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.CategoryTheory.Limits.Shapes.Pullback.Connected | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 16
} | {
"line": 74,
"column": 16
} | [
{
"pp": "I : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} I\ninst✝¹ : IsConnected I\ninst✝ : Category.{v_2, u_2} C\nF G : I ⥤ C\nα : F ⟶ G\ncF : Cocone F\ncG : Cocone G\nf✝ : cF ⟶ (Cocone.precompose α).obj cG\nhf : ∀ (i : I), IsPushout (cF.ι.app i) (α.app i) f✝.hom (cG.ι.app i)\nhcF : IsColimit cF\ns : ... | [
"I : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} I\ninst✝¹ : IsConnected I\ninst✝ : Category.{v_2, u_2} C\nF G : I ⥤ C\nα : F ⟶ G\ncF : Cocone F\ncG : Cocone G\nf✝ : cF ⟶ (Cocone.precompose α).obj cG\nhf : ∀ (i : I), IsPushout (cF.ι.app i) (α.app i) f✝.hom (cG.ι.app i)\nhcF : IsColimit cF\ns : Cocone G\nj ... | this _ j | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 295,
"column": 2
} | {
"line": 299,
"column": 60
} | {
"line": 300,
"column": 2
} | [
{
"pp": "case intro\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = U... | [
"case intro\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | have hφ : Function.Injective φ := by
dsimp [φ]
refine .comp ?_ (Algebra.TensorProduct.piRight _ Γ(S, US) _ _).injective
exact .piMap fun i ↦ (hV _).comp <| CommRingCat.isPushout_tensorProduct _ _ _
|>.flip.isoPushout.commRingCatIsoToRingEquiv.injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 278,
"column": 42
} | {
"line": 278,
"column": 59
} | {
"line": 278,
"column": 60
} | [
{
"pp": "n : Type u\nS T : Scheme\nf : S ⟶ T\ni : n\n⊢ (ConcreteCategory.hom (Scheme.Hom.appTop (map n f))) (coord T i) = (toSpecMvPolyIntEquiv n) (toSpecMvPoly n S) i",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Nat.instMulZeroClas... | [
"n : Type u\nS T : Scheme\nf : S ⟶ T\ni : n\n⊢ coord S i = (toSpecMvPolyIntEquiv n) (toSpecMvPoly n S) i"
] | map_appTop_coord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 69
} | {
"line": 337,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | [
"R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal... | rw [coeffSubmodule_mapRingHom_comp, ← Submodule.map_pow] at hT₂span | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 49
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case refine_1\nR : CommRingCat\nH✝ : ∀ (I : Ideal ↑R), I.IsPrime → ∃ f ∉ I, ∀ (x y : ↑R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0\np : ↥(Spec R)\nf : ↑R\nhf : f ∉ p.asIdeal\nH : ∀ (x y : ↑R), y * x = 0 → y ∉ p.asIdeal → ∃ n, f ^ n * x = 0\n⊢ IsAffineOpen (PrimeSpectrum.basicOpen f)",
"ppTerm": "?ref... | [
"case refine_2\nR : CommRingCat\nH✝ : ∀ (I : Ideal ↑R), I.IsPrime → ∃ f ∉ I, ∀ (x y : ↑R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0\np : ↥(Spec R)\nf : ↑R\nhf : f ∉ p.asIdeal\nH : ∀ (x y : ↑R), y * x = 0 → y ∉ p.asIdeal → ∃ n, f ^ n * x = 0\n⊢ Function.Injective ⇑(ConcreteCategory.hom ((Spec R).presheaf.germ (PrimeSp... | · rw [← basicOpen_eq_of_affine]
exact (isAffineOpen_top (Spec R)).basicOpen _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 45
} | {
"line": 222,
"column": 45
} | [
{
"pp": "case e_a\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : f x = g x\nH : Scheme.Hom.stalkMap f x = Y.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap g x\nV : Y.Opens\nhV : V ∈ Y.affineOpens\nhxV : f x ∈ ↑V\nhxV' : g x ∈ V\nU : X.Opens\nhxU : x ∈ U\nleft✝ : IsAffineOpen U\nhUV : ... | [
"case e_a\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : f x = g x\nH : Scheme.Hom.stalkMap f x = Y.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap g x\nV : Y.Opens\nhV : V ∈ Y.affineOpens\nhxV : f x ∈ ↑V\nhxV' : g x ∈ V\nU : X.Opens\nhxU : x ∈ U\nleft✝ : IsAffineOpen U\nhUV : U ≤ f ⁻¹ᵁ V ... | ← cancel_mono (X.presheaf.germ U x hxU) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 361,
"column": 8
} | {
"line": 366,
"column": 79
} | {
"line": 367,
"column": 6
} | [
{
"pp": "case e'_2.e'_4\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀]... | [] | dsimp only [e₁]
rw [Set.preimage_sdiff, preimage_comap_zeroLocus, preimage_comap_zeroLocus,
Set.image_singleton, Pi.smul_def, ← Set.smul_set_range, Set.range_comp]
congr 1
refine (PrimeSpectrum.zeroLocus_smul_of_isUnit (.map _ ?_) _).symm
exact isUnit_iff_exists_inv'.mpr ⟨_, Is... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 361,
"column": 8
} | {
"line": 366,
"column": 79
} | {
"line": 367,
"column": 6
} | [
{
"pp": "case e'_2.e'_4\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀]... | [] | dsimp only [e₁]
rw [Set.preimage_sdiff, preimage_comap_zeroLocus, preimage_comap_zeroLocus,
Set.image_singleton, Pi.smul_def, ← Set.smul_set_range, Set.range_comp]
congr 1
refine (PrimeSpectrum.zeroLocus_smul_of_isUnit (.map _ ?_) _).symm
exact isUnit_iff_exists_inv'.mpr ⟨_, Is... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 63
} | {
"line": 176,
"column": 0
} | [
{
"pp": "case e_a.e_a\nX Y : Scheme\nf : X.PartialMap Y\nU : X.Opens\nhU : Dense ↑U\nhU' : U ≤ f.domain\nx : ↥X\nhx : x ∈ U\ne : ⟨x, ⋯⟩ = (X.homOfLE hU') ⟨x, hx⟩\n⊢ Hom.stalkMap f.domain.ι ⟨x, ⋯⟩ ≫ (↑f.domain).presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap (X.homOfLE hU') ⟨x, hx⟩ =\n X.presheaf.stalkSpecializes ... | [] | rw [← Hom.stalkSpecializes_stalkMap_assoc, Hom.stalkMap_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AlgebraicIndependent.Adjoin | {
"line": 47,
"column": 20
} | {
"line": 47,
"column": 28
} | {
"line": 47,
"column": 28
} | [
{
"pp": "case hC\nι : Type u_1\nF : Type u_2\nE : Type u_3\nx : ι → E\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhx : AlgebraicIndependent F x\ni : FractionRing (MvPolynomial ι F) →ₐ[F] E := IsFractionRing.liftAlgHom ⋯\nx✝ : F\n⊢ (((↑i).comp (algebraMap (MvPolynomial ι F) (FractionRing (MvPolynom... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AlgebraicIndependent.Adjoin | {
"line": 47,
"column": 20
} | {
"line": 47,
"column": 28
} | {
"line": 47,
"column": 28
} | [
{
"pp": "case hX\nι : Type u_1\nF : Type u_2\nE : Type u_3\nx : ι → E\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhx : AlgebraicIndependent F x\ni : FractionRing (MvPolynomial ι F) →ₐ[F] E := IsFractionRing.liftAlgHom ⋯\ni✝ : ι\n⊢ ((↑i).comp (algebraMap (MvPolynomial ι F) (FractionRing (MvPolynomi... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Normal.Closure | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 37
} | {
"line": 171,
"column": 37
} | [
{
"pp": "case inl\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\nh : Normal F L\nh✝ : IsEmpty (K →ₐ[F] L)\n⊢ Normal F ↥(normalClosure F K L)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq... | [
"case inl\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\nh : Normal F L\nh✝ : IsEmpty (K →ₐ[F] L)\n⊢ Normal F ↥⊥"
] | rw [normalClosure, iSup_of_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 871,
"column": 84
} | {
"line": 872,
"column": 36
} | {
"line": 874,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ M.IsBasis I X ↔ M.Indep I ∧ I ⊆ X ∧ ∀ (J : Set α), M.Indep J → I ⊆ J → J ⊆ X → I = J",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left"... | [] | by
rw [isBasis_iff', and_iff_left hX] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 59
} | {
"line": 233,
"column": 8
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [
"α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Ind... | refine hIe <| indep_compact _ fun J hJss hJfin ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 234,
"column": 19
} | {
"line": 234,
"column": 24
} | {
"line": 234,
"column": 25
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [
"α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Ind... | hImax | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 28
} | {
"line": 68,
"column": 28
} | [
{
"pp": "α : Type u_1\n⊢ (emptyOn α)✶ = emptyOn α",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"EmptyCollection.emptyCol... | [
"α : Type u_1\n⊢ (emptyOn α)✶.E = ∅"
] | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 28
} | {
"line": 75,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ M = emptyOn α ∨ M.Nonempty",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"Or",
"EmptyCollection.e... | [
"α : Type u_1\nM : Matroid α\n⊢ M.E = ∅ ∨ M.Nonempty"
] | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 28
} | {
"line": 79,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : IsEmpty α\nM : Matroid α\n⊢ M = emptyOn α",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"EmptyCollection.emptyCo... | [
"α : Type u_1\ninst✝ : IsEmpty α\nM : Matroid α\n⊢ M.E = ∅"
] | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 251,
"column": 15
} | {
"line": 251,
"column": 35
} | {
"line": 251,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nN : Matroid β\nf : α → β\n⊢ N.comap f ↾ f ⁻¹' N.E = N.comap f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.restrict_eq_self_iff",
"congrArg",
"Matroid.E",
"id",
"Set.preimage",
"propext",
... | [
"α : Type u_1\nβ : Type u_2\nN : Matroid β\nf : α → β\n⊢ f ⁻¹' N.E = (N.comap f).E"
] | restrict_eq_self_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 53
} | {
"line": 404,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nB : Set α\nhB : B ⊆ M.E\n⊢ (∃ B₀, M.IsBase B₀ ∧ f '' B = f '' B₀) ↔ M.IsBase B",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Exists",
"Matroid.IsBase",
"And",
"And.intro",
"Iff... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nB : Set α\nhB : B ⊆ M.E\nx✝ : ∃ B₀, M.IsBase B₀ ∧ f '' B = f '' B₀\nJ : Set α\nhJ : M.IsBase J\nhIJ : f '' B = f '' J\n⊢ M.IsBase B"
] | refine ⟨fun ⟨J, hJ, hIJ⟩ ↦ ?_, fun h ↦ ⟨B, h, rfl⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Map | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 74
} | {
"line": 412,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nI : Set α\nM : Matroid α\nX : Set α\nhIX : M.IsBasis I X\nf : α → β\nhf : InjOn f M.E\ne : α\nhe : e ∈ X\nhe' : f e ∉ f '' I\nhss : insert e I ⊆ M.E\n⊢ (M.map f hf).Dep (insert (f e) (f '' I))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"α : Type u_1\nβ : Type u_2\nI : Set α\nM : Matroid α\nX : Set α\nhIX : M.IsBasis I X\nf : α → β\nhf : InjOn f M.E\ne : α\nhe : e ∈ X\nhe' : f e ∉ f '' I\nhss : insert e I ⊆ M.E\n⊢ ¬(M.map f hf).Indep (insert (f e) (f '' I))"
] | rw [← not_indep_iff (by simpa [← image_insert_eq] using image_mono hss)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 536,
"column": 42
} | {
"line": 538,
"column": 20
} | {
"line": 540,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nB : Set α\nhB : M.IsBase B\nf : α ↪ β\n⊢ (M.mapEmbedding f).IsBase (⇑f '' B)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.mapEmbedding",
"Exists",
"Matroid.IsBase",
"id"... | [] | by
rw [Matroid.mapEmbedding, map_isBase_iff]
exact ⟨B, hB, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Map | {
"line": 676,
"column": 52
} | {
"line": 676,
"column": 87
} | {
"line": 676,
"column": 87
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nhN : N.E = M.E\nh : M.restrictSubtype M.E = N.restrictSubtype M.E\nI : Set α\nhI : I ⊆ M.E\n⊢ (N.restrictSubtype M.E).Indep (M.E ↓∩ I) ↔ N.Indep I",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
... | [
"α : Type u_1\nM N : Matroid α\nhN : N.E = M.E\nh : M.restrictSubtype M.E = N.restrictSubtype M.E\nI : Set α\nhI : I ⊆ M.E\n⊢ N.Indep I ↔ N.Indep I",
"α : Type u_1\nM N : Matroid α\nhN : N.E = M.E\nh : M.restrictSubtype M.E = N.restrictSubtype M.E\nI : Set α\nhI : I ⊆ M.E\n⊢ I ⊆ M.E"
] | restrictSubtype_indep_iff_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 922,
"column": 2
} | {
"line": 922,
"column": 95
} | {
"line": 923,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS B : Set α\nhS : M.Spanning S\n⊢ M.IsBasis B S ↔ M.IsBase B ∧ B ⊆ S",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Matroid.IsBase",
"HasSubset.Subset",
"Matroid.IsBasis.subset",
"And",
"And.right",
"And.left",
... | [
"case refine_1\nα : Type u_2\nM : Matroid α\nS B : Set α\nhS : M.Spanning S\nh : M.IsBasis B S\n⊢ M.IsBase B",
"case refine_2\nα : Type u_2\nM : Matroid α\nS B : Set α\nhS : M.Spanning S\nh : M.IsBase B ∧ B ⊆ S\n⊢ S ⊆ M.closure B"
] | refine ⟨fun h ↦ ⟨?_, h.subset⟩, fun h ↦ h.1.indep.isBasis_of_subset_of_subset_closure h.2 ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 956,
"column": 19
} | {
"line": 956,
"column": 73
} | {
"line": 956,
"column": 73
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX B : Set α\nhB : M.IsBase B\nhX : M.Spanning X\nhXB : X ⊆ B\nB' : Set α\nhB' : M.IsBase B'\nhB'X : B' ⊆ X\n⊢ B ⊆ X",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HasSubset.Subset.trans",
"id",
"Set... | [] | by rwa [← hB'.eq_of_subset_isBase hB (hB'X.trans hXB)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 980,
"column": 19
} | {
"line": 980,
"column": 46
} | {
"line": 980,
"column": 47
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX R I : Set α\nhI : (M ↾ R).IsBasis' I X\nhI' : M.IsBasis' I (X ∩ R)\nhIR : I ⊆ R\ne : α\n⊢ e ∈ (M ↾ R).E ∧ ((M ↾ R).Indep (insert e I) → e ∈ I) ↔ e ∈ M.closure I ∧ e ∈ R ∨ e ∈ R \\ M.E",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_2\nM : Matroid α\nX R I : Set α\nhI : (M ↾ R).IsBasis' I X\nhI' : M.IsBasis' I (X ∩ R)\nhIR : I ⊆ R\ne : α\n⊢ e ∈ (M ↾ R).E ∧ ((M ↾ R).Indep (insert e I) → e ∈ I) ↔\n (e ∈ M.E ∧ (M.Indep (insert e I) → e ∈ I)) ∧ e ∈ R ∨ e ∈ R \\ M.E"
] | hI'.indep.mem_closure_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 42
} | {
"line": 528,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC : Set α\ninst✝ : M.Finitary\nhC : M.IsCircuit C\nJ : Set α\nhJC : J ⊆ C\nhJfin : J.Finite\nhJ : ¬M.Indep J\n⊢ C.Finite",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.Finite",
"id",
"Matroid.Is... | [] | rwa [← hC.eq_of_not_indep_subset hJ hJC] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1020,
"column": 62
} | {
"line": 1020,
"column": 89
} | {
"line": 1021,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\n⊢ ∀ (x : α), f x ∈ M.E ∧ ((M.comap f).Indep (insert x I) → x ∈ I) ↔ f x ∈ M.closure (f '' I)",
"ppTerm": "?m.55",
"assigned": true,
"usedC... | [
"α : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\n⊢ ∀ (x : α),\n f x ∈ M.E ∧ ((M.comap f).Indep (insert x I) → x ∈ I) ↔ f x ∈ M.E ∧ (M.Indep (insert (f x) (f '' I)) → f x ∈ f '' I)"
] | hI'.indep.mem_closure_iff', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 592,
"column": 51
} | {
"line": 592,
"column": 96
} | {
"line": 592,
"column": 96
} | [
{
"pp": "α : Type u_1\nX : Set α\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nhX : X ⊆ M.E\nI : Set α\nhI : M.IsBasis I X\n⊢ (f '' I).encard = I.encard",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"congrArg",
"Matroid.E",
... | [
"α : Type u_1\nX : Set α\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nhX : X ⊆ M.E\nI : Set α\nhI : M.IsBasis I X\n⊢ I.encard = I.encard"
] | (hf.mono hI.indep.subset_ground).encard_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 26
} | {
"line": 648,
"column": 35
} | {
"line": 648,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 26
} | {
"line": 648,
"column": 35
} | {
"line": 648,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 26
} | {
"line": 648,
"column": 35
} | {
"line": 648,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 60
} | {
"line": 648,
"column": 69
} | {
"line": 648,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Eq.... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 60
} | {
"line": 648,
"column": 69
} | {
"line": 648,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Eq.... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 60
} | {
"line": 648,
"column": 69
} | {
"line": 648,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Eq.... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 649,
"column": 24
} | {
"line": 649,
"column": 33
} | {
"line": 649,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"ppTerm": "?m.336",
"assigned": true,
"usedConstants": [
... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 649,
"column": 24
} | {
"line": 649,
"column": 33
} | {
"line": 649,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"ppTerm": "?m.336",
"assigned": true,
"usedConstants": [
... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 649,
"column": 24
} | {
"line": 649,
"column": 33
} | {
"line": 649,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"ppTerm": "?m.336",
"assigned": true,
"usedConstants": [
... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 759,
"column": 4
} | {
"line": 759,
"column": 73
} | {
"line": 760,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nM₁ M₂ : Matroid α\nhE : M₁.E = M₂.E\nhl : M₁.loops = M₂.loops\nhc : M₁.coloops = M₂.coloops\nh : ∀ I ⊆ M₁.E, Disjoint I (M₁.loops ∪ M₁.coloops) → (M₁.Indep I ↔ M₂.Indep I)\nI : Set α\nhI : I ⊆ M₁.E\nhdj : Disjoint I M₁.loops\n⊢ Disjoint (I \\ M₁.coloops) M₁.loops ∧ Disjoint (I \... | [] | exact ⟨disjoint_of_subset_left sdiff_subset hdj, disjoint_sdiff_left⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 70,
"column": 23
} | {
"line": 71,
"column": 41
} | {
"line": 71,
"column": 42
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\nhf : Transcendental R f\nthis : Transcendental (↥(supported R ∅)) ((Polynomial.aeval (X i)) f)\ng : R ≃ₐ[R] ↥(supported R ∅) := (Algebra.botEquivOfInjective ⋯).symm.trans ((supported R ∅).equivOfEq ⊥ ⋯).symm\n⊢ ¬IsAlgebraic R ((Polynomial... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\nhf : Transcendental R f\nthis : Transcendental (↥(supported R ∅)) ((Polynomial.aeval (X i)) f)\ng : R ≃ₐ[R] ↥(supported R ∅) := (Algebra.botEquivOfInjective ⋯).symm.trans ((supported R ∅).equivOfEq ⊥ ⋯).symm\n⊢ ¬IsAlgebraic (↥(supported R ∅)) ((RingH... | ← isAlgebraic_ringHom_iff_of_comp_eq g (RingHom.id (MvPolynomial σ R))
Function.injective_id (by ext1; rfl), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 796,
"column": 2
} | {
"line": 796,
"column": 53
} | {
"line": 797,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ni : I\nU : (D.obj i).Opens\nhU : IsCompact ↑U\ns : ↑Γ(D.obj i, U)\nhs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0\nthis ... | [
"I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ni : I\nU : (D.obj i).Opens\nhU : IsCompact ↑U\ns : ↑Γ(D.obj i, U)\nhs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0\nthis : CompactSpa... | have H : (D.map (𝟙 _) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 92
} | {
"line": 355,
"column": 0
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nA' : Type v\ninst✝¹ : CommRing A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nhf : Surjective ⇑f\n⊢ trdeg R A' ≤ trdeg R A",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ca... | [] | rw [← (trdeg R A).lift_id, ← (trdeg R A').lift_id]; exact lift_trdeg_le_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 92
} | {
"line": 355,
"column": 0
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nA' : Type v\ninst✝¹ : CommRing A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nhf : Surjective ⇑f\n⊢ trdeg R A' ≤ trdeg R A",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ca... | [] | rw [← (trdeg R A).lift_id, ← (trdeg R A').lift_id]; exact lift_trdeg_le_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 328,
"column": 27
} | {
"line": 328,
"column": 66
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case inl.ha\nR : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : NoZeroDivisors A\ns t : Set A\na : A\nh : (matroid R A).IsBasis s t\nh✝ : Subsingleton A\n⊢ ¬IsAlgebraic (↥(adjoin R s)) a",
"ppTerm": "?inl.ha",
"assigned":... | [] | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 328,
"column": 27
} | {
"line": 328,
"column": 66
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case inl.hb\nR : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : NoZeroDivisors A\ns t : Set A\na : A\nh : (matroid R A).IsBasis s t\nh✝ : Subsingleton A\n⊢ ¬IsAlgebraic (↥(adjoin R t)) a",
"ppTerm": "?inl.hb",
"assigned":... | [] | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 899,
"column": 2
} | {
"line": 901,
"column": 97
} | {
"line": 902,
"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.pt... | obtain ⟨k, fk, hk⟩ := IsCofiltered.inf_exists S
(σ.attach.image₂ (fun (x y : σ) ↦ ⟨j x.1 y.1, i x.1, hjS x.2 y.2, hiS x.2, fjx x y⟩) σ.attach ∪
σ.attach.image₂ (fun (x y : σ) ↦ ⟨j x.1 y.1, i y.1, hjS x.2 y.2, hiS y.2, fjy x y⟩) σ.attach) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | {
"line": 419,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisi... | [] | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | {
"line": 419,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisi... | [] | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | {
"line": 419,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisi... | [] | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAc... | [] | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAc... | [] | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAc... | [] | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 478,
"column": 58
} | {
"line": 485,
"column": 80
} | {
"line": 487,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf : F[X]\nq : ℕ\nhF : ExpChar F q\nn : ℕ\n⊢ ((expand F (q ^ n)) f).natSepDegree = f.natSepDegree",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Iff.mpr",
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"P... | [] | by
obtain - | hprime := hF
· simp only [one_pow, expand_one]
haveI := Fact.mk hprime
classical
simpa only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_def, map_expand,
Fintype.card_coe] using Fintype.card_eq.2
⟨(f.map (algebraMap F (AlgebraicClosure F))).rootsExpandPowEquivRoots q n⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Galois.Basic | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 27
} | {
"line": 169,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois (↥⊥) E ↔ IsGalois F E",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
... | [] | constructor
· intro h
exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
· intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 27
} | {
"line": 169,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois (↥⊥) E ↔ IsGalois F E",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
... | [] | constructor
· intro h
exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
· intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 997,
"column": 4
} | {
"line": 997,
"column": 64
} | {
"line": 998,
"column": 4
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ninst✝ : c.pt.IsQuasiAffine\nx : ↥c.pt\ni... | [
"I : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ninst✝ : c.pt.IsQuasiAffine\nx : ↥c.pt\ni : I\nU : To... | obtain ⟨k, fki, fkj, -⟩ := IsCofilteredOrEmpty.cone_objs i j | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 119,
"column": 41
} | {
"line": 120,
"column": 90
} | {
"line": 122,
"column": 0
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\ni : ι\nV : Opens (X i)\nhV : IsCompact ↑V\n⊢ IsCompactOpenCovered f (f i '' ↑V)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Iff.of_eq",
... | [] | by
refine ⟨{i}, Set.finite_singleton i, fun j hj ↦ hj ▸ V, by rintro i rfl; simpa, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 308,
"column": 78
} | {
"line": 310,
"column": 7
} | {
"line": 312,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.twoTorsionPolynomial.discr = 16 * W.Δ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"WeierstrassCurve.b₄._proof_1",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | by
simp only [b₂, b₄, b₆, b₈, Δ, twoTorsionPolynomial, Cubic.discr]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 392,
"column": 44
} | {
"line": 393,
"column": 48
} | {
"line": 395,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsElliptic\nh : W.c₄ = 0\n⊢ W.j = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.j",
... | [] | by
rw [j_eq_zero_iff', h, zero_pow three_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 20
} | {
"line": 158,
"column": 21
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg... | [
"R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 ↔\n y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆"
] | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 20
} | {
"line": 166,
"column": 21
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ↔ (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).Equation 0 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 ↔\n (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).Equation 0 0"
] | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 216,
"column": 19
} | {
"line": 216,
"column": 33
} | {
"line": 216,
"column": 34
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ∧ (evalEval x y W.polynomialX ≠ 0 ∨ evalEval x y W.polynomialY ≠ 0) ↔\n W.Equation x y ∧ (W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) ≠ 0 ∨ 2 * y + W.a₁ * x + W.a₃ ≠ 0)",
"ppTerm": "?m.83",
"assigned": true,
"usedCo... | [
"R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 ∧\n (evalEval x y W.polynomialX ≠ 0 ∨ evalEval x y W.polynomialY ≠ 0) ↔\n y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 ∧\n (W.a₁ * y... | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Norm | {
"line": 64,
"column": 83
} | {
"line": 71,
"column": 70
} | {
"line": 73,
"column": 0
} | [
{
"pp": "S : Type u_2\nι : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\nF : Type u_4\ninst✝⁴ : Field F\ninst✝³ : Algebra F[X] S\ninst✝² : Finite ι\ninst✝¹ : Algebra F S\ninst✝ : IsScalarTower F F[X] S\nb : Basis ι F[X] S\nf : S\nhf : f ≠ 0\n⊢ finrank F (S ⧸ span {f}) = ((Algebra.norm F[X]) f).natDegree",... | [] | by
haveI := Fintype.ofFinite ι
have h := span_singleton_eq_bot.not.2 hf
rw [natDegree_eq_of_degree_eq
(degree_eq_degree_of_associated <| associated_norm_prod_smith b hf)]
rw [natDegree_prod _ _ fun i _ => smithCoeffs_ne_zero b _ h i, finrank_quotient_eq_sum F h b]
congr with i
exact (AdjoinRoot.powerB... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 216,
"column": 13
} | {
"line": 216,
"column": 27
} | {
"line": 216,
"column": 28
} | [
{
"pp": "case nat.succ.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℕ\n⊢ preNormEDS b c d (2 * ↑(m + 1 + 1 + 1)) =\n preNormEDS b c d (↑(m + 1 + 1 + 1) - 1) ^ 2 * preNormEDS b c d ↑(m + 1 + 1 + 1) *\n preNormEDS b c d (↑(m + 1 + 1 + 1) + 2) -\n preNormEDS b c d (↑(m + 1 + 1 + 1) - 2)... | [
"case nat.succ.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℕ\n⊢ preNormEDS b c d (2 * (↑m + 1 + 1 + 1)) =\n preNormEDS b c d (↑m + 1 + 1 + 1 - 1) ^ 2 * preNormEDS b c d (↑m + 1 + 1 + 1) *\n preNormEDS b c d (↑m + 1 + 1 + 1 + 2) -\n preNormEDS b c d (↑m + 1 + 1 + 1 - 2) * preNormEDS b c ... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 9
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n) =\n preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -\n preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2\nm : ℕ\n⊢ preNormEDS... | [] | simp_rw [mul_neg, ← sub_neg_eq_add, ← neg_sub', ← neg_add', preNormEDS_neg, ih]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 9
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n) =\n preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -\n preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2\nm : ℕ\n⊢ preNormEDS... | [] | simp_rw [mul_neg, ← sub_neg_eq_add, ← neg_sub', ← neg_add', preNormEDS_neg, ih]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 231,
"column": 13
} | {
"line": 231,
"column": 27
} | {
"line": 231,
"column": 28
} | [
{
"pp": "case nat.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nn✝ : ℕ\n⊢ preNormEDS b c d (2 * ↑(n✝ + 1 + 1) + 1) =\n (preNormEDS b c d (↑(n✝ + 1 + 1) + 2) * preNormEDS b c d ↑(n✝ + 1 + 1) ^ 3 * if Even ↑(n✝ + 1 + 1) then b else 1) -\n preNormEDS b c d (↑(n✝ + 1 + 1) - 1) * preNormEDS b c d (↑(... | [
"case nat.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nn✝ : ℕ\n⊢ preNormEDS b c d (2 * (↑n✝ + 1 + 1) + 1) =\n (preNormEDS b c d (↑n✝ + 1 + 1 + 2) * preNormEDS b c d (↑n✝ + 1 + 1) ^ 3 * if Even (↑n✝ + 1 + 1) then b else 1) -\n preNormEDS b c d (↑n✝ + 1 + 1 - 1) * preNormEDS b c d (↑n✝ + 1 + 1 + 1) ^... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 237,
"column": 13
} | {
"line": 237,
"column": 27
} | {
"line": 237,
"column": 28
} | [
{
"pp": "case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n + 1) =\n (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -\n preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b\nm : ... | [
"case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n + 1) =\n (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -\n preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b\nm : ℕ\n⊢ preNorm... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 470,
"column": 13
} | {
"line": 470,
"column": 27
} | {
"line": 470,
"column": 28
} | [
{
"pp": "case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nk : ℤ\nih :\n ∀ (n : ℕ),\n complEDS b c d k (2 * ↑n + 1) =\n complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -\n complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b... | [
"case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nk : ℤ\nih :\n ∀ (n : ℕ),\n complEDS b c d k (2 * ↑n + 1) =\n complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -\n complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 337,
"column": 33
} | {
"line": 337,
"column": 42
} | {
"line": 337,
"column": 43
} | [
{
"pp": "case pos\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Nonsingular x₁ y₁\nh₂ : W.Nonsingular x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nhx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁\n⊢ W.Nonsingular x₁ (W.slope x₁ x₂ y₁ y₂ * 0 + y₁)",
"ppTerm": "?pos✝... | [
"case pos\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Nonsingular x₁ y₁\nh₂ : W.Nonsingular x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nhx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁\n⊢ W.Nonsingular x₁ (0 + y₁)"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 463,
"column": 25
} | {
"line": 463,
"column": 37
} | {
"line": 463,
"column": 38
} | [
{
"pp": "R : Type r\nS : Type s\nF : Type u\nK : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Field F\ninst✝⁹ : Field K\nW' : Affine R\ninst✝⁸ : Algebra R S\ninst✝⁷ : DecidableEq F\ninst✝⁶ : DecidableEq K\ninst✝⁵ : Algebra R F\ninst✝⁴ : Algebra S F\ninst✝³ : IsScalarTower R S F\ninst✝² : Algebr... | [
"R : Type r\nS : Type s\nF : Type u\nK : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Field F\ninst✝⁹ : Field K\nW' : Affine R\ninst✝⁸ : Algebra R S\ninst✝⁷ : DecidableEq F\ninst✝⁶ : DecidableEq K\ninst✝⁵ : Algebra R F\ninst✝⁴ : Algebra S F\ninst✝³ : IsScalarTower R S F\ninst✝² : Algebra R K\ninst✝... | ← map_slope, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | {
"line": 469,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRi... | [] | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | {
"line": 469,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRi... | [] | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | {
"line": 469,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRi... | [] | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 356,
"column": 65
} | {
"line": 356,
"column": 74
} | {
"line": 356,
"column": 75
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ y₁ : F\nh₁ : W.Equation x₁ y₁\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\nh₂ : W.Equation x₁ y₁\nhy : y₁ ≠ W.negY x₁ y₁\ny : F := (y₁ - W.negY x₁ y₁) ^ 2\nhxy : (y₁ - W.negY x₁ y₁) ^ 2 ≠ 0\n⊢ 1 ... | [
"F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ y₁ : F\nh₁ : W.Equation x₁ y₁\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\nh₂ : W.Equation x₁ y₁\nhy : y₁ ≠ W.negY x₁ y₁\ny : F := (y₁ - W.negY x₁ y₁) ^ 2\nhxy : (y₁ - W.negY x₁ y₁) ^ 2 ≠ 0\n⊢ 1 + 0 = 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 575,
"column": 55
} | {
"line": 577,
"column": 7
} | {
"line": 579,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharTwoJEqZeroNF\n⊢ W.b₈ = -W.a₄ ^ 2",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroCl... | [] | by
rw [b₈, a₁_of_isCharTwoJEqZeroNF, a₂_of_isCharTwoJEqZeroNF]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | {
"line": 562,
"column": 0
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | [] | rw [← map_Ψ₃, map_baseChange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | {
"line": 562,
"column": 0
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | [] | rw [← map_Ψ₃, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | {
"line": 562,
"column": 0
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | [] | rw [← map_Ψ₃, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 218,
"column": 51
} | {
"line": 218,
"column": 56
} | {
"line": 218,
"column": 57
} | [
{
"pp": "case even.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p}... | [
"case even.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree... | h₁.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 22
} | {
"line": 403,
"column": 23
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Affine R\np q : R[X]\n⊢ ((CoordinateRing.basis W').repr (p • 1) + (CoordinateRing.basis W').repr (q • (mk W') Y)) 0 *\n ((CoordinateRing.basis W').repr ((q * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)) • 1) +\n (CoordinateRing.basis W').repr... | [
"R : Type r\ninst✝ : CommRing R\nW' : Affine R\np q : R[X]\n⊢ (((CoordinateRing.basis W').repr (p • 1)) 0 + ((CoordinateRing.basis W').repr (q • (mk W') Y)) 0) *\n (((CoordinateRing.basis W').repr ((q * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)) • 1)) 1 +\n ((CoordinateRing.basis W').repr ((... | Finsupp.add_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 228,
"column": 74
} | {
"line": 228,
"column": 79
} | {
"line": 228,
"column": 80
} | [
{
"pp": "case odd.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ... | [
"case odd.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_... | h₁.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 248,
"column": 61
} | {
"line": 248,
"column": 70
} | {
"line": 248,
"column": 71
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | [
"case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 255,
"column": 62
} | {
"line": 255,
"column": 71
} | {
"line": 255,
"column": 72
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | [
"case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 340,
"column": 4
} | {
"line": 341,
"column": 55
} | {
"line": 342,
"column": 4
} | [
{
"pp": "case succ.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := ⋯\nh : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) := ⋯\nn : ℕ\nhd : (n + 1) ^ 2 - 1 = 2 * expDegree ... | [
"case succ.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n\nh : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=\n fun {n} ↦ natDegr... | rw [coeff_mul_add_eq_of_natDegree_le (dp h.1), coeff_pow_of_natDegree_le h.1, h.2,
apply_ite₂ coeff, coeff_Ψ₂Sq, coeff_one_zero, hc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 273,
"column": 59
} | {
"line": 273,
"column": 68
} | {
"line": 273,
"column": 69
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | [
"case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 277,
"column": 37
} | {
"line": 277,
"column": 46
} | {
"line": 277,
"column": 47
} | [
{
"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\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nt... | [
"F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthis : Invert... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 280,
"column": 60
} | {
"line": 280,
"column": 69
} | {
"line": 280,
"column": 70
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | [
"case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 301,
"column": 33
} | {
"line": 301,
"column": 42
} | {
"line": 301,
"column": 43
} | [
{
"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\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nt... | [
"F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthis : Invert... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 91
} | {
"line": 411,
"column": 92
} | [
{
"pp": "case succ.succ.left\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n}... | [
"case succ.succ.left.refine_1\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p... | refine natDegree_sub_le_of_le (dm (dm natDegree_X_le (dp h.1)) ?_) (dm (dm h.1 h.1) ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 727,
"column": 6
} | {
"line": 727,
"column": 71
} | {
"line": 729,
"column": 0
} | [
{
"pp": "case neg\nR : Type r\nS : Type s\nA F : Type u\nB K : Type v\nL : Type w\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\nW' : Affine R\nW : Affine F\ninst✝² : DecidableEq F\ninst✝¹ : DecidableEq K\ninst✝ : Decida... | [] | exact (CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal' h₁ h₂ hxy).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 751,
"column": 4
} | {
"line": 751,
"column": 36
} | {
"line": 752,
"column": 4
} | [
{
"pp": "case mp\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nP : W.Point\nhP : toClass P = 0\n⊢ P = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"WeierstrassCurve.Affine.Point.instAddZe... | [
"case mp.zero\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nhP : toClass zero = 0\n⊢ zero = 0",
"case mp.some\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx✝ y✝ : F\nh : W.Equation x✝ y✝\nright✝ : evalEval x✝ y✝ W.polynomialX ≠ 0 ∨ evalEval x✝ y✝ W.polynomialY ≠ 0\nhP ... | rcases P with (_ | ⟨_, _, h, _⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 100,
"column": 17
} | {
"line": 100,
"column": 26
} | {
"line": 100,
"column": 27
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - W'.a₁ * P x * 0 - W'.a₃ * 0 ^ 3 = -P y",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroup... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - 0 - W'.a₃ * 0 ^ 3 = -P y"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 100,
"column": 61
} | {
"line": 100,
"column": 70
} | {
"line": 100,
"column": 71
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - W'.a₃ * 0 = -P y",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - 0 = -P y"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 436,
"column": 64
} | {
"line": 436,
"column": 73
} | {
"line": 436,
"column": 74
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhPz : P z = 0\nhPx : P x = 0\nhP : 2 * 0 ≠ 0\n⊢ False",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toS... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhPz : P z = 0\nhPx : P x = 0\nhP : 0 ≠ 0\n⊢ False"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 818,
"column": 2
} | {
"line": 818,
"column": 97
} | {
"line": 820,
"column": 0
} | [
{
"pp": "case some.some\nR : Type r\nS : Type s\nF : Type u\nK : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Field F\ninst✝⁹ : Field K\nW' : Affine R\ninst✝⁸ : DecidableEq F\ninst✝⁷ : DecidableEq K\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R F\ninst✝⁴ : Algebra S F\ninst✝³ : IsScalarTower R S F\... | [] | · simpa only [some.injEq] using ⟨f.injective (some.inj h).left, f.injective (some.inj h).right⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 114,
"column": 39
} | {
"line": 114,
"column": 48
} | {
"line": 114,
"column": 49
} | [
{
"pp": "case a\nR : Type u_4\nS : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Field S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\nB : BilinForm S M\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : DecidableEq ι\nhB : B.Nondegenerate... | [
"case a\nR : Type u_4\nS : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Field S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\nB : BilinForm S M\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : DecidableEq ι\nhB : B.Nondegenerate\nb : Basis ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 60,
"column": 88
} | {
"line": 62,
"column": 99
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : IsDomain R\ninst✝⁷ : Algebra R S\nK : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : IsIntegrallyClosed R\ninst✝² : IsDomain S\ninst✝¹ : Algebra K S\ninst✝ : IsScalarTower R K S\ns :... | [] | by
let L := FractionRing S
rw [← isIntegrallyClosed_eq_field_fractions K L hs, algebraMap_eq (IsFractionRing.injective S L)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 222,
"column": 2
} | {
"line": 234,
"column": 29
} | {
"line": 236,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : K[X]\nh0 : p ≠ 0\nh : IsUnit (integerNormalization R⁰ p).primPart\n⊢ IsUnit p",
"ppTerm": "?m.26",
"assigned": true,
... | [] | rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
obtain ⟨c, c0, hc⟩ := integerNormalization_spec R⁰ p
rw [Algebra.smul_def, algebraMap_apply] at hc
apply isUnit_of_mul_isUnit_right
rw [← hc, (integerNormalization R⁰ p).eq_C_content_mul_primPart, ← hu, ← map_mul, isUnit_iff]
refine
⟨algebraMap R K ((integerNor... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 222,
"column": 2
} | {
"line": 234,
"column": 29
} | {
"line": 236,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : K[X]\nh0 : p ≠ 0\nh : IsUnit (integerNormalization R⁰ p).primPart\n⊢ IsUnit p",
"ppTerm": "?m.26",
"assigned": true,
... | [] | rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
obtain ⟨c, c0, hc⟩ := integerNormalization_spec R⁰ p
rw [Algebra.smul_def, algebraMap_apply] at hc
apply isUnit_of_mul_isUnit_right
rw [← hc, (integerNormalization R⁰ p).eq_C_content_mul_primPart, ← hu, ← map_mul, isUnit_iff]
refine
⟨algebraMap R K ((integerNor... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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