module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 146,
"column": 4
} | {
"line": 150,
"column": 45
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case refine_2\nP : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Sche... | [] | · simp only [← CategoryTheory.Over.forget_map]
rw [MorphismProperty.Comma.toCommaMorphism_eq_hom,
← MorphismProperty.Comma.forget_map, ← Functor.comp_map]
rw [← PreservesPullback.iso_hom_fst, P.cancel_left_of_respectsIso]
exact P.pullback_fst _ _ (𝒰.map_prop j) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 205,
"column": 44
} | {
"line": 205,
"column": 56
} | {
"line": 205,
"column": 57
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\ninst✝⁶ : P.IsStableUnderComposition\nX✝ : Scheme\n𝒰✝ : Cover (precoverage P) X✝\n𝒱✝ : (x : 𝒰✝.I₀) → Cover (precoverage P) (𝒰✝.X x)\ninst✝⁵ : X✝.Over S\ninst✝⁴ : Cover.Over S 𝒰✝\ni... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 113,
"column": 4
} | {
"line": 114,
"column": 11
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\nUs : (x : S) → x ∈ U → Set S\nhU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U\nhUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a... | [
"case refine_1.refine_2\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\nUs : (x : S) → x ∈ U → Set S\nhU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U\nhUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a\nhUo : ∀ (x... | · simp [Opens.forall]
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Category.TopCat.EffectiveEpi | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 72
} | {
"line": 70,
"column": 2
} | [
{
"pp": "B X : TopCat\nπ : X ⟶ B\n⊢ EffectiveEpi π ↔ IsQuotientMap ⇑(ConcreteCategory.hom π)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCategory",
"ContinuousMap",
"Nonempty.intro",
"TopCat.str",
"T... | [
"B X : TopCat\nπ : X ⟶ B\nx✝ : EffectiveEpi π\n⊢ IsQuotientMap ⇑(ConcreteCategory.hom π)"
] | refine ⟨fun _ ↦ ?_, fun hπ ↦ ⟨⟨effectiveEpiStructOfQuotientMap π hπ⟩⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 86,
"column": 9
} | {
"line": 86,
"column": 88
} | {
"line": 86,
"column": 88
} | [
{
"pp": "C✝ : Type u_1\ninst✝³ : Category.{v_1, u_1} C✝\nC : Type u_2\ninst✝² : Category.{v_2, u_2} C\nI : Type u_3\nZ Y : I → C\nX : C\ng : (i : I) → Z i ⟶ Y i\nf : (i : I) → Y i ⟶ X\ninst✝¹ : EffectiveEpiFamily Z fun i ↦ g i ≫ f i\ninst✝ : ∀ (i : I), Epi (g i)\nW : C\nφ : (a : I) → Y a ⟶ W\nh : ∀ {Z : C} (a₁ ... | [] | simpa [assoc] using h i₁ i₂ (g₁ ≫ g i₁) (g₂ ≫ g i₂) (by simpa [assoc] using eq) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 86,
"column": 9
} | {
"line": 86,
"column": 88
} | {
"line": 86,
"column": 88
} | [
{
"pp": "C✝ : Type u_1\ninst✝³ : Category.{v_1, u_1} C✝\nC : Type u_2\ninst✝² : Category.{v_2, u_2} C\nI : Type u_3\nZ Y : I → C\nX : C\ng : (i : I) → Z i ⟶ Y i\nf : (i : I) → Y i ⟶ X\ninst✝¹ : EffectiveEpiFamily Z fun i ↦ g i ≫ f i\ninst✝ : ∀ (i : I), Epi (g i)\nW : C\nφ : (a : I) → Y a ⟶ W\nh : ∀ {Z : C} (a₁ ... | [] | simpa [assoc] using h i₁ i₂ (g₁ ≫ g i₁) (g₂ ≫ g i₂) (by simpa [assoc] using eq) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 86,
"column": 9
} | {
"line": 86,
"column": 88
} | {
"line": 86,
"column": 88
} | [
{
"pp": "C✝ : Type u_1\ninst✝³ : Category.{v_1, u_1} C✝\nC : Type u_2\ninst✝² : Category.{v_2, u_2} C\nI : Type u_3\nZ Y : I → C\nX : C\ng : (i : I) → Z i ⟶ Y i\nf : (i : I) → Y i ⟶ X\ninst✝¹ : EffectiveEpiFamily Z fun i ↦ g i ≫ f i\ninst✝ : ∀ (i : I), Epi (g i)\nW : C\nφ : (a : I) → Y a ⟶ W\nh : ∀ {Z : C} (a₁ ... | [] | simpa [assoc] using h i₁ i₂ (g₁ ≫ g i₁) (g₂ ≫ g i₂) (by simpa [assoc] using eq) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.EffectiveEpi.Preserves | {
"line": 74,
"column": 4
} | {
"line": 75,
"column": 10
} | {
"line": 76,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nD : Type u_2\ninst✝¹ : Category.{v_2, u_2} D\ne : C ≌ D\nB : C\nα : Type u_3\nX : α → C\nπ : (a : α) → X a ⟶ B\ninst✝ : EffectiveEpiFamily X π\nW✝ : D\nε : (a : α) → e.functor.obj (X a) ⟶ W✝\nh :\n ∀ {Z : D} (a₁ a₂ : α) (g₁ : Z ⟶ e.functor.obj (X a₁)) (g₂ ... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nD : Type u_2\ninst✝¹ : Category.{v_2, u_2} D\ne : C ≌ D\nB : C\nα : Type u_3\nX : α → C\nπ : (a : α) → X a ⟶ B\ninst✝ : EffectiveEpiFamily X π\nW✝ : D\nε : (a : α) → e.functor.obj (X a) ⟶ W✝\nh :\n ∀ {Z : D} (a₁ a₂ : α) (g₁ : Z ⟶ e.functor.obj (X a₁)) (g₂ : Z ⟶ e.func... | · rw [← congrArg e.inverse.map (hm a)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 89
} | {
"line": 535,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nA : ExistsHomHomCompEqCompAux D t f\n⊢ Set.range ⇑A.g ⊆ ↑(Schem... | [] | simpa [ExistsHomHomCompEqCompAux.hii', g] using! A.exists_index.choose_spec.choose_spec | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 89
} | {
"line": 535,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nA : ExistsHomHomCompEqCompAux D t f\n⊢ Set.range ⇑A.g ⊆ ↑(Schem... | [] | simpa [ExistsHomHomCompEqCompAux.hii', g] using! A.exists_index.choose_spec.choose_spec | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 89
} | {
"line": 535,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nA : ExistsHomHomCompEqCompAux D t f\n⊢ Set.range ⇑A.g ⊆ ↑(Schem... | [] | simpa [ExistsHomHomCompEqCompAux.hii', g] using! A.exists_index.choose_spec.choose_spec | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 752,
"column": 8
} | {
"line": 752,
"column": 70
} | {
"line": 753,
"column": 8
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ni : I\ninst✝ : CompactSpace ↥(D.obj i)\ns : ↑Γ(D.obj i, ⊤)\nhs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0\nx : ↥(D.ob... | [
"case e'_2\nI : Type u\ninst✝³ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ni : I\ninst✝ : CompactSpace ↥(D.obj i)\ns : ↑Γ(D.obj i, ⊤)\nhs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0\nx : ↥(D.obj... | convert congr((c.pt.presheaf.map (homOfLE le_top).op).hom $hs) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 28
} | {
"line": 243,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\ninst✝ : W.IsElliptic\n⊢ ↑(C • W).Δ'⁻¹ = ↑C.u ^ 12 * ↑W.Δ'⁻¹",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"congrArg",
... | [
"R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\ninst✝ : W.IsElliptic\n⊢ ↑(C.u ^ 12 * W.Δ'⁻¹) = ↑C.u ^ 12 * ↑W.Δ'⁻¹"
] | inv_variableChange_Δ', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 306,
"column": 47
} | {
"line": 306,
"column": 59
} | {
"line": 306,
"column": 60
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Affine R\nx₁ x₂ y₁ ℓ : R\nhx' : W'.Equation (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ)\nhx : eval (W'.addX x₁ x₂ ℓ) (derivative (W'.addPolynomial x₁ y₁ ℓ)) ≠ 0\n⊢ evalEval (W'.addX x₁ x₂ ℓ) (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁) W'.polynomialX ≠ 0 ∨\n evalEval (W'.addX... | [
"R : Type r\ninst✝ : CommRing R\nW' : Affine R\nx₁ x₂ y₁ ℓ : R\nhx' : W'.Equation (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ)\nhx : eval (W'.addX x₁ x₂ ℓ) (derivative (W'.addPolynomial x₁ y₁ ℓ)) ≠ 0\n⊢ evalEval (W'.addX x₁ x₂ ℓ) (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁)\n (C (C W'.a₁) * Y - C (C 3 * X ^ 2 + C (2 * W'.... | polynomialX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 415,
"column": 97
} | {
"line": 417,
"column": 10
} | {
"line": 419,
"column": 0
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Affine R\nf : R →+* S\nx₁ y₁ x₂ ℓ : R\n⊢ (W'.map f).negAddY (f x₁) (f x₂) (f y₁) (f ℓ) = f (W'.negAddY x₁ x₂ y₁ ℓ)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCom... | [] | by
simp only [negAddY, map_addX]
map_simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 448,
"column": 25
} | {
"line": 448,
"column": 36
} | {
"line": 448,
"column": 37
} | [
{
"pp": "R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Affine R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsScala... | [
"R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Affine R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsScalarTower R S B... | ← map_addX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 339,
"column": 52
} | {
"line": 339,
"column": 77
} | {
"line": 339,
"column": 78
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\n⊢ span {(mk W) (Y - C (linePolynomial x₁ y₁ (W.slope ... | [
"F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\n⊢ span {(mk W) (Y - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)... | sub_sub_sub_cancel_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 25
} | {
"line": 342,
"column": 2
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\n⊢ span {(mk W) (Y - C (linePolynomial x₁ y₁ (W.slope ... | [
"F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\n⊢ (span {XClass W x₁} ⊔ (span {XClass W x₂} ⊔ span {(mk W) (Y - W... | apply congr_arg (_ ∘ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 741,
"column": 4
} | {
"line": 741,
"column": 98
} | {
"line": 742,
"column": 4
} | [
{
"pp": "case even\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nb c d : R\nm✝ : ℕ\nih : ∀ k < 2 * (m✝ + 3), f (preNormEDS' b c d k) = preNormEDS' (f b) (f c) (f d) k\n⊢ f (preNormEDS' b c d (2 * (m✝ + 3))) = preNo... | [
"case even\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nb c d : R\nm✝ : ℕ\nih : ∀ k < 2 * (m✝ + 3), f (preNormEDS' b c d k) = preNormEDS' (f b) (f c) (f d) k\n⊢ f (preNormEDS' b c d (m✝ + 2)) ^ 2 * f (preNormEDS' b c ... | simp only [preNormEDS'_even, preNormEDS'_odd, apply_ite f, map_pow, map_mul, map_sub, map_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 741,
"column": 4
} | {
"line": 741,
"column": 98
} | {
"line": 742,
"column": 4
} | [
{
"pp": "case odd\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nb c d : R\nm✝ : ℕ\nih : ∀ k < 2 * (m✝ + 2) + 1, f (preNormEDS' b c d k) = preNormEDS' (f b) (f c) (f d) k\n⊢ f (preNormEDS' b c d (2 * (m✝ + 2) + 1)) ... | [
"case odd\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nb c d : R\nm✝ : ℕ\nih : ∀ k < 2 * (m✝ + 2) + 1, f (preNormEDS' b c d k) = preNormEDS' (f b) (f c) (f d) k\n⊢ ((f (preNormEDS' b c d (m✝ + 4)) * f (preNormEDS' b c... | simp only [preNormEDS'_even, preNormEDS'_odd, apply_ite f, map_pow, map_mul, map_sub, map_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 7
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ W.c₆ = -64 * W.a₂ ^ 3 + 288 * W.a₂ * W.a₄ - 864 * W.a₆",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",... | [] | rw [c₆, b₂_of_isCharNeTwoNF, b₄_of_isCharNeTwoNF, b₆_of_isCharNeTwoNF]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 7
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ W.c₆ = -64 * W.a₂ ^ 3 + 288 * W.a₂ * W.a₄ - 864 * W.a₆",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",... | [] | rw [c₆, b₂_of_isCharNeTwoNF, b₄_of_isCharNeTwoNF, b₆_of_isCharNeTwoNF]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 673,
"column": 2
} | {
"line": 673,
"column": 53
} | {
"line": 674,
"column": 2
} | [
{
"pp": "case a₁\nF : Type u_2\ninst✝¹ : Field F\ninst✝ : CharP F 2\nW : WeierstrassCurve F\nha₁ : W.a₁ ≠ 0\n⊢ (W.toCharTwoJNeZeroNF ha₁ • W).a₁ = 1",
"ppTerm": "?a₁",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",... | [
"case a₃\nF : Type u_2\ninst✝¹ : Field F\ninst✝ : CharP F 2\nW : WeierstrassCurve F\nha₁ : W.a₁ ≠ 0\n⊢ (W.toCharTwoJNeZeroNF ha₁ • W).a₃ = 0",
"case a₄\nF : Type u_2\ninst✝¹ : Field F\ninst✝ : CharP F 2\nW : WeierstrassCurve F\nha₁ : W.a₁ ≠ 0\n⊢ (W.toCharTwoJNeZeroNF ha₁ • W).a₄ = 0"
] | · simp [toCharTwoJNeZeroNF, ha₁, variableChange_a₁] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1198,
"column": 4
} | {
"line": 1210,
"column": 75
} | {
"line": 1211,
"column": 2
} | [
{
"pp": "case inr\nI : Type u\ninst✝⁵ : Category.{u, u} I\ninst✝⁴ : IsCofiltered I\nR : CommRingCat\ninst✝³ : IsAffine (Spec R)\nS : CommRingCat\ninst✝² : IsAffine (Spec S)\nφ : R ⟶ S\ninst✝¹ : LocallyOfFinitePresentation (Spec.map φ)\nD : I ⥤ CommRingCatᵒᵖ\nc : Cone (D ⋙ Scheme.Spec)\nhc : IsLimit c\ninst✝ : ∀... | [] | have inst : IsAffine c.pt := isAffine_of_isLimit _ hc
let e' : (D ⋙ Scheme.Spec).op ⋙ Γ ≅ D.leftOp := D.leftOp.isoWhiskerLeft SpecΓIdentity
let c' := coneOfCoconeLeftOp ((Cocone.precompose e'.inv).obj (Γ.mapCocone c.op))
have inst : ∀ i, IsAffine ((D ⋙ Scheme.Spec).op.obj i).unop := by dsimp; infer_instance... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1198,
"column": 4
} | {
"line": 1210,
"column": 75
} | {
"line": 1211,
"column": 2
} | [
{
"pp": "case inr\nI : Type u\ninst✝⁵ : Category.{u, u} I\ninst✝⁴ : IsCofiltered I\nR : CommRingCat\ninst✝³ : IsAffine (Spec R)\nS : CommRingCat\ninst✝² : IsAffine (Spec S)\nφ : R ⟶ S\ninst✝¹ : LocallyOfFinitePresentation (Spec.map φ)\nD : I ⥤ CommRingCatᵒᵖ\nc : Cone (D ⋙ Scheme.Spec)\nhc : IsLimit c\ninst✝ : ∀... | [] | have inst : IsAffine c.pt := isAffine_of_isLimit _ hc
let e' : (D ⋙ Scheme.Spec).op ⋙ Γ ≅ D.leftOp := D.leftOp.isoWhiskerLeft SpecΓIdentity
let c' := coneOfCoconeLeftOp ((Cocone.precompose e'.inv).obj (Γ.mapCocone c.op))
have inst : ∀ i, IsAffine ((D ⋙ Scheme.Spec).op.obj i).unop := by dsimp; infer_instance... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 165,
"column": 2
} | {
"line": 166,
"column": 54
} | {
"line": 167,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE E' : WeierstrassCurve F\ninst✝⁴ : E.IsElliptic\ninst✝³ : E'.IsElliptic\ninst✝² : CharP F 3\ninst✝¹ : E.IsShortNF\ninst✝ : E'.IsShortNF\nha₄ : E.a₄ ≠ 0\nha₄' : E'.a₄ ≠ 0\nthis : NeZero 4\nu : F\nhu : u ^ 4 = E.a₄ / E'.a₄\n⊢ ∃ C, C • E = E'",
... | [
"F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE E' : WeierstrassCurve F\ninst✝⁴ : E.IsElliptic\ninst✝³ : E'.IsElliptic\ninst✝² : CharP F 3\ninst✝¹ : E.IsShortNF\ninst✝ : E'.IsShortNF\nha₄ : E.a₄ ≠ 0\nha₄' : E'.a₄ ≠ 0\nthis : NeZero 4\nu : F\nhu : u ^ 4 = E.a₄ / E'.a₄\nr : F\nhr : 1 * r ^ 3 + E.a₄ * r + (... | obtain ⟨r, hr⟩ := IsSepClosed.exists_root_C_mul_X_pow_add_C_mul_X_add_C' 3 3
1 _ (E.a₆ - u ^ 6 * E'.a₆) (by simp) (by simp) ha₄ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 18
} | {
"line": 306,
"column": 19
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\n⊢ W'.polynomialX = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z ^ 2 + C W'.a₄ * Z ^ 4)",
"ppTerm": "?m.150",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Weierstra... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\n⊢ (pderiv x) W'.polynomial = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z ^ 2 + C W'.a₄ * Z ^ 4)"
] | polynomialX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 162,
"column": 40
} | {
"line": 162,
"column": 44
} | {
"line": 162,
"column": 45
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\nhy : P y = W.negY P\nhy' : (eval P) W.polynomialY = P y - W.negY P\n⊢ W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (eval P) W.polynomialY ≠ 0) ↔ W.Equation P ∧ (eval P) W.polynomialX ≠ 0",
"ppTerm": "?m.65",
"assigned": ... | [
"F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\nhy : P y = W.negY P\nhy' : (eval P) W.polynomialY = P y - W.negY P\n⊢ W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ P y - W.negY P ≠ 0) ↔ W.Equation P ∧ (eval P) W.polynomialX ≠ 0"
] | hy', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 293,
"column": 6
} | {
"line": 297,
"column": 83
} | {
"line": 298,
"column": 6
} | [
{
"pp": "case pos\nF : Type u\ninst✝ : Field F\nW : Jacobian 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⊢ W.Nonsingular (W.add P Q)",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case neg\nF : Type u\ninst✝ : Field F\nW : Jacobian 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⊢ W.Nonsingular (W.add P Q)"
] | · by_cases hy : P y * Q z ^ 3 = Q y * P z ^ 3
· simp only [add_of_Y_eq hPz hQz hxy.left hy hxy.right, nonsingular_smul _ <|
isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, nonsingular_zero]
· simp only [add_of_Y_ne hP.left hQ.left hPz hQz hxy.left hy,
nonsingular_smul _ <... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Nonarchimedean.Basic | {
"line": 127,
"column": 93
} | {
"line": 135,
"column": 10
} | {
"line": 137,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\ninst✝ : NonarchimedeanRing R\nU : OpenAddSubgroup R\n⊢ ∃ V, ↑V * ↑V ⊆ ↑U",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"HMul.hMul",
... | [] | by
let ⟨V, H⟩ := prod_self_subset <| (U.isOpen.preimage continuous_mul).mem_nhds <| by
simpa only [Set.mem_preimage, Prod.snd_zero, mul_zero] using! U.zero_mem
use V
rintro v ⟨a, ha, b, hb, hv⟩
have hy := H (Set.mk_mem_prod ha hb)
simp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy
rw [SetLike.mem_c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 11
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ B j\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
... | [] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 11
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ B j\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
... | [] | use k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 11
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ B j\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
... | [] | use k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 11
} | {
"line": 109,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni k : ι\nhk : ↑(B k) * ↑(B k) ⊆ ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGr... | [] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 11
} | {
"line": 109,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni k : ι\nhk : ↑(B k) * ↑(B k) ⊆ ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGr... | [] | use k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 11
} | {
"line": 109,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni k : ι\nhk : ↑(B k) * ↑(B k) ⊆ ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGr... | [] | use k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 11
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x₀ * x) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 11
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x₀ * x) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 11
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x₀ * x) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 11
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x * x₀) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 11
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x * x₀) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 11
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case h.left\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\nx₀ : A\ni k : ι\nhk : ↑(B k) ⊆ (fun x ↦ x * x₀) ⁻¹' ↑(B i)\n⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Add... | [] | use k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 11
} | {
"line": 260,
"column": 4
} | [
{
"pp": "case h.left\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ ... | [] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 11
} | {
"line": 260,
"column": 4
} | [
{
"pp": "case h.left\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ ... | [] | use k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 11
} | {
"line": 260,
"column": 4
} | [
{
"pp": "case h.left\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\ni j k : ι\nhk : B k ≤ B i ⊓ ... | [] | use k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 500,
"column": 6
} | {
"line": 500,
"column": 70
} | {
"line": 500,
"column": 71
} | [
{
"pp": "F : 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\n ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^... | [
"F : 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⊢ Affine.Point.some\n (W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)\n ... | toAffine_some <| nonsingular_add_of_Z_ne_zero hP hQ hPz hQz hxy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 16
} | {
"line": 507,
"column": 2
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nhQz : Q z = 0\n⊢ W'.addX P Q = (-(Q x * P z)) ^ 2 * P x",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congr... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nhQz : Q z = 0\n⊢ P x * Q x ^ 2 * P z ^ 2 - 2 * P y * Q y * P z * 0 + P x ^ 2 * Q x * 0 ^ 2 - W'.a₁ * P x * Q y * P z ^ 2 * 0 -\n W'.a₁ * P y * Q x * P z * 0 ^ 2 +\n 2 * W'.a₂ * P x * Q x * P z ^ 2 * 0 ^ 2 -\n ... | rw [addX, hQz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 525,
"column": 6
} | {
"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.le... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.WithZeroTopology | {
"line": 104,
"column": 58
} | {
"line": 106,
"column": 23
} | {
"line": 108,
"column": 0
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nx : Γ₀\nh : x ≠ 0\n⊢ (𝓝 x).HasBasis (fun x ↦ True) fun x_1 ↦ {x}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pure.pure",
"WithZeroTopology.topologicalSpace",
"Eq.mpr",
"Filter.hasBasis_pure",
... | [] | by
rw [nhds_of_ne_zero h]
exact hasBasis_pure _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.WithZeroTopology | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 37
} | {
"line": 165,
"column": 4
} | [
{
"pp": "α : Type u_1\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\nx y : Γ₀\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (x, y)) (𝓝 ((x, y).1 * (x, y).2))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"WithZeroTopology.topologicalSpace",... | [
"case inr\nα : Type u_1\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\nx y : Γ₀\nthis : ∀ (x y : Γ₀), x ≤ y → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (x, y)) (𝓝 ((x, y).1 * (x, y).2))\nhle : ¬x ≤ y\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (x, y)) (𝓝 ((x, y).1 * (x, y).2))",
... | wlog hle : x ≤ y generalizing x y | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 35
} | {
"line": 160,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ (𝓤 R).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 - p.1) < ↑γ}",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val"... | [
"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ (Filter.comap (fun x ↦ x.2 - x.1) (𝓝 0)).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 - p.1) < ↑γ}"
] | uniformity_eq_comap_nhds_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 114,
"column": 63
} | {
"line": 120,
"column": 51
} | {
"line": 122,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Ring R\ninst✝⁴ : ValuativeRel R\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\nv : Valuation R Γ₀\ninst✝¹ : v.Compatible\ninst✝ : IsTopologicalAddGroup R\nH : ∀ {s : Set R}, s ∈ 𝓝 0 ↔ ∃ γ, {z | v.restrict z < ↑γ} ⊆ s\n⊢ IsValuativeTopolo... | [] | by
apply of_mem_nhds_iff_vle v (fun {s x} ↦ ?_)
rw [← vadd_mem_nhds_vadd_iff (g := -x)]
simp only [vadd_eq_add, neg_add_cancel, H, subset_vadd_set_iff, neg_neg]
suffices ∀ (γ : (ValueGroup₀ (.ofClass v))ˣ), (x +ᵥ {z | v.restrict z < ↑γ}) =
{a | v.restrict (-x + a) < ↑γ} by simp_all [neg_add_eq_sub]
simp [... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 35
} | {
"line": 233,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Ring R\ninst✝⁴ : ValuativeRel R\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\n_u : UniformSpace R\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ (𝓤 R).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 -... | [
"R : Type u_1\ninst✝⁵ : Ring R\ninst✝⁴ : ValuativeRel R\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\n_u : UniformSpace R\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ (Filter.comap (fun x ↦ x.2 - x.1) (𝓝 0)).HasBasis (fun x ↦ True) fun γ ↦... | uniformity_eq_comap_nhds_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 799,
"column": 25
} | {
"line": 799,
"column": 36
} | {
"line": 799,
"column": 37
} | [
{
"pp": "R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Jacobian R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsSca... | [
"R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Jacobian R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsScalarTower R S... | ← map_addX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 49
} | {
"line": 304,
"column": 2
} | [
{
"pp": "case h\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝² : Field K\nv : Valuation K Γ\ninst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup\ninst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup\ng : Γˣ := ⋯\nhg : g = Subgroup.genLTOne (MonoidWithZeroHom.ofCla... | [
"case h\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝² : Field K\nv : Valuation K Γ\ninst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup\ninst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup\ng : Γˣ := ⋯\nhg : g = Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueG... | simp only [MonoidWithZeroHom.coe_ofClass] at hπ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 31
} | {
"line": 65,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx y : K\nγ : Γ₀ˣ\ny_ne : y ≠ 0\nh✝ : v (x - y) < min (↑γ * (v y * v y)) (v y)\nhyp1 : v (x - y) < ↑γ * (v y * v y)\nhyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ\nhyp2 : v (x - y) < v y\nkey : ... | [] | exact v.zero_iff.1 key.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 585,
"column": 2
} | {
"line": 601,
"column": 52
} | {
"line": 603,
"column": 0
} | [
{
"pp": "R : 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'\nh : v.IsEquiv w\n⊢ UniformContinuous ⇑(RingHom.id R)",
"ppTerm": "?m.31",
"assigned": true,
"usedCon... | [] | have h_val : ((Valued.mk' v).v).IsEquiv (Valued.mk' w).v := h
have h_res : v.restrict.IsEquiv w.restrict := h_val.restrict
refine @uniformContinuous_of_continuousAt_zero _ _ (Valued.mk' w).toUniformSpace _ _
_ (Valued.mk' v).toUniformSpace _ _ _ _ (RingHom.id R) ?_
simp_rw [ContinuousAt, map_zero, (Valued.has... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 585,
"column": 2
} | {
"line": 601,
"column": 52
} | {
"line": 603,
"column": 0
} | [
{
"pp": "R : 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'\nh : v.IsEquiv w\n⊢ UniformContinuous ⇑(RingHom.id R)",
"ppTerm": "?m.31",
"assigned": true,
"usedCon... | [] | have h_val : ((Valued.mk' v).v).IsEquiv (Valued.mk' w).v := h
have h_res : v.restrict.IsEquiv w.restrict := h_val.restrict
refine @uniformContinuous_of_continuousAt_zero _ _ (Valued.mk' w).toUniformSpace _ _
_ (Valued.mk' v).toUniformSpace _ _ _ _ (RingHom.id R) ?_
simp_rw [ContinuousAt, map_zero, (Valued.has... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 29
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case h.a₆\nR : Type u_1\ninst✝² : CommRing R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nW : WeierstrassCurve K\nh₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁\nh₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂\nh₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃\nh₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄\nh₆ : ∃ r₆, (algebraMap R... | [] | apply h₆.choose_spec.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 29
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case h.a₆\nR : Type u_1\ninst✝² : CommRing R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nW : WeierstrassCurve K\nh₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁\nh₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂\nh₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃\nh₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄\nh₆ : ∃ r₆, (algebraMap R... | [] | apply h₆.choose_spec.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 29
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case h.a₆\nR : Type u_1\ninst✝² : CommRing R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nW : WeierstrassCurve K\nh₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁\nh₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂\nh₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃\nh₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄\nh₆ : ∃ r₆, (algebraMap R... | [] | apply h₆.choose_spec.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 15
} | {
"line": 221,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nh : IsIntegral R W\nr : R\nhr : (algebraMap R K) r = W.Δ\n⊢ (valuation K (maximalIdeal R)) W.Δ ≤ 1",
... | [
"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nW : WeierstrassCurve K\nh : IsIntegral R W\nr : R\nhr : (algebraMap R K) r = W.Δ\n⊢ (valuation K (maximalIdeal R)) ((algebraMap R K) r) ≤ 1"
... | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Reduction | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 29
} | {
"line": 333,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nW : WeierstrassCurve K\ninst✝ : IsMinimal R W\n⊢ IsMinimal R W ∧ (valuation K (IsDiscreteValuationRing.maximalIdeal R)) W.Δ = 1 ... | [
"R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nW : WeierstrassCurve K\ninst✝ : IsMinimal R W\n⊢ IsMinimal R W ∧ (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (i... | ← integralModel_Δ_eq R W, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 319,
"column": 6
} | {
"line": 320,
"column": 76
} | {
"line": 321,
"column": 6
} | [
{
"pp": "case hp\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ IsClosed {x | extension (x.1 * x.2) = extension x.1 * extension x.2}",
"ppTerm... | [
"case hp\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nc1 : Continuous fun x ↦ extension (x.1 * x.2)\n⊢ IsClosed {x | extension (x.1 * x.2) = extension... | have c1 : Continuous fun x : hat K × hat K => Valued.extension (x.1 * x.2) :=
Valued.continuous_extension.comp (continuous_fst.mul continuous_snd) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 29
} | {
"line": 171,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\na : R\n⊢ (coeff n) ((monomial n) a) = a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"L... | [] | classical
rw [monomial_def]
exact Pi.single_eq_same _ _ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 29
} | {
"line": 171,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\na : R\n⊢ (coeff n) ((monomial n) a) = a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"L... | [] | classical
rw [monomial_def]
exact Pi.single_eq_same _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 29
} | {
"line": 171,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\na : R\n⊢ (coeff n) ((monomial n) a) = a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"L... | [] | classical
rw [monomial_def]
exact Pi.single_eq_same _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 218,
"column": 73
} | {
"line": 219,
"column": 27
} | {
"line": 221,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\n⊢ (coeff n) X = if n = 1 then 1 else 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"id",
... | [] | by
rw [X_eq, coeff_monomial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 25
} | {
"line": 502,
"column": 25
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\ns t : σ\nH : ¬single s 1 = single t 1\nh : 1 = 0\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"NeZero.one",
"AddCommMonoidWithO... | [] | exact one_ne_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 519,
"column": 12
} | {
"line": 519,
"column": 22
} | {
"line": 519,
"column": 23
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\nS : Type u_3\nT : Type u_4\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn : σ →₀ ℕ\n⊢ f ((coeff n) 1) = (coeff n) 1",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAd... | [
"σ : Type u_1\nR : Type u_2\nS : Type u_3\nT : Type u_4\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn : σ →₀ ℕ\n⊢ f (if n = 0 then 1 else 0) = (coeff n) 1"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 267,
"column": 2
} | {
"line": 272,
"column": 29
} | {
"line": 274,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ emultiplicity v.asIdeal (Ideal.span {r}) ≤ ↑n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"WithZero.exp_ad... | [] | rw [← ENat.lt_coe_add_one_iff, ← ENat.coe_one, ← ENat.coe_add, emultiplicity_lt_iff_not_dvd,
← intValuation_le_pow_iff_dvd, not_le, Nat.cast_add, Nat.cast_one, neg_add, exp_add,
exp_neg 1, mul_inv_lt_iff₀ (by simp)]
by_cases hv : v.intValuation r = 0
· simp [hv]
· rw [lt_mul_exp_iff_le hv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 267,
"column": 2
} | {
"line": 272,
"column": 29
} | {
"line": 274,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ emultiplicity v.asIdeal (Ideal.span {r}) ≤ ↑n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"WithZero.exp_ad... | [] | rw [← ENat.lt_coe_add_one_iff, ← ENat.coe_one, ← ENat.coe_add, emultiplicity_lt_iff_not_dvd,
← intValuation_le_pow_iff_dvd, not_le, Nat.cast_add, Nat.cast_one, neg_add, exp_add,
exp_neg 1, mul_inv_lt_iff₀ (by simp)]
by_cases hv : v.intValuation r = 0
· simp [hv]
· rw [lt_mul_exp_iff_le hv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 658,
"column": 2
} | {
"line": 658,
"column": 17
} | {
"line": 659,
"column": 6
} | [
{
"pp": "case inr.succ\nR : Type u_2\ninst✝ : CommSemiring R\nφ : R⟦X⟧\nn' : ℕ\nih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)\nhn : n' + 1 > 0\n⊢ (coeff 1) (φ ^ (n' + 1)) = ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)",
"ppTerm": "?inr.succ",
"assigned": t... | [] | | succ n' ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 246,
"column": 2
} | {
"line": 249,
"column": 19
} | {
"line": 251,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf g : MvPowerSeries σ R\n⊢ min (weightedOrder w f) (weightedOrder w g) ≤ weightedOrder w (f + g)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWith... | [] | apply le_weightedOrder w
simp +contextual only
[coeff_eq_zero_of_lt_weightedOrder w, lt_min_iff, map_add, add_zero,
imp_true_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 246,
"column": 2
} | {
"line": 249,
"column": 19
} | {
"line": 251,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf g : MvPowerSeries σ R\n⊢ min (weightedOrder w f) (weightedOrder w g) ≤ weightedOrder w (f + g)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWith... | [] | apply le_weightedOrder w
simp +contextual only
[coeff_eq_zero_of_lt_weightedOrder w, lt_min_iff, map_add, add_zero,
imp_true_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 441,
"column": 2
} | {
"line": 442,
"column": 30
} | {
"line": 444,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nf : MvPowerSeries σ R\nn : ℕ\n⊢ f.order = ↑n ↔ (∃ d, (coeff d) f ≠ 0 ∧ degree d = n) ∧ ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | [] | simp_rw [degree_eq_weight_one]
exact weightedOrder_eq_nat _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 441,
"column": 2
} | {
"line": 442,
"column": 30
} | {
"line": 444,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nf : MvPowerSeries σ R\nn : ℕ\n⊢ f.order = ↑n ↔ (∃ d, (coeff d) f ≠ 0 ∧ degree d = n) ∧ ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | [] | simp_rw [degree_eq_weight_one]
exact weightedOrder_eq_nat _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 625,
"column": 2
} | {
"line": 625,
"column": 88
} | {
"line": 626,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\np : ℕ\nhf : ↑p = weightedOrder w f\nhf' : (weightedHomogeneousComponent w p) f = 0\n⊢ False",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"No... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\np : ℕ\nhf : ↑p = weightedOrder w f\nhf' : (weightedHomogeneousComponent w p) f = 0\nd : σ →₀ ℕ\nhd : (coeff d) f ≠ 0 ∧ ↑((weight w) d) = weightedOrder w f\n⊢ False"
] | obtain ⟨d, hd⟩ := f.exists_coeff_ne_zero_and_weightedOrder w (by rw [← hf, toNat_coe]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 856,
"column": 6
} | {
"line": 857,
"column": 30
} | {
"line": 859,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\nK : Type u_2\nS : Type u_3\ninst✝⁴ : Field K\ninst✝³ : CommSemiring S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝ : Algebra S K\nr : S\nx : WithVal (valuation K v)\n⊢ r • ↑x = (Completion.coe... | [] | simp [Algebra.smul_def, Completion.algebraMap_def, WithVal.algebraMap_right_apply,
Completion.coeRingHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 856,
"column": 6
} | {
"line": 857,
"column": 30
} | {
"line": 859,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\nK : Type u_2\nS : Type u_3\ninst✝⁴ : Field K\ninst✝³ : CommSemiring S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝ : Algebra S K\nr : S\nx : WithVal (valuation K v)\n⊢ r • ↑x = (Completion.coe... | [] | simp [Algebra.smul_def, Completion.algebraMap_def, WithVal.algebraMap_right_apply,
Completion.coeRingHom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 856,
"column": 6
} | {
"line": 857,
"column": 30
} | {
"line": 859,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\nK : Type u_2\nS : Type u_3\ninst✝⁴ : Field K\ninst✝³ : CommSemiring S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝ : Algebra S K\nr : S\nx : WithVal (valuation K v)\n⊢ r • ↑x = (Completion.coe... | [] | simp [Algebra.smul_def, Completion.algebraMap_def, WithVal.algebraMap_right_apply,
Completion.coeRingHom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 69
} | {
"line": 302,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : Semiring R\nf : MvPowerSeries σ R\nh : constantCoeff f = 0\nn m : ℕ\nhm : n + 1 ≤ m\n⊢ ↑n < m • f.order",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"instHSMul",
"instAddMonoidWithOneE... | [
"σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : Semiring R\nf : MvPowerSeries σ R\nh : constantCoeff f = 0\nn m : ℕ\nhm : n + 1 ≤ m\n⊢ ↑m ≤ m • f.order"
] | refine (ENat.coe_lt_coe.mpr (Nat.add_one_le_iff.mp hm)).trans_le ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 987,
"column": 4
} | {
"line": 987,
"column": 55
} | {
"line": 988,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : 1 < Valued.v a\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"ppTerm": "?neg✝",
... | [
"case neg\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : 1 < Valued.v a\nϖ : R\nhϖ : v.intValuation ϖ = exp (-1)\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v"... | obtain ⟨ϖ, hϖ⟩ := intValuation_exists_uniformizer v | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 15
} | {
"line": 339,
"column": 2
} | [
{
"pp": "case inr\nσ : Type u_3\nR : Type u_4\ninst✝³ : TopologicalSpace R\ninst✝² : CommSemiring R\nι : Type u_5\nf : ι → MvPowerSeries σ R\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nw : σ → ℕ\nhempty : Nonempty ι\nd : σ →₀ ℕ\nh : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)\ni ... | [
"case inr\nσ : Type u_3\nR : Type u_4\ninst✝³ : TopologicalSpace R\ninst✝² : CommSemiring R\nι : Type u_5\nf : ι → MvPowerSeries σ R\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nw : σ → ℕ\nhempty : Nonempty ι\nd : σ →₀ ℕ\nh : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)\ni : ι\nhi : ∀ ... | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.RingTheory.PowerSeries.PiTopology | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 15
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : CommSemiring R\nι : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → R⟦X⟧\nhempty : Nonempty ι\nn : ℕ\nh : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order\ni : ι\nhi : ∀ (b : ι), i ≤ b → ↑n < (f b).order\ns : Fi... | [
"case inr\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : CommSemiring R\nι : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → R⟦X⟧\nhempty : Nonempty ι\nn : ℕ\nh : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order\ni : ι\nhi : ∀ (b : ι), i ≤ b → ↑n < (f b).order\ns : Finset ι\nhs :... | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 418,
"column": 2
} | {
"line": 431,
"column": 32
} | {
"line": 433,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nf g : R⟦X⟧\n⊢ (f * g).divXPowOrder = f.divXPowOrder * g.divXPowOrder",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Semigroup.toMul",
"Trans.trans",
"MvPowerSeries.ins... | [] | by_cases! h : f = 0 ∨ g = 0
· rcases h with (h | h) <;> simp [h]
apply X_pow_mul_cancel (k := f.order.toNat + g.order.toNat)
calc
_ = X ^ ((f * g).order.toNat) * (f * g).divXPowOrder := by
rw [order_mul, ENat.toNat_add (order_eq_top.not.mpr h.1) (order_eq_top.not.mpr h.2)]
_ = f * g := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 418,
"column": 2
} | {
"line": 431,
"column": 32
} | {
"line": 433,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nf g : R⟦X⟧\n⊢ (f * g).divXPowOrder = f.divXPowOrder * g.divXPowOrder",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Semigroup.toMul",
"Trans.trans",
"MvPowerSeries.ins... | [] | by_cases! h : f = 0 ∨ g = 0
· rcases h with (h | h) <;> simp [h]
apply X_pow_mul_cancel (k := f.order.toNat + g.order.toNat)
calc
_ = X ^ ((f * g).order.toNat) * (f * g).divXPowOrder := by
rw [order_mul, ENat.toNat_add (order_eq_top.not.mpr h.1) (order_eq_top.not.mpr h.2)]
_ = f * g := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 276,
"column": 23
} | {
"line": 276,
"column": 51
} | {
"line": 278,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : UniformSpace R\nS : Type u_3\ninst✝⁸ : CommRing S\ninst✝⁷ : UniformSpace S\nφ : R →+* S\na : σ → S\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : IsUniformAddGroup R\ninst✝⁴ : IsUniformAddGroup S\ninst✝³ : CompleteSpace S\ninst✝² : T2Space S\ninst... | [] | exact continuous_eval₂ hφ ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 146,
"column": 54
} | {
"line": 153,
"column": 5
} | {
"line": 155,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\nf : A⟦X⟧\nhf : HasSubst f\nn : ℕ\n⊢ ∀ᶠ (m : ℕ) in Filter.atTop, ∀ n' ≤ n, (coeff n') (f ^ m) = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"RingHom.instRingHomClass",
"le_ref... | [] | by
obtain ⟨k, hk⟩ := id hf
refine Filter.eventually_of_mem (Filter.Ici_mem_atTop (k * (n + 1))) fun m hm n' hn' ↦
coeff_of_lt_order _ ?_
obtain ⟨m, rfl⟩ := le_iff_exists_add.mp (Set.mem_Ici.mp hm)
grw [pow_add, ← order_mul_ge, pow_mul, ← le_order_pow_of_constantCoeff_eq_zero _
(by rwa [map_pow]), ← _roo... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 346,
"column": 2
} | {
"line": 355,
"column": 13
} | {
"line": 356,
"column": 2
} | [
{
"pp": "f : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\n⊢ LSeriesSummable f s",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [
"f : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\nhsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖\n⊢ LSeriesSummable f s"
] | have hsum : Summable fun n : ℕ ↦ ‖(C : ℂ) / n ^ (s + (1 - x))‖ := by
simp_rw [div_eq_mul_inv, norm_mul, ← cpow_neg]
have hsx : -s.re + x - 1 < -1 := by linarith only [hs]
refine Summable.mul_left _ <|
Summable.of_norm_bounded_eventually_nat (g := fun n ↦ (n : ℝ) ^ (-s.re + x - 1)) ?_ ?_
· simpa
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 742,
"column": 2
} | {
"line": 743,
"column": 70
} | {
"line": 744,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\na : σ → R\nf : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ ∀ b ∈ ⋯.toFinset, b ≠ n → (coeff b) f * (coeff n) (b.prod fun s e ↦ (a s • X s) ^ e) = 0",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\na : σ → R\nf : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ n ∉ ⋯.toFinset → (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e) = 0"
] | · intro b hb hbn
rw [← monomial_eq, coeff_monomial, if_neg (Ne.symm hbn), mul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.EGauge | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 48
} | {
"line": 204,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns : Set E\nx : E\na : 𝕜\ny : E\nhy : y ∈ s\nhxy : (fun x ↦ a • x) y = c • x\nhc : c ≠ 0\n⊢ (fun x ↦ (c⁻¹ * a) • x) y = x",
"ppTerm": "?inr",
"assigned": true,
"usedC... | [] | simp only [mul_smul, hxy, inv_smul_smul₀ hc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.EGauge | {
"line": 255,
"column": 2
} | {
"line": 283,
"column": 49
} | {
"line": 285,
"column": 0
} | [
{
"pp": "𝕜 : 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)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0)... | [] | refine le_antisymm ?_ (iSup₂_le fun i hi ↦ le_egauge_pi hi _ _)
refine le_of_forall_gt fun r hr ↦ ?_
have : ∀ i ∈ I, ∃ c : 𝕜, x i ∈ c • U i ∧ ‖c‖ₑ < r := fun i hi ↦
egauge_lt_iff.mp <| (le_iSup₂ i hi).trans_lt hr
choose! c hc hcr using this
obtain ⟨c₀, hc₀, hc₀I, hc₀r⟩ :
∃ c₀ : 𝕜, (c₀ ≠ 0 ∨ I = univ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.EGauge | {
"line": 255,
"column": 2
} | {
"line": 283,
"column": 49
} | {
"line": 285,
"column": 0
} | [
{
"pp": "𝕜 : 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)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0)... | [] | refine le_antisymm ?_ (iSup₂_le fun i hi ↦ le_egauge_pi hi _ _)
refine le_of_forall_gt fun r hr ↦ ?_
have : ∀ i ∈ I, ∃ c : 𝕜, x i ∈ c • U i ∧ ‖c‖ₑ < r := fun i hi ↦
egauge_lt_iff.mp <| (le_iSup₂ i hi).trans_lt hr
choose! c hc hcr using this
obtain ⟨c₀, hc₀, hc₀I, hc₀r⟩ :
∃ c₀ : 𝕜, (c₀ ≠ 0 ∨ I = univ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 143,
"column": 71
} | {
"line": 146,
"column": 77
} | {
"line": 148,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommGroup F\ninst✝¹ : TopologicalSpace F\ninst✝ : Module 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\n⊢ f =o[𝕜; l] g ↔ ∀ U ... | [] | by
simp only [isLittleOTVS_iff, ← ENNReal.coe_zero, ENNReal.nhds_coe, ← NNReal.bot_eq_zero,
(nhds_bot_basis_Iic.map _).tendsto_right_iff]
simp +contextual [ENNReal.div_le_iff_le_mul, pos_iff_ne_zero, EventuallyLE] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 62
} | {
"line": 678,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf : f =o[𝕜; l] 1\n⊢ Tendsto f l (𝓝 0)",
"ppTerm": "?mp",
"assigned": true... | [
"case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf :\n ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id i) (f x) ≤ ... | rw [(basis_sets _).isLittleOTVS_iff nhds_basis_ball] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 20
} | {
"line": 683,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf :\n ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id... | [
"case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf :\n ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id i) (f x) ≤ ... | norm_cast at hr₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 258,
"column": 4
} | {
"line": 259,
"column": 67
} | {
"line": 260,
"column": 4
} | [
{
"pp": "case mp\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\nh : p.radius = ∞\nr : ℝ≥0\n⊢ Summable fun n ↦ ‖p n‖ * ↑r ^ n",
... | [
"case mp\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\nh : p.radius = ∞\nr : ℝ≥0\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhp : ∀ (n : ℕ), ... | obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 764,
"column": 59
} | {
"line": 764,
"column": 78
} | {
"line": 765,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c... | [
"case refine_1\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c‖₊\nh :\n ∀... | norm_cast at hε hδ₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 776,
"column": 49
} | {
"line": 776,
"column": 81
} | {
"line": 777,
"column": 6
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c‖₊\nh :\n ∀ (i... | [] | simp only [div_eq_mul_inv]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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