module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1135,
"column": 10
} | {
"line": 1135,
"column": 21
} | [
{
"pp": "I : Type u\ninst✝⁶ : Category.{u, u} I\ninst✝⁵ : IsCofiltered I\nR : CommRingCat\ninst✝⁴ : IsAffine (Spec R)\nS : CommRingCat\ninst✝³ : IsAffine (Spec S)\nφ : R ⟶ S\ninst✝² : LocallyOfFinitePresentation (Spec.map φ)\nD : I ⥤ CommRingCatᵒᵖ\nc : Cone (D ⋙ Scheme.Spec)\nhc : IsLimit c\ninst✝¹ : ∀ (i : I),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 447,
"column": 4
} | {
"line": 447,
"column": 33
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nn : ℤ\nhn : n = 0\n⊢ W.Φ n ≠ 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Ne",
"Int",
"Polynomial",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1138,
"column": 4
} | {
"line": 1138,
"column": 71
} | [
{
"pp": "case inr\nI : Type u\ninst✝⁶ : Category.{u, u} I\ninst✝⁵ : IsCofiltered I\nR : CommRingCat\ninst✝⁴ : IsAffine (Spec R)\nS : CommRingCat\ninst✝³ : IsAffine (Spec S)\nφ : R ⟶ S\ninst✝² : LocallyOfFinitePresentation (Spec.map φ)\nD : I ⥤ CommRingCatᵒᵖ\nc : Cone (D ⋙ Scheme.Spec)\nhc : IsLimit c\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1145,
"column": 30
} | {
"line": 1145,
"column": 41
} | [
{
"pp": "case a\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\ninst✝ : ∀ (i : I), IsAffine ((D ⋙ Scheme.Spec).obj i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1146,
"column": 32
} | {
"line": 1146,
"column": 43
} | [
{
"pp": "I : 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\ninst✝ : ∀ (i : I), IsAffine ((D ⋙ Scheme.Spec).obj i)\ne : ((Fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1147,
"column": 7
} | {
"line": 1147,
"column": 18
} | [
{
"pp": "I : 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\ninst✝ : ∀ (i : I), IsAffine ((D ⋙ Scheme.Spec).obj i)\ne : ((Fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 30
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nn : ℕ\na b c : k\nhn : ↑n = 0\nhn' : 2 ≤ n\nhb : b ≠ 0\nf : k[X] := C a * X ^ n + C b * X + C c\nhdeg : f.degree ≠ 0\nhsep : f.Separable\nx : k\nhx : f.IsRoot x\n⊢ a * x ^ n + b * x + c = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 77
} | [
{
"pp": "case h\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\nhn' : 0 < n\nthis : (X ^ n - C x).degree ≠ 0\nhx : ¬x = 0\nz : k\nhz : (X ^ n - C x).IsRoot z\n⊢ z ^ n = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 183,
"column": 17
} | {
"line": 183,
"column": 41
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1210,
"column": 8
} | {
"line": 1210,
"column": 24
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1214,
"column": 6
} | {
"line": 1215,
"column": 48
} | [
{
"pp": "case refine_3\nI : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1235,
"column": 36
} | {
"line": 1235,
"column": 47
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Basic | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 36
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nhα : F⟮α⟯ = ⊤\niso : ↥F⟮α⟯ ≃ₐ[F] E :=\n { toFun := fun e ↦ ↑e, invFun := fun e ↦ ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1236,
"column": 25
} | {
"line": 1236,
"column": 77
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1243,
"column": 4
} | {
"line": 1244,
"column": 36
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1250,
"column": 6
} | {
"line": 1250,
"column": 21
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Basic | {
"line": 552,
"column": 2
} | {
"line": 559,
"column": 38
} | [
{
"pp": "case base\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\np : F[X]\nsp : Polynomial.IsSplittingField F E p\nhp : p.Separable\nhFE : FiniteDimensional F E\nthis : DecidableEq E := ⋯\ns : Set E := ⋯\nadjoin_root : adjoin F s = ⊤\nP : IntermediateField F E → Prop := ⋯... | · have key := IntermediateField.card_algHom_adjoin_integral F (K := E)
(show IsIntegral F (0 : E) from isIntegral_zero)
rw [IsSeparable, minpoly.zero, Polynomial.natDegree_X] at key
specialize key Polynomial.separable_X (Polynomial.Splits.X.map (algebraMap F E))
rw [← @Subalgebra.finrank_bot F E _ _ _... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1261,
"column": 4
} | {
"line": 1261,
"column": 91
} | [
{
"pp": "case refine_1\nI : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 196,
"column": 4
} | {
"line": 198,
"column": 30
} | [
{
"pp": "case of_j_ne_zero.of_j_eq_zero\nF : 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\nheq : E.j = E'.j\nC : VariableChange F\ninst✝¹ : (C • E).IsCharThreeJNeZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').I... | · have h := (C • E).j_ne_zero_of_isCharThreeJNeZeroNF_of_char_three
rw [variableChange_j, heq, ← variableChange_j E' C', j_of_isShortNF_of_char_three] at h
exact False.elim (h rfl) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 68
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.dblXYZ P =\n W.dblZ P •\n ![W.toAffine.addX (P x / P z ^ 2) ... | dblXYZ_of_Z_ne_zero hP hQ hPz hQz hx hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.WithZero | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | [
{
"pp": "case coe.zero\ne : ℝ≥0\nhe : 1 < e\na✝ : Multiplicative ℤ\n⊢ 0 < ↑a✝ → (toNNReal ⋯) 0 < (toNNReal ⋯) ↑a✝",
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 251,
"column": 2
} | {
"line": 252,
"column": 64
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : SeparatelyContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\nthis : Filter.Tendsto (fun y ↦ y * (x⁻¹ * g)) (𝓝 x) (𝓝 g)\n⊢ ↑H ∈ 𝓝 x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 482,
"column": 6
} | {
"line": 482,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\nU V : Set G\nx✝¹ : (fun S ↦ ∃ T ∈ 𝓝 1, S * T ⊆ W) U\nx✝ : (fun S ↦ ∃ T ∈ 𝓝 1, S * T ⊆ W) V\nT₁ : Set G\nhT₁ : T₁ ∈ 𝓝 1\nmem1 : U * T₁ ⊆ W\nT₂ : Set G\n... | rw [union_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 528,
"column": 6
} | {
"line": 528,
"column": 54
} | [
{
"pp": "case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nx✝ : G\nha : x✝ ∈ ⋃ n, V ^ (n + 1)\nk : ℕ\nhk : x✝ ∈ V ^ (k + 1)\n⊢ x✝⁻¹ ∈ V⁻¹ ^ (k + 1)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.Basic | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\ninst✝ : NonarchimedeanRing R\nU : OpenAddSubgroup R\n⊢ 0 ∈ (fun p ↦ p.1 * p.2) ⁻¹' ↑U",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 63,
"column": 58
} | {
"line": 63,
"column": 85
} | [
{
"pp": "A : Type u_3\nι : Type u_4\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y ↦ x * y) ⁻¹' ↑(B i)\nx : A\ni j : ι\nhj : ↑(B j) ⊆ (fun y ↦ x * y) ⁻¹' ↑(B i)\n⊢ ↑(B j) ⊆ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 19
} | [
{
"pp": "case h.right.a\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\na : A\ns : Set A\ni : ι\nhi : {b | b - a ∈ B i} ⊆ s\nb : A\nb_in : b ∈ ↑(B i)\n⊢ (fun y ↦ a + y) b ∈ {b | b - a ∈ B i}",
"usedConstants": [
"Eq.mpr",
"AddGr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 17
} | [
{
"pp": "case right\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni : ι\nx✝ : TopologicalSpace A := hB.topology\na : A\na_in : a ∈ (B i).carrier\nb : A\nb_in : b ∈ {b | b - a ∈ B i}\n⊢ b ∈ (B i).carrier",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 458,
"column": 30
} | {
"line": 458,
"column": 64
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 ≠ Q x * P z ^ 2\n⊢ P x / P z ^ 2 ≠ Q x / Q z ^ 2",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"Field.toDivision... | by rwa [ne_eq, ← X_eq_iff hPz hQz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.WithZeroTopology | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 63
} | [
{
"pp": "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\nthis : Tendsto ((fun p ↦ p.1 * p.2) ∘ Prod.swap) (𝓝 (x, y)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx : R\n⊢ 0 ≤ᵥ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 458,
"column": 2
} | {
"line": 460,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 ≠ Q x * P z ^ 2\n⊢ W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3) =\n (P y * Q z ^ 3 - Q y * P z ^ 3) / (P z * Q z * addZ P Q)",
... | rw [Affine.slope_of_X_ne <| by rwa [ne_eq, ← X_eq_iff hPz hQz],
div_sub_div _ _ (pow_ne_zero 2 hPz) (pow_ne_zero 2 hQz), mul_comm <| _ ^ 2, addZ]
simp [field] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 458,
"column": 2
} | {
"line": 460,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 ≠ Q x * P z ^ 2\n⊢ W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3) =\n (P y * Q z ^ 3 - Q y * P z ^ 3) / (P z * Q z * addZ P Q)",
... | rw [Affine.slope_of_X_ne <| by rwa [ne_eq, ← X_eq_iff hPz hQz],
div_sub_div _ _ (pow_ne_zero 2 hPz) (pow_ne_zero 2 hQz), mul_comm <| _ ^ 2, addZ]
simp [field] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 309,
"column": 2
} | {
"line": 312,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx : ↥(posSubmonoid R)\n⊢ ↑x ≠ 0",
"usedConstants": [
"ValuativeRel.vlt",
"False",
"congrArg",
"CommSemiring.toSemiring",
"Mathlib.Tactic.Contrapose.contrapose₃",
"ValuativeRel.not_vlt_zero._simp_1",
... | have := x.prop
rw [posSubmonoid_def] at this
contrapose this
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 309,
"column": 2
} | {
"line": 312,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx : ↥(posSubmonoid R)\n⊢ ↑x ≠ 0",
"usedConstants": [
"ValuativeRel.vlt",
"False",
"congrArg",
"CommSemiring.toSemiring",
"Mathlib.Tactic.Contrapose.contrapose₃",
"ValuativeRel.not_vlt_zero._simp_1",
... | have := x.prop
rw [posSubmonoid_def] at this
contrapose this
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 329,
"column": 8
} | {
"line": 329,
"column": 19
} | [
{
"pp": "case left\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z r : R\nu : ↥(posSubmonoid R)\ns : R\nv : ↥(posSubmonoid R)\nt : R\nw : ↥(posSubmonoid R)\nh1 : r * ↑v ≤ᵥ s * ↑u\nh2 : s * ↑u ≤ᵥ r * ↑v\nh3 : s * ↑w ≤ᵥ t * ↑v\nh4 : t * ↑v ≤ᵥ s * ↑w\nthis✝ : r * ↑v * ↑w ≤ᵥ s * ↑w * ↑u\nthis : r ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 19
} | [
{
"pp": "case right\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z r : R\nu : ↥(posSubmonoid R)\ns : R\nv : ↥(posSubmonoid R)\nt : R\nw : ↥(posSubmonoid R)\nh1 : r * ↑v ≤ᵥ s * ↑u\nh2 : s * ↑u ≤ᵥ r * ↑v\nh3 : s * ↑w ≤ᵥ t * ↑v\nh4 : t * ↑v ≤ᵥ s * ↑w\nthis✝ : t * ↑v * ↑u ≤ᵥ s * ↑u * ↑w\nthis : t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 405,
"column": 15
} | {
"line": 405,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx : R\ny : ↥(posSubmonoid R)\nh : ValueGroupWithZero.mk x y = 0\n⊢ x ≤ᵥ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 406,
"column": 42
} | {
"line": 406,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx : R\ny : ↥(posSubmonoid R)\nh : x ≤ᵥ 0\n⊢ x * ↑1 ≤ᵥ 0 * ↑y",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"Membership.mem",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 13
} | [
{
"pp": "case mk.mk\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : ValuativeRel R\nα : Type u_2\ninst✝ : Mul α\nf : R → ↥(posSubmonoid R) → α\nhf : ∀ (x y : R) (t s : ↥(posSubmonoid R)), x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → f x s = f y t\nhdist : ∀ (a b : R) (r s : ↥(posSubmonoid R)), f (a * b) (r * s) = f a r *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 466,
"column": 21
} | {
"line": 470,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z : R\na b c : ValueGroupWithZero R\n⊢ a * b * c = a * (b * c)",
"usedConstants": [
"Semigroup.toMul",
"Submonoid.mul",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulMemClass.toSemigroup",
"congrAr... | by
induction a using ValueGroupWithZero.ind
induction b using ValueGroupWithZero.ind
induction c using ValueGroupWithZero.ind
simp [mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 515,
"column": 6
} | {
"line": 515,
"column": 17
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z✝ x y z w : R\nt s u v : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nh₃ : z * ↑u ≤ᵥ w * ↑v\nh₄ : w * ↑v ≤ᵥ z * ↑u\nhw : w ≤ᵥ 0\nhz : ¬z ≤ᵥ 0\n⊢ z * ↑u ≤ᵥ 0 * ↑u",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 518,
"column": 6
} | {
"line": 518,
"column": 17
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z✝ x y z w : R\nt s u v : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nh₃ : z * ↑u ≤ᵥ w * ↑v\nh₄ : w * ↑v ≤ᵥ z * ↑u\nhw : ¬w ≤ᵥ 0\nhz : z ≤ᵥ 0\n⊢ w * ↑v ≤ᵥ 0 * ↑v",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 47
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z✝ x y z w : R\nt s u v : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nh₃ : z * ↑u ≤ᵥ w * ↑v\nh₄ : w * ↑v ≤ᵥ z * ↑u\nhw : ¬w ≤ᵥ 0\nhz : ¬z ≤ᵥ 0\n⊢ (fun a s b t ↦ a * ↑t ≤ᵥ b * ↑s) x s z v = (fun a s b t ↦ a *... | refine propext ⟨fun h => ?_, fun h => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 603,
"column": 6
} | {
"line": 603,
"column": 17
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z x y : R\nt s : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nhx : x ≤ᵥ 0\nhy : ¬y ≤ᵥ 0\n⊢ y * ↑s ≤ᵥ 0 * ↑s",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 606,
"column": 6
} | {
"line": 606,
"column": 17
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z x y : R\nt s : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nhx : ¬x ≤ᵥ 0\nhy : y ≤ᵥ 0\n⊢ x * ↑t ≤ᵥ 0 * ↑t",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 609,
"column": 8
} | {
"line": 609,
"column": 30
} | [
{
"pp": "case neg.h₁\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z x y : R\nt s : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nhx : ¬x ≤ᵥ 0\nhy : ¬y ≤ᵥ 0\n⊢ ↑s * ↑⟨y, hy⟩ ≤ᵥ ↑t * ↑⟨x, hx⟩",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 610,
"column": 8
} | {
"line": 610,
"column": 30
} | [
{
"pp": "case neg.h₂\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z x y : R\nt s : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nhx : ¬x ≤ᵥ 0\nhy : ¬y ≤ᵥ 0\n⊢ ↑t * ↑⟨x, hx⟩ ≤ᵥ ↑s * ↑⟨y, hy⟩",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 786,
"column": 24
} | {
"line": 786,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx x✝ : R\nh : x✝ ∈ {x | x ≤ᵥ 0}\n⊢ x • x✝ ∈ {x | x ≤ᵥ 0}",
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"CommSemiring.toSemiring",
"DistribMulAction.toDistribSMul",
"AddMonoid.toAddZeroClass",
"se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 793,
"column": 2
} | {
"line": 793,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ : R\n⊢ x✝ ∈ supp R ↔ x✝ ∈ (valuation R).supp",
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 23
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\na : R\nha : a ≤ᵥ 1\nn m : ℕ\nhnm : n ≤ n + m\n⊢ a ^ (n + m) ≤ᵥ a ^ n",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"pow_add",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 892,
"column": 2
} | {
"line": 892,
"column": 23
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\na : R\nha : 1 ≤ᵥ a\nn m : ℕ\nhnm : n ≤ n + m\n⊢ a ^ n ≤ᵥ a ^ (n + m)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"pow_add",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 944,
"column": 2
} | {
"line": 944,
"column": 13
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ninst✝ : ValuativeRel K\nx : K\nhx : x ≠ 0\n⊢ 1 ≤ᵥ x⁻¹ ↔ x ≤ᵥ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 949,
"column": 2
} | {
"line": 949,
"column": 13
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ninst✝ : ValuativeRel K\nx : K\nhx : x ≠ 0\n⊢ x⁻¹ ≤ᵥ 1 ↔ 1 ≤ᵥ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1002,
"column": 32
} | {
"line": 1002,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\ns : (ValueGroupWithZero R)ˣ\nh : 1 ≠ s\n⊢ ↑s ≠ 1",
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"congrArg",
"Units",
"id",
"Ne",
"Units.instOne",
"LinearOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1003,
"column": 32
} | {
"line": 1003,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nr s : (ValueGroupWithZero R)ˣ\nh : r ≠ s\nhr : r ≠ 1\n⊢ ↑r ≠ 1",
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"congrArg",
"Units",
"id",
"Ne",
"Units.instOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1011,
"column": 28
} | {
"line": 1011,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\n⊢ v r ≠ 0",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1013,
"column": 6
} | {
"line": 1013,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\nhγ : v r ≠ 0\n⊢ v r ≤ v ↑s → v r < v ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1016,
"column": 19
} | {
"line": 1016,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr✝ : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\nhγ : v r ≠ 0\nhγ' : v r ≤ v ↑s → v r ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1016,
"column": 19
} | {
"line": 1016,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr✝ : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\nhγ : v r ≠ 0\nhγ' : v r ≤ v ↑s → v r ... | simpa using hγ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1016,
"column": 19
} | {
"line": 1016,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr✝ : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\nhγ : v r ≠ 0\nhγ' : v r ≤ v ↑s → v r ... | simpa using hγ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1016,
"column": 19
} | {
"line": 1016,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\ns : ↥(posSubmonoid R)\nhr✝ : ValueGroupWithZero.mk r s ≠ 0\nhr' : ValueGroupWithZero.mk r s ≠ 1\nhγ : v r ≠ 0\nhγ' : v r ≤ v ↑s → v r ... | simpa using hγ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1019,
"column": 9
} | {
"line": 1019,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nr : R\nhr : v r ≠ 0\nhr' : v r ≠ 1\n⊢ (valuation R) r ≠ 1",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1028,
"column": 25
} | {
"line": 1028,
"column": 53
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ninst✝ : ValuativeRel K\nx : K\ny : ↥(posSubmonoid K)\n⊢ ValueGroupWithZero.mk x y = (valuation K) x / (valuation K) ↑y",
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"instHDiv",
"GroupWithZer... | ValueGroupWithZero.mk_eq_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1074,
"column": 25
} | {
"line": 1076,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : IsDiscrete R\ninst✝ : IsNontrivial R\n⊢ 0 < uniformizer R",
"usedConstants": [
"Iff.mpr",
"ValuativeRel.instLinearOrderValueGroupWithZero",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Exists",
... | by
obtain ⟨γ, hγ, hγ'⟩ := IsNontrivial.exists_lt_one (R := R)
exact hγ.trans_le (le_uniformizer_iff.mpr hγ') | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.UniformRing | {
"line": 134,
"column": 17
} | {
"line": 134,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝⁹ : Ring α\ninst✝⁸ : UniformSpace α\ninst✝⁷ : IsTopologicalRing α\ninst✝⁶ : IsUniformAddGroup α\nβ : Type u\ninst✝⁵ : UniformSpace β\ninst✝⁴ : Ring β\ninst✝³ : IsUniformAddGroup β\ninst✝² : IsTopologicalRing β\nf : α →+* β\nhf✝ : Continuous ⇑f\ninst✝¹ : CompleteSpace β\ninst✝ : T0Spa... | by simp_rw [← coe_zero, extension_coe hf, f.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.UniformRing | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 32
} | [
{
"pp": "α : Type u_1\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : IsTopologicalRing α\ninst✝¹¹ : IsUniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : IsUniformAddGroup β\ninst✝⁷ : IsTopologicalRing β\nf : α →+* β\nhf : Continuous ⇑f\nA : Type u_2\ninst✝⁶ : Ring A\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1188,
"column": 4
} | {
"line": 1188,
"column": 15
} | [
{
"pp": "R : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\na : R\nr : ↥(posSubmonoid R)\nb : R\ns : ↥(posSubmonoid R)\nh : v (a * ↑s) < v (b * ↑r)\n⊢ v.toMonoidWithZeroHom a * v.toMonoidWithZeroHom ↑s < v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1278,
"column": 2
} | {
"line": 1278,
"column": 13
} | [
{
"pp": "R : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nγ : ValueGroupWithZero R\nx : R\n⊢ v.restrict x < (orderMonoidIso v) γ ↔ (valuation R) x < γ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 31
} | [
{
"pp": "case inr.inl\nK : Type u\ninst✝² : DivisionRing K\nΓ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : MulArchimedean Γ₀\nv : Valuation K Γ₀\nx : K\nhx : v x ≠ 0\nr : Γ₀ˣ\nhr : ↑r ≠ 0\nh : ∀ (x : K), v x ≠ 0 → ↑r < v x\nH : Units.mk0 (v x) hx = 1\n⊢ v x = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1314,
"column": 7
} | {
"line": 1314,
"column": 84
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : ValuativeRel A\ninst✝² : ValuativeRel B\ninst✝¹ : Algebra A B\ninst✝ : ValuativeExtension A B\nx✝ : ↥(posSubmonoid A)\na : A\nha : a ∈ posSubmonoid A\n⊢ (algebraMap A B) a ∈ posSubmonoid B",
"usedConstants": [
"Val... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 104,
"column": 8
} | {
"line": 104,
"column": 19
} | [
{
"pp": "case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : (ValueGroup₀ v)ˣ\nγx : Γ₀ˣ\nHx : v x = ↑γx\nu : (ValueGroup₀ v)ˣ := Units.mk0 ((restrict₀ v) x) ⋯\nhu_def : u = Units.mk0 ((restrict₀ v) x) ⋯\nhu : embedding ↑u⁻¹ = ↑γx⁻¹\ny : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 19
} | [
{
"pp": "case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : (ValueGroup₀ v)ˣ\nγx : Γ₀ˣ\nHx : v x = ↑γx\nu : (ValueGroup₀ v)ˣ := Units.mk0 ((restrict₀ v) x) ⋯\nhu_def : u = Units.mk0 ((restrict₀ v) x) ⋯\nhu : embedding ↑u⁻¹ = ↑γx⁻¹\ny : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 39,
"column": 34
} | {
"line": 39,
"column": 45
} | [
{
"pp": "R : Type ?u.3208\nS : Type ?u.3211\nM : Type ?u.3214\ninst✝⁶ : CommRing R\ninst✝⁵ : Field S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Algebra R S\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nB : BilinForm S M\nN : Submodule R M\na b : M\nha : a ∈ {x | ∀ y ∈ N, (B x) y ∈ 1}\nhb : b ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 215,
"column": 44
} | {
"line": 215,
"column": 55
} | [
{
"pp": "K : Type u\ninst✝³ : DivisionRing K\nΓ₀ : Type v\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : MulArchimedean Γ₀\ninst✝ : Valued K Γ₀\nr : Γ₀\nhr : r ≠ 0\nh : ∀ (x : K), v x ≠ 0 → r < v x\n⊢ ∀ (x : K), x ≠ 0 → v x = 1",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrdere... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | [
{
"pp": "case a.a\nR : Type u_1\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\ninst✝¹ : IsDomain R\nhB : B.Nondegenerate\ninst✝ : IsTorsionFree R S\nN : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 15
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x = 0}",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrdere... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 205,
"column": 64
} | {
"line": 205,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nr : K[X]\nhr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r\nd' : R[X]\nhr' : Polynomi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 40
} | [
{
"pp": "case neg\nR : Type u_3\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : DecidableEq T\nx : S\ny : T\nσ : S →+* T\nhy : eval₂ σ y (minpolyDiv R x * (X - C x)) = 0\nh : ¬σ x = y\n⊢ eval₂ σ y (minpolyDiv R x) = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 25
} | [
{
"pp": "case hy\nR : Type u_2\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nx : S\ninst✝¹ : IsDomain S\ninst✝ : DecidableEq S\ny : S\nhy : (aeval y) (minpoly R x) = 0\n⊢ eval₂ ((RingHom.id S).comp (algebraMap R S)) y (minpoly R x) = 0",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 86,
"column": 2
} | {
"line": 98,
"column": 7
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\ni : ℕ\n⊢ (minpolyDiv R x).coeff i ∈ R[x]",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.recAux",
"AddSubmonoidCla... | by_contra H
have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ R[x] := by
intro j; induction j with
| zero => exact H
| succ j IH =>
intro H; apply IH
rw [coeff_minpolyDiv]
refine add_mem ?_ (mul_mem H (self_mem_adjoin_singleton R x))
exact Subalgebra.algebraMap_mem _ _
apply this (nat... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 86,
"column": 2
} | {
"line": 98,
"column": 7
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\ni : ℕ\n⊢ (minpolyDiv R x).coeff i ∈ R[x]",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.recAux",
"AddSubmonoidCla... | by_contra H
have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ R[x] := by
intro j; induction j with
| zero => exact H
| succ j IH =>
intro H; apply IH
rw [coeff_minpolyDiv]
refine add_mem ?_ (mul_mem H (self_mem_adjoin_singleton R x))
exact Subalgebra.algebraMap_mem _ _
apply this (nat... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 15
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : IsIntegral R x\na✝ : Nontrivial S\nthis : (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff = 1\n⊢ (minpolyDiv R x).Monic",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : IsIntegral R x\na✝ : Nontrivial S\nthis : (minpolyDiv R x * (X - C x)).leadingCoeff = (map (algebraMap R S) (minpoly R x)).leadingCoeff\n⊢ (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff ≠ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nhx : IsIntegral R x\ninst✝ : Nontrivial S\n⊢ (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff ≠ 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 60
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\nf : ι' → ι\nhf : Injective f\np q : MvPolynomial ι' R\n⊢ (aeval (x ∘ f)) p = (aeval (x ∘ f)) q → p = q",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\n⊢ AlgebraicIndependent R Subtype.val",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 29
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns t : Set ι\nH : AlgebraicIndepOn R x t\nhst : s ⊆ t\n⊢ AlgebraicIndepOn R x s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nhs : IsIntegral R s\np : R[X]\nhp : minpoly R s ∣ p\n⊢ (Polynomial.aeval s) p = 0",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 13
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB₁ B₂ : Set α\ne : α\nhB₁ : M.IsBase B₁\nhB₂ : M.IsBase B₂\nhxB₁ : e ∈ B₁\nhxB₂ : e ∉ B₂\n⊢ ∃ y, (y ∈ B₂ ∧ y ∉ B₁) ∧ M.IsBase (insert y (B₁ \\ {e}))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 27
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\np : R[X]\nhirr : Irreducible p\nhp : (Polynomial.aeval x) p = 0\nisUnit : IsUnit p.leadingCoeff\n⊢ IsInt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 627,
"column": 6
} | {
"line": 627,
"column": 22
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB B' : Set α\ne : α\nhB : M.IsBase B\nhB' : M.IsBase B'\nh : B \\ B' = {e}\nf : α\nhf : f ∈ B' \\ B\nhb : M.IsBase (insert f B \\ {e})\nhne : f ≠ e\n⊢ insert f B \\ {e} ⊆ B'",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.diff_subset_iff",
"Set.i... | diff_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 642,
"column": 65
} | {
"line": 642,
"column": 78
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nB : Set α\ne f : α\nhB : M.IsBase B\nhf : f ∉ B\nhI : M.Indep (insert f (B \\ {e}))\nB' : Set α\nhB' : M.IsBase B'\nhfB : f ∈ B'\nh : B \\ B' = {e}\nhx : B' \\ B = {f}\n⊢ M.IsBase (B' \\ B ∪ B ⊓ B')",
"usedConstants": [
"BooleanAlgebra.toGeneralizedBoole... | inf_eq_inter, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 657,
"column": 2
} | {
"line": 658,
"column": 9
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nI : Set α\ne f : α\nhe : e ∉ I\nhf : f ∉ I\nheI : M.IsBase (insert e I)\nhfI : M.Indep (insert f I)\nhef : e ≠ f\n⊢ M.IsBase (insert f I)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 38
} | [
{
"pp": "case a.refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 66
} | [
{
"pp": "case a.refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 :=... | scaleRoots_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 197,
"column": 14
} | {
"line": 198,
"column": 58
} | [
{
"pp": "case a.refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 :=... | ← inv_mul_cancel_left₀ (b := algebraMap S L s)
(a := algebraMap K L (algebraMap R K r)) (by simpa), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\nhx : IsIntegral R x\nP : R[X]\nhP₁ : (Minpoly.toAdjoin R x) ((AdjoinRoot.mk (minpoly R x)) P) = 0\n⊢ (Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1020,
"column": 2
} | {
"line": 1020,
"column": 13
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\nI : Set α\nhI : M.Indep I\ne : α\nhe : e ∈ {x | M.IsBasis I (insert x I)} \\ I\nhu : M.Indep (insert e I)\n⊢ e ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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