"AddMonoidWithOne.unary",
"Distr... | [] | by rw [← coe_add, ← coe_mul, ← coe_mul, ← coe_mul, ← coe_add, mul_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 12
} | {
"line": 986,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na : K\n⊢ 0 <ᵥ a ↔ a ≠ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ValuativeRel.vlt",
"congrArg",
"ValuativeRel.vle_zero_iff._simp_1",
"DivisionRing.toDivisionSemiring",
"ValuativeRe... | [] | simp [vlt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 12
} | {
"line": 986,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na : K\n⊢ 0 <ᵥ a ↔ a ≠ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ValuativeRel.vlt",
"congrArg",
"ValuativeRel.vle_zero_iff._simp_1",
"DivisionRing.toDivisionSemiring",
"ValuativeRe... | [] | simp [vlt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 12
} | {
"line": 986,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na : K\n⊢ 0 <ᵥ a ↔ a ≠ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ValuativeRel.vlt",
"congrArg",
"ValuativeRel.vle_zero_iff._simp_1",
"DivisionRing.toDivisionSemiring",
"ValuativeRe... | [] | simp [vlt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1008,
"column": 2
} | {
"line": 1008,
"column": 23
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na b : K\nhb : b ≠ 0\n⊢ 1 ≤ᵥ a / b ↔ b ≤ᵥ a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"DivisionRing.toDivisi... | [] | simp [vle_div_iff hb] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1008,
"column": 2
} | {
"line": 1008,
"column": 23
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na b : K\nhb : b ≠ 0\n⊢ 1 ≤ᵥ a / b ↔ b ≤ᵥ a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"DivisionRing.toDivisi... | [] | simp [vle_div_iff hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1008,
"column": 2
} | {
"line": 1008,
"column": 23
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝¹ : DivisionRing K\ninst✝ : ValuativeRel K\na b : K\nhb : b ≠ 0\n⊢ 1 ≤ᵥ a / b ↔ b ≤ᵥ a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"DivisionRing.toDivisi... | [] | simp [vle_div_iff hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 16
} | {
"line": 572,
"column": 4
} | [
{
"pp": "case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : With... | [
"case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : WithVal v\nhr₀ :... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 32
} | {
"line": 128,
"column": 4
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\n⊢ ↥(closure {x}) ... | [
"G : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G' := max y y⁻¹\nhy' : y... | set y' := max y y⁻¹ with hy' | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 151,
"column": 22
} | {
"line": 151,
"column": 60
} | {
"line": 151,
"column": 60
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nhsurj : Function.Surjective ⇑v\nx : K\nh : x ≠ 0\nh0 : v x ≠ 0\n⊢ Tendsto (⇑v) (𝓝 x) (𝓝 (v x))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"WithZ... | [
"case inr\nK : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nhsurj : Function.Surjective ⇑v\nx : K\nh : x ≠ 0\nh0 : v x ≠ 0\n⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v x_1 = v x"
] | WithZeroTopology.tendsto_of_ne_zero h0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 56
} | {
"line": 542,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na : R\nn✝ : ℕ\n⊢ (coeff n✝) (mk fun n ↦ a ^ n * (coeff n) 0) = (coeff n✝) 0",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
"SemilinearMapCla... | [] | simp only [map_zero, PowerSeries.coeff_mk, mul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 26
} | {
"line": 793,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na : R\n⊢ ↑(C a) = PowerSeries.C a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Polynomial.coe_monomial",
"LinearMap.instFunLike",
"Polynomial.monomial",
"instOfNatNat",
"MvPowerSeries.in... | [
"R : Type u_1\ninst✝ : Semiring R\na : R\nthis : ↑((monomial 0) a) = (PowerSeries.monomial 0) a\n⊢ ↑(C a) = PowerSeries.C a"
] | have := coe_monomial 0 a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 920,
"column": 14
} | {
"line": 920,
"column": 30
} | {
"line": 921,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) r",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Linea... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) (r * 1)"
] | rw [← mul_one r] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 920,
"column": 14
} | {
"line": 920,
"column": 30
} | {
"line": 921,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) r",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Linea... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) (r * 1)"
] | rw [← mul_one r] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 920,
"column": 14
} | {
"line": 920,
"column": 30
} | {
"line": 921,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) r",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Linea... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ne : σ →₀ ℕ\nr : R\n| (MvPowerSeries.monomial e) (r * 1)"
] | rw [← mul_one r] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 314,
"column": 33
} | {
"line": 314,
"column": 81
} | {
"line": 316,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\na : R\ni : σ →₀ ℕ\nhi : ↑((weight w) i) < weightedOrder w f\n⊢ (coeff i) (a • f) = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul"... | [] | by simp [coeff_eq_zero_of_lt_weightedOrder _ hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 322,
"column": 33
} | {
"line": 322,
"column": 81
} | {
"line": 324,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\nS : Type u_3\ninst✝ : Semiring S\nφ : R →+* S\ni : σ →₀ ℕ\nhi : ↑((weight w) i) < weightedOrder w f\n⊢ (coeff i) ((map φ) f) = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"RingHom.instRi... | [] | by simp [coeff_eq_zero_of_lt_weightedOrder _ hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 32
} | {
"line": 450,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\nd : σ →₀ ℕ\na : R\ninst✝ : Decidable (a = 0)\n⊢ ((monomial d) a).order = if a = 0 then ⊤ else ↑((weight fun x ↦ 1) d)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"MvPowerSeries.weightedOrder_monomial",
"instOfNatNat"... | [] | exact weightedOrder_monomial _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Trunc | {
"line": 147,
"column": 4
} | {
"line": 149,
"column": 85
} | {
"line": 150,
"column": 4
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ns : Finset (σ →₀ ℕ)\nhs : IsLowerSet ↑s\nk : ℕ\np : MvPowerSeries σ R\nn : ℕ\nhmn : 1 ≤ n\nih : (truncFinset R s) (↑((truncFinset R s) p) ^ n) = (truncFinset R s) (p ^ n)\nx : σ →₀ ℕ\nhx : x ∈ s\n⊢ MvPolynomial.coeff x ((truncFinset R s) (↑(... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ns : Finset (σ →₀ ℕ)\nhs : IsLowerSet ↑s\nk : ℕ\np : MvPowerSeries σ R\nn : ℕ\nhmn : 1 ≤ n\nih : (truncFinset R s) (↑((truncFinset R s) p) ^ n) = (truncFinset R s) (p ^ n)\nx : σ →₀ ℕ\nhx : x ∉ s\n⊢ MvPolynomial.coeff x ((truncFinset R s) (↑((truncFinset... | · rw [coeff_truncFinset_of_mem _ hx, coeff_truncFinset_of_mem _ hx, pow_succ,
← coeff_truncFinset_mul_truncFinset_eq_coeff_mul hs _ _ hx, ih, truncFinset_truncFinset
(by rfl), pow_succ, coeff_truncFinset_mul_truncFinset_eq_coeff_mul hs _ _ hx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.ENat | {
"line": 52,
"column": 63
} | {
"line": 53,
"column": 62
} | {
"line": 55,
"column": 0
} | [
{
"pp": "x : ℕ∞\nhx : x ≠ ⊤\n⊢ IsOpen {x}",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"congrArg",
"ENat.nhds_eq_pure",
"nhds",
"Set.instSingletonSet",
"id",
"Filter.instPure",
"ENat.instTopologicalSpace",
... | [] | by
rw [isOpen_singleton_iff_nhds_eq_pure, ENat.nhds_eq_pure hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 210,
"column": 2
} | {
"line": 213,
"column": 47
} | {
"line": 215,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\ninst✝² : UniformSpace R\nS : Type u_3\ninst✝¹ : CommRing S\ninst✝ : UniformSpace S\nφ : R →+* S\na : σ → S\nf : MvPolynomial σ R\n⊢ eval₂ φ a ↑f = MvPolynomial.eval₂ φ a f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have : ∃ p : MvPolynomial σ R, (p : MvPowerSeries σ R) = f := ⟨f, rfl⟩
rw [eval₂, dif_pos this]
congr
rw [← MvPolynomial.coe_inj, this.choose_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 210,
"column": 2
} | {
"line": 213,
"column": 47
} | {
"line": 215,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\ninst✝² : UniformSpace R\nS : Type u_3\ninst✝¹ : CommRing S\ninst✝ : UniformSpace S\nφ : R →+* S\na : σ → S\nf : MvPolynomial σ R\n⊢ eval₂ φ a ↑f = MvPolynomial.eval₂ φ a f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have : ∃ p : MvPolynomial σ R, (p : MvPowerSeries σ R) = f := ⟨f, rfl⟩
rw [eval₂, dif_pos this]
congr
rw [← MvPolynomial.coe_inj, this.choose_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 389,
"column": 2
} | {
"line": 390,
"column": 6
} | {
"line": 392,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ divXPowOrder 1 = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PowerSeries.divXPowOrd... | [] | ext k
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 389,
"column": 2
} | {
"line": 390,
"column": 6
} | {
"line": 392,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ divXPowOrder 1 = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PowerSeries.divXPowOrd... | [] | ext k
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 103,
"column": 18
} | {
"line": 103,
"column": 47
} | {
"line": 103,
"column": 48
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nf g : PowerSeries R\nhq : 1 < q\nk : ℕ\nhs :\n Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })\n (Finset.antidiagonal k) ⊆\n (q ^ k).divisorsAntidiagonal\ni j : ℕ\nhab : (q ^ i, q ^ j) ∈ (q ... | [
"case pos\nR : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nf g : PowerSeries R\nhq : 1 < q\nk : ℕ\nhs :\n Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })\n (Finset.antidiagonal k) ⊆\n (q ^ k).divisorsAntidiagonal\ni j : ℕ\nhab : (q ^ i, q ^ j).1 * (q ^ i, q ^ j... | Nat.mem_divisorsAntidiagonal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Cardinality | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 16
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case succ\nc : ℝ\nh1 : 0 < c\nh2 : c < 1 / 2\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ k < 0, f k = g k\nfn : f 0 = false\ngn : g 0 = true\nf_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n\... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 33
} | {
"line": 190,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : PowerSeries R\nhf : PowerSeries.constantCoeff f = 1\nm n : ℕ\nhmn : m.Coprime n\np k : ℕ\nhp : Nat.Prime p\nhk : 0 < k\n⊢ ((ofPowerSeries (p ^ k)) f) (m * n) = ((ofPowerSeries (p ^ k)) f) m * ((ofPowerSeries (p ^ k)) f) n",
"ppTerm": "?m.70",
"assigned": tr... | [
"R : Type u_1\ninst✝ : CommRing R\nf : PowerSeries R\nhf : PowerSeries.constantCoeff f = 1\nm n : ℕ\nhmn : m.Coprime n\np k : ℕ\nhp : Nat.Prime p\nhk : 0 < k\n⊢ ((ofPowerSeries p) (PowerSeries.subst (PowerSeries.X ^ k) f)) (m * n) =\n ((ofPowerSeries p) (PowerSeries.subst (PowerSeries.X ^ k) f)) m *\n ((ofP... | rw [ofPowerSeries_pow p hk.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 631,
"column": 8
} | {
"line": 631,
"column": 20
} | {
"line": 631,
"column": 21
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : C... | [
"σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : CommRing T✝\n... | coeff_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 27
} | {
"line": 637,
"column": 4
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : C... | [
"σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : CommRing T✝\n... | simp only [coeff_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 27
} | {
"line": 664,
"column": 4
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_9\ninst✝ : CommSemiring R\nx : MvPowerSeries σ R\nn : σ →₀ ℕ\nh : ¬n = 0\n⊢ ((coeff n) fun n ↦ (n.prod fun s m ↦ 0 ^ m) * (coeff n) x) = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Semiring.toModule",
... | [
"case neg\nσ : Type u_1\nR : Type u_9\ninst✝ : CommSemiring R\nx : MvPowerSeries σ R\nn : σ →₀ ℕ\nh : ¬n = 0\n⊢ (n.prod fun s m ↦ 0 ^ m) * x n = 0"
] | simp only [coeff_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.EGauge | {
"line": 264,
"column": 6
} | {
"line": 268,
"column": 43
} | {
"line": 269,
"column": 4
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nU : (i : ι) → Set (E i)\nx : (i : ι) → E i\nr : ℝ≥0∞\nc : ι → 𝕜\nhr₀ : 0 < r\nhI : ∅.Finite\nhU : ∀ i ∈ ∅, Balanced 𝕜 (U i)\nhI₀ : ∅ = uni... | [] | obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
· use 0
simp [@eq_comm _ ∅, hι, hr₀]
· rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
exact ⟨c₀, .inl hc₀, by simp, hc₀r⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.EGauge | {
"line": 264,
"column": 6
} | {
"line": 268,
"column": 43
} | {
"line": 269,
"column": 4
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nU : (i : ι) → Set (E i)\nx : (i : ι) → E i\nr : ℝ≥0∞\nc : ι → 𝕜\nhr₀ : 0 < r\nhI : ∅.Finite\nhU : ∀ i ∈ ∅, Balanced 𝕜 (U i)\nhI₀ : ∅ = uni... | [] | obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
· use 0
simp [@eq_comm _ ∅, hι, hr₀]
· rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
exact ⟨c₀, .inl hc₀, by simp, hc₀r⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 471,
"column": 6
} | {
"line": 484,
"column": 18
} | {
"line": 485,
"column": 6
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\n⊢ ∑ᶠ (d : ℕ), (coeff d) P * (coeff (n + 1 +... | [
"R : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\n⊢ ∑ᶠ (d : ℕ), (coeff d) P * ((coeff (n + 1 + 1)) (B ^ d... | · refine finsum_congr fun i ↦ ?_
· congr 1
obtain (_ | _ | i) := i
· simp
· simp [← sub_eq_add_neg]
· simp only [add_assoc, Nat.reduceAdd]
rw [add_comm B, add_pow, map_sum, Finset.sum_eq_single (a := 0)]
· simp
· rintro (_ | _ | j) hj h... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.EGauge | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 50
} | {
"line": 344,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nx : E\nhc : 1 < ‖c‖\nthis : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }\nh₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0\n⊢ ‖x‖ₑ ≠ 0",
"ppTerm": "?m.81",
"assigned": t... | [] | simpa [enorm, ← NNReal.coe_eq_zero] using h₀ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.EGauge | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 50
} | {
"line": 344,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nx : E\nhc : 1 < ‖c‖\nthis : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }\nh₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0\n⊢ ‖x‖ₑ ≠ 0",
"ppTerm": "?m.81",
"assigned": t... | [] | simpa [enorm, ← NNReal.coe_eq_zero] using h₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.EGauge | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 50
} | {
"line": 344,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nx : E\nhc : 1 < ‖c‖\nthis : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }\nh₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0\n⊢ ‖x‖ₑ ≠ 0",
"ppTerm": "?m.81",
"assigned": t... | [] | simpa [enorm, ← NNReal.coe_eq_zero] using h₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.TangentCone.DimOne | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 78
} | {
"line": 32,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\ns : Set 𝕜\nx : 𝕜\nhx : AccPt x (𝓟 s)\ny : 𝕜\n⊢ y ∈ tangentConeAt 𝕜 s x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instHDiv",
"instSMulOfMul",
"NormedRing.toRing",
"AddGroupWithOne.toAddGroup",
"... | [
"case hd₀\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\ns : Set 𝕜\nx : 𝕜\nhx : AccPt x (𝓟 s)\ny : 𝕜\n⊢ Tendsto (fun x_1 ↦ x_1 - x) (𝓝[≠] x) (𝓝 0)",
"case hds\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\ns : Set 𝕜\nx : 𝕜\nhx : AccPt x (𝓟 s)\ny : 𝕜\n⊢ ∃ᶠ (n : 𝕜) in 𝓝[≠] x, x + (n - x) ∈ s",
"case h... | apply mem_tangentConeAt_of_frequently (𝓝[≠] x) (fun z ↦ y / (z - x)) (· - x) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\ninst✝ : T1Spa... | [] | exact hasFDerivWithinAt_congr_set_nhdsNE h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\ninst✝ : T1Spa... | [] | exact hasFDerivWithinAt_congr_set_nhdsNE h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\ninst✝ : T1Spa... | [] | exact hasFDerivWithinAt_congr_set_nhdsNE h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 26
} | {
"line": 144,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf₀ f₁ : E → F\nf' : E →L[𝕜] F\nx : E\ns : Set E\nh : f₀ =ᶠ[𝓝[s] x] f₁\... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Const | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 17
} | {
"line": 338,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\ns : Set E\nhf : Injective ⇑(fderivWithin 𝕜 f s x)\n⊢ ... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\ns : Set E\nhf : Injective ⇑(fderivWithin 𝕜 f s x)\na✝ : Nontrivia... | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 250,
"column": 92
} | {
"line": 253,
"column": 6
} | {
"line": 255,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ deriv f x = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.m... | [] | by
unfold deriv
rw [fderiv_zero_of_not_differentiableAt h]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 80
} | {
"line": 628,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nF : Type u_5\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : Module 𝕜 F\nl : Filter α\ng : α → F\nι : Type u_7\nE : ι → Type u_8\ninst✝³ : (i : ι) → AddCommGroup (E i)\ninst✝² : (i : ι) → Module 𝕜 (E i)\ninst✝¹ : (i : ι... | [
"α : Type u_1\n𝕜 : Type u_3\nF : Type u_5\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : Module 𝕜 F\nl : Filter α\ng : α → F\nι : Type u_7\nE : ι → Type u_8\ninst✝³ : (i : ι) → AddCommGroup (E i)\ninst✝² : (i : ι) → Module 𝕜 (E i)\ninst✝¹ : (i : ι) → Topologi... | rcases (hIf.eventually_all.mpr this).exists_mem_of_smallSets with ⟨V, hV₀, hV⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 80
} | {
"line": 651,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nF : Type u_5\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : Module 𝕜 F\nl : Filter α\ng : α → F\nι : Type u_7\nE : ι → Type u_8\ninst✝³ : (i : ι) → AddCommGroup (E i)\ninst✝² : (i : ι) → Module 𝕜 (E i)\ninst✝¹ : (i : ι... | [
"α : Type u_1\n𝕜 : Type u_3\nF : Type u_5\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : Module 𝕜 F\nl : Filter α\ng : α → F\nι : Type u_7\nE : ι → Type u_8\ninst✝³ : (i : ι) → AddCommGroup (E i)\ninst✝² : (i : ι) → Module 𝕜 (E i)\ninst✝¹ : (i : ι) → Topologi... | rcases (hIf.eventually_all.mpr this).exists_mem_of_smallSets with ⟨V, hV₀, hV⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 21
} | {
"line": 422,
"column": 4
} | [
{
"pp": "case a\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nC : ℝ\n⊢ ∃ i', ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑... | [
"case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nC : ℝ\n⊢ ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ (_ : ∀ (n : ... | use ‖p 1‖ ⊔ C / r | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Analytic.Basic | {
"line": 213,
"column": 47
} | {
"line": 213,
"column": 88
} | {
"line": 213,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall ... | [] | by simpa only [add_sub_cancel] using hy.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 308,
"column": 6
} | {
"line": 308,
"column": 21
} | {
"line": 309,
"column": 6
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nr : ℝ≥0∞\nhr : r = 0\nhc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)\nh : ∀ (x : ℕ), ∃ x_1, ∃ (_ : x... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nr : ℝ≥0∞\nhr : r = 0\nhc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)\nh : ∀ (x : ℕ), ∃ x_1, ∃ (_ : x ≤ x_1), c x_1 = 0\nti... | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Analytic.Basic | {
"line": 645,
"column": 27
} | {
"line": 645,
"column": 77
} | {
"line": 646,
"column": 4
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeri... | [] | exact ((p (i + k)).le_opNorm _).trans_eq (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Composition | {
"line": 677,
"column": 84
} | {
"line": 693,
"column": 85
} | {
"line": 695,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nq : FormalMultilinearSeries 𝕜 F G\n... | [] | by
-- we expand the composition, using the multilinearity of `q` to expand along each coordinate.
suffices H :
(∑ n ∈ Finset.range M,
∑ r ∈ Fintype.piFinset fun i : Fin n => Finset.Ico 1 N,
q n fun i : Fin n => p (r i) fun _j => z) =
∑ i ∈ compPartialSumTarget 0 M N, q.compAlongCompositi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Inverse | {
"line": 400,
"column": 10
} | {
"line": 400,
"column": 35
} | {
"line": 400,
"column": 35
} | [
{
"pp": "case h\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\nc : Composition k\nhd : (2 ≤ k ∧ k < n + 1) ∧ 1 < c.length\nj : Fin c.length\nthis : c ≠ Composition.single k ⋯\n⊢ c.blocksFun j < n",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Preor... | [
"case h\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\nc : Composition k\nhd : (2 ≤ k ∧ k < n + 1) ∧ 1 < c.length\nj : Fin c.length\nthis : ∀ (i : Fin c.length), c.blocksFun i < k\n⊢ c.blocksFun j < n"
] | Composition.ne_single_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Basic | {
"line": 934,
"column": 2
} | {
"line": 934,
"column": 68
} | {
"line": 935,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall ... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBa... | refine ⟨(x + ·)⁻¹' (insert x s) ∩ Metric.eball (0 : E) r', ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.