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.Dynamics.TopologicalEntropy.Semiconj | {
"line": 213,
"column": 2
} | {
"line": 215,
"column": 74
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : UniformSpace Y\nS : X → X\nT : Y → Y\nφ : X → Y\nh : Semiconj φ S T\nF G : Set X\nh' : UniformContinuousOn φ G\nhF : F ⊆ G\nhG : MapsTo S G G\n⊢ coverEntropy T (φ '' F) ≤ coverEntropy (MapsTo.restrict S G G hG) (val ⁻¹' F)",
"usedConstant... | have hφ : Semiconj (G.restrict φ) (hG.restrict S G G) T := by
intro x
rw [G.restrict_apply, G.restrict_apply, hG.val_restrict_apply, h.eq x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.AbelRuffini | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 38
} | [
{
"pp": "case h\nF : Type u_1\ninst✝ : Field F\nn : ℕ\na : F\nh : (map (RingHom.id F) (X ^ n - 1)).Splits\nha : ¬a = 0\nha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0\nhn : ¬n = 0\nhn' : 0 < n\nhn'' : X ^ n - C a ≠ 0\nhn''' : X ^ n - 1 ≠ 0\nmem_range : ∀ {c : (X ^ n - C a).SplittingField}, c ^ n = 1 → ... | rw [hc, mul_div_cancel₀ (σ b) hb'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.LanguageMap | {
"line": 523,
"column": 4
} | {
"line": 523,
"column": 14
} | [
{
"pp": "case refine_1.inr\nL : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\nn : ℕ\nx : Fin n → M\nc : (constantsOn ↑A).Functions n\n⊢ f.toFun (funMap (Sum.inr c) x) = fu... | | inr c => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.ModelTheory.Basic | {
"line": 538,
"column": 6
} | {
"line": 538,
"column": 36
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nP : Type u_1\ninst✝¹ : L.Structure P\nQ : Type u_2\ninst✝ : L.Structure Q\nf : M ≃[L] N\nn : ℕ\nf' : L.Functions n\nx : Fin n → N\n⊢ f.symm.toFun (funMap f' x) = funMap f' (f.symm.toFun ∘ x)",
"use... | simp only [Equiv.toFun_as_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Basic | {
"line": 543,
"column": 6
} | {
"line": 543,
"column": 36
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nP : Type u_1\ninst✝¹ : L.Structure P\nQ : Type u_2\ninst✝ : L.Structure Q\nf : M ≃[L] N\nn : ℕ\nr : L.Relations n\nx : Fin n → N\n⊢ RelMap r (f.symm.toFun ∘ x) ↔ RelMap r x",
"usedConstants": [
... | simp only [Equiv.toFun_as_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 43,
"column": 19
} | {
"line": 43,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ + t₂) = x + y",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 43,
"column": 19
} | {
"line": 43,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ + t₂) = x + y",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 43,
"column": 19
} | {
"line": 43,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ + t₂) = x + y",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 45,
"column": 19
} | {
"line": 45,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ * t₂) = x * y",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 45,
"column": 19
} | {
"line": 45,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ * t₂) = x * y",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 45,
"column": 19
} | {
"line": 45,
"column": 27
} | [
{
"pp": "α : Type u_1\np x y : FreeCommRing α\nx✝¹ : ∃ t, Term.realize FreeCommRing.of t = x\nx✝ : ∃ t, Term.realize FreeCommRing.of t = y\nt₁ : ring.Term α\nht₁ : Term.realize FreeCommRing.of t₁ = x\nt₂ : ring.Term α\nht₂ : Term.realize FreeCommRing.of t₂ = y\n⊢ Term.realize FreeCommRing.of (t₁ * t₂) = x * y",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Syntax | {
"line": 590,
"column": 13
} | {
"line": 590,
"column": 61
} | [
{
"pp": "case equal\nL : Language\nα : Type u'\nn n✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x t ↦ Term.relabel (Sum.map id (natAdd 0)) t) (fun x ↦ id) (fun x ↦ castLE ⋯) (equal t₁✝ t₂✝) =\n castLE ⋯ (equal t₁✝ t₂✝)",
"usedConstants": [
"FirstOrder.Language.BoundedFormula.mapTermRel",... | simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Syntax | {
"line": 590,
"column": 13
} | {
"line": 590,
"column": 61
} | [
{
"pp": "case equal\nL : Language\nα : Type u'\nn n✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x t ↦ Term.relabel (Sum.map id (natAdd 0)) t) (fun x ↦ id) (fun x ↦ castLE ⋯) (equal t₁✝ t₂✝) =\n castLE ⋯ (equal t₁✝ t₂✝)",
"usedConstants": [
"FirstOrder.Language.BoundedFormula.mapTermRel",... | simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Syntax | {
"line": 590,
"column": 13
} | {
"line": 590,
"column": 61
} | [
{
"pp": "case equal\nL : Language\nα : Type u'\nn n✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x t ↦ Term.relabel (Sum.map id (natAdd 0)) t) (fun x ↦ id) (fun x ↦ castLE ⋯) (equal t₁✝ t₂✝) =\n castLE ⋯ (equal t₁✝ t₂✝)",
"usedConstants": [
"FirstOrder.Language.BoundedFormula.mapTermRel",... | simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Syntax | {
"line": 591,
"column": 11
} | {
"line": 591,
"column": 59
} | [
{
"pp": "case rel\nL : Language\nα : Type u'\nn n✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x t ↦ Term.relabel (Sum.map id (natAdd 0)) t) (fun x ↦ id) (fun x ↦ castLE ⋯) (rel R✝ ts✝) =\n castLE ⋯ (rel R✝ ts✝)",
"usedConstants": [
"Eq.mpr",
"FirstOrder... | simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Syntax | {
"line": 666,
"column": 11
} | {
"line": 666,
"column": 95
} | [
{
"pp": "case h.rel\nL : Language\nL' : Language\nα : Type u'\nn : ℕ\nL'' : Language\nφ : L' →ᴸ L''\nψ : L →ᴸ L'\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\n⊢ (φ.comp ψ).onBoundedFormula (rel R✝ ts✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (rel R✝ ts✝)",
"usedConstants": [
... | simp only [onBoundedFormula, comp_onRelation, comp_onTerm, Function.comp_apply]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Syntax | {
"line": 666,
"column": 11
} | {
"line": 666,
"column": 95
} | [
{
"pp": "case h.rel\nL : Language\nL' : Language\nα : Type u'\nn : ℕ\nL'' : Language\nφ : L' →ᴸ L''\nψ : L →ᴸ L'\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\n⊢ (φ.comp ψ).onBoundedFormula (rel R✝ ts✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (rel R✝ ts✝)",
"usedConstants": [
... | simp only [onBoundedFormula, comp_onRelation, comp_onTerm, Function.comp_apply]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Algebra.Field.CharP | {
"line": 79,
"column": 48
} | {
"line": 79,
"column": 56
} | [
{
"pp": "case neg.inl\np : ℕ\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : CompatibleRing K\nhp0 : ¬p = 0\nhp : ¬Nat.Prime p\nH : CharP K p\nh✝ : Nat.Prime p\n⊢ False",
"usedConstants": [
"False",
"Nat.Prime",
"congrArg",
"False.elim",
"Eq.mp",
"not_true_eq_false",
"True... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Algebra.Field.CharP | {
"line": 79,
"column": 48
} | {
"line": 79,
"column": 56
} | [
{
"pp": "case neg.inr\np : ℕ\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : CompatibleRing K\nhp0 : ¬p = 0\nhp : ¬Nat.Prime p\nH : CharP K p\nh✝ : p = 0\n⊢ False",
"usedConstants": [
"False",
"congrArg",
"False.elim",
"Eq.mp",
"not_true_eq_false",
"instOfNatNat",
"Nat",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Semantics | {
"line": 307,
"column": 46
} | {
"line": 307,
"column": 55
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nl : ℕ\nv : α → M\nxs : Fin l → M\nR : L.Relations 1\nt : L.Term (α ⊕ Fin l)\n⊢ (RelMap R fun i ↦ realize (Sum.elim v xs) (![t] i)) ↔ RelMap R ![realize (Sum.elim v xs) t]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Firs... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 315,
"column": 46
} | {
"line": 315,
"column": 55
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nl : ℕ\nv : α → M\nxs : Fin l → M\nR : L.Relations 2\nt₁ t₂ : L.Term (α ⊕ Fin l)\n⊢ (RelMap R fun i ↦ realize (Sum.elim v xs) (![t₁, t₂] i)) ↔\n RelMap R ![realize (Sum.elim v xs) t₁, realize (Sum.elim v xs) t₂]",
"usedConstants": [
... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Ultraproducts | {
"line": 149,
"column": 63
} | {
"line": 149,
"column": 72
} | [
{
"pp": "α : Type u_1\nM : α → Type u_2\nu : Ultrafilter α\nL : Language\ninst✝¹ : (a : α) → L.Structure (M a)\ninst✝ : ∀ (a : α), Nonempty (M a)\nβ : Type u_3\nφ : L.Formula β\nx : β → (a : α) → M a\n⊢ BoundedFormula.Realize φ (fun i ↦ Quotient.mk' (x i)) default ↔\n BoundedFormula.Realize φ (fun i ↦ Quotie... | iff_eq_eq | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Ultraproducts | {
"line": 157,
"column": 32
} | {
"line": 157,
"column": 41
} | [
{
"pp": "α : Type u_1\nM : α → Type u_2\nu : Ultrafilter α\nL : Language\ninst✝¹ : (a : α) → L.Structure (M a)\ninst✝ : ∀ (a : α), Nonempty (M a)\nφ : L.Sentence\n⊢ Formula.Realize φ default ↔ Formula.Realize φ fun i ↦ Quotient.mk' fun a ↦ default i",
"usedConstants": [
"Eq.mpr",
"Inhabited.defa... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 424,
"column": 35
} | {
"line": 424,
"column": 44
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → M\nxs : Fin (n + 1) → M\n⊢ φ.Realize v (xs ∘ fun i ↦ if ↑i < n then i.castSucc else i.succ) ↔ φ.Realize v (xs ∘ castSucc)",
"usedConstants": [
"Eq.mpr",
"Fin.succ",
"congrArg",
... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 36
} | {
"line": 448,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₁✝.v... | by simp [hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 80
} | {
"line": 448,
"column": 93
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₂✝.v... | by simp [hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Semantics | {
"line": 453,
"column": 36
} | {
"line": 453,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\nf : ↥(rel R✝ ts✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(rel R✝ ts✝).freeVarFinset), v (f a) =... | by simp [hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Semantics | {
"line": 557,
"column": 39
} | {
"line": 557,
"column": 48
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nv : α → M\nR : L.Relations 1\nt : L.Term α\n⊢ (RelMap R fun i ↦ Term.realize v (![t] i)) ↔ RelMap R ![Term.realize v t]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"FirstOrder.Language.Term",
"id",
"instOfNat... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 564,
"column": 39
} | {
"line": 564,
"column": 48
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nv : α → M\nR : L.Relations 2\nt₁ t₂ : L.Term α\n⊢ (RelMap R fun i ↦ Term.realize v (![t₁, t₂] i)) ↔ RelMap R ![Term.realize v t₁, Term.realize v t₂]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"FirstOrder.Language.Term",... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 851,
"column": 6
} | {
"line": 851,
"column": 25
} | [
{
"pp": "case all.e__v.h\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nn n✝ : ℕ\nf✝ : L.BoundedFormula α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), f✝.toFormula.Realize v ↔ f✝.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n f✝.toFormula.Realize (Sum.elim (v ∘ Sum.inl) ... | rcases x with _ | x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.ModelTheory.Encoding | {
"line": 296,
"column": 18
} | {
"line": 296,
"column": 43
} | [
{
"pp": "L : Language\nα : Type u'\n⊢ lift.{max (max u' v) u, max (max u' v) u} #(List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)) ≤\n lift.{max (max u u') v, max (max u u') v} (max ℵ₀ (lift.{max u v, u'} #α + lift.{u', max u v} L.card))",
"usedConstants": [
"Sum.nonemptyRight",
... | mk_list_eq_max_mk_aleph0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Substructures | {
"line": 432,
"column": 4
} | {
"line": 433,
"column": 32
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS✝ : L.Substructure M\ns : Set M\nφ : M →[L] N\nS : L.Substructure N\nn : ℕ\nf : L.Functions n\nx : Fin n → M\nhx : ∀ (i : Fin n), x i ∈ ⇑φ ⁻¹' ↑S\n⊢ funMap f x ∈ ⇑φ ⁻¹' ↑S",
... | rw [mem_preimage, φ.map_fun]
exact S.fun_mem f (φ ∘ x) hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Substructures | {
"line": 432,
"column": 4
} | {
"line": 433,
"column": 32
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS✝ : L.Substructure M\ns : Set M\nφ : M →[L] N\nS : L.Substructure N\nn : ℕ\nf : L.Functions n\nx : Fin n → M\nhx : ∀ (i : Fin n), x i ∈ ⇑φ ⁻¹' ↑S\n⊢ funMap f x ∈ ⇑φ ⁻¹' ↑S",
... | rw [mem_preimage, φ.map_fun]
exact S.fun_mem f (φ ∘ x) hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Definability | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 25
} | [
{
"pp": "case intro.intro.mpr\nM : Type w\nA : Set M\nL : Language\ninst✝² : L.Structure M\nα : Type u₁\nβ : Type u_1\ns : Set (β → M)\nh✝ : A.Definable L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n A.Definable L\n ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'\n ... | rintro ⟨y, ys, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.ModelTheory.Definability | {
"line": 664,
"column": 2
} | {
"line": 668,
"column": 6
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nf : M → M\nh : TermDefinable₁ L f\n⊢ A.Definable₂ L (Function.graph f)",
"usedConstants": [
"Set.TermDefinable₁.termDefinable",
"Set.ext",
"Eq.mpr",
"Unit.unit",
"Set.Definable₂._proof_1",
"FirstOrder.La... | obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L
use t.relabel (Option.elim · 1 (fun _ ↦ 0))
ext v
convert! Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Definability | {
"line": 664,
"column": 2
} | {
"line": 668,
"column": 6
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nf : M → M\nh : TermDefinable₁ L f\n⊢ A.Definable₂ L (Function.graph f)",
"usedConstants": [
"Set.TermDefinable₁.termDefinable",
"Set.ext",
"Eq.mpr",
"Unit.unit",
"Set.Definable₂._proof_1",
"FirstOrder.La... | obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L
use t.relabel (Option.elim · 1 (fun _ ↦ 0))
ext v
convert! Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 161,
"column": 8
} | {
"line": 161,
"column": 22
} | [
{
"pp": "case val.a.refine_2\np : ℕ\nκ : Cardinal.{u_2}\nhκ : ℵ₀ < κ\nM : Type u_2\nstruc✝¹ : Language.ring.Structure M\nis_model✝¹ : M ⊨ Theory.ACF p\nnonempty'✝¹ : Nonempty M\nN : Type u_2\nstruc✝ : Language.ring.Structure N\nis_model✝ : N ⊨ Theory.ACF p\nnonempty'✝ : Nonempty N\nhM : #↑{ Carrier := M, struc ... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.FreeCommRing | {
"line": 95,
"column": 24
} | {
"line": 95,
"column": 32
} | [
{
"pp": "case pos\nι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\nf : (i : ι) × ↥(monoms i) ⊕ κ → R\ni : ι\nm : ↥(monoms i)\na✝ : m ∈ (monoms i).attach\nh0 : f (Sum.inl ⟨i, m⟩) = 0\n⊢ f (Sum.inl ⟨i, m⟩) * ∏ x ∈ (↑m).sup... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPolynomial.FreeCommRing | {
"line": 95,
"column": 24
} | {
"line": 95,
"column": 32
} | [
{
"pp": "case neg\nι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\nf : (i : ι) × ↥(monoms i) ⊕ κ → R\ni : ι\nm : ↥(monoms i)\na✝ : m ∈ (monoms i).attach\nh0 : ¬f (Sum.inl ⟨i, m⟩) = 0\n⊢ f (Sum.inl ⟨i, m⟩) * ∏ x ∈ (↑m).su... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Algebra.Ring.Definability | {
"line": 37,
"column": 6
} | {
"line": 37,
"column": 33
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : CompatibleRing K\nS : Finset (MvPolynomial ι K)\np' : ↥S → FreeCommRing ((i : ↥S) × ↥(↑i).support ⊕ ι) := genericPolyMap fun p ↦ (↑p).support\nthis✝ : DecidableEq ι := Classical.decEq ι\nthis : DecidableEq K := Classical.decEq K\n⊢ ∃ φ, zeroLocus K ... | MvPolynomial.zeroLocus_span | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 34
} | [
{
"pp": "case a.inr\nφ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\np : ℕ\nhp : p ∈ {q | Nat.Prime q}\nq : ℕ\nproperty✝ : Nat.Prime q\nhq : ⟨q, property✝⟩ ∉ {⟨p, hp⟩}\nK : Type\nstruc✝ : Language.ring.Structure K\nis_model✝ : K ⊨ Theory.ACF ↑⟨q, property✝... | let _ := fieldOfModelACF q K | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Nullstellensatz | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 28
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\nσ : Type u_3\nI : Ideal (MvPolynomial σ k)\np : MvPolynomial σ k\nhp : p ∈ I.radical\nx : σ → K\nhx : x ∈ zeroLocus K I\n⊢ p ∈ vanishingIdeal k {x}",
"usedConstants": [
"Submodule",
"Nat.instMulZeroClas... | rw [radical_eq_sInf] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 10
} | [
{
"pp": "case h\na b : Cardinal.{u_1}\nhb : a * b ≠ 0\n⊢ b ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"eq_false",
"Cardinal",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.Finite.Extension | {
"line": 57,
"column": 2
} | {
"line": 59,
"column": 55
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\ninst✝⁴ : Finite k\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : CharP k p\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Algebra (ZMod p) k\nthis : Algebra (ZMod p) k := ZMod.algebra k p\n⊢ Module.finrank (ZMod p) (GaloisField p (Module.finrank (ZMod p) k * n)) = Module.finrank (ZMo... | convert!
GaloisField.finrank p (n := Module.finrank (ZMod p) k * n) <|
mul_ne_zero Module.finrank_pos.ne' <| NeZero.ne n | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 105,
"column": 93
} | {
"line": 111,
"column": 23
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : Field K\na b : σ → K\nh : a ≠ b\n⊢ (eval a) (indicator b) = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Finsupp.instAddZeroClass",
"Finset.mem_univ",
"Eq.mpr",
"GroupWithZero.toMono... | by
obtain ⟨i, hi⟩ : ∃ i, a i ≠ b i := by rwa [Ne, funext_iff, not_forall] at h
simp only [indicator, map_prod, map_sub, map_one, map_pow, eval_X, eval_C,
Finset.prod_eq_zero_iff]
refine ⟨i, Finset.mem_univ _, ?_⟩
rw [FiniteField.pow_card_sub_one_eq_one, sub_self]
rwa [Ne, sub_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Differential.Basic | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 16
} | [
{
"pp": "case cons.hb\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ (Multiset.map f s✝).prod ≠ 0",
"used... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.Differential.Basic | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 16
} | [
{
"pp": "case cons.hb\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ (Multiset.map f s✝).prod ≠ 0",
"used... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Basic | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 16
} | [
{
"pp": "case cons.hb\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ (Multiset.map f s✝).prod ≠ 0",
"used... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Differential.Basic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 14
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ ∀ x ∈ s✝, f x ≠ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.Differential.Basic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 14
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ ∀ x ∈ s✝, f x ≠ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Basic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 14
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\nf : ι → R\na✝ : ι\ns✝ : Multiset ι\nh₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum\nh : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0\n⊢ ∀ x ∈ s✝, f x ≠ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Differential.Basic | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nι : Type u_2\ns : Finset ι\nf : ι → R\nh : ∀ x ∈ s, f x = 0\n⊢ (∏ x ∈ s, f x)′ / ∏ x ∈ s, f x = ∑ x ∈ s, (f x)′ / f x",
"usedConstants": [
"Derivation",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithO... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Discriminant | {
"line": 164,
"column": 2
} | {
"line": 166,
"column": 6
} | [
{
"pp": "K : Type u\nL : Type v\nE : Type z\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K E\ninst✝² : Module.Finite K L\ninst✝¹ : IsAlgClosed E\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\ne : Fin pb.dim ≃ (L →ₐ[K] E)\n⊢ (algebraMap K E) (discr K ⇑pb.b... | rw [discr_powerBasis_eq_prod _ _ _ e]
congr; ext i; congr; ext j
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Discriminant | {
"line": 164,
"column": 2
} | {
"line": 166,
"column": 6
} | [
{
"pp": "K : Type u\nL : Type v\nE : Type z\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K E\ninst✝² : Module.Finite K L\ninst✝¹ : IsAlgClosed E\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\ne : Fin pb.dim ≃ (L →ₐ[K] E)\n⊢ (algebraMap K E) (discr K ⇑pb.b... | rw [discr_powerBasis_eq_prod _ _ _ e]
congr; ext i; congr; ext j
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Invariant.Basic | {
"line": 180,
"column": 41
} | {
"line": 180,
"column": 54
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nG : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : Group G\ninst✝² : MulSemiringAction G B\ninst✝¹ : IsInvariant A B G\ninst✝ : Finite G\nval✝ : Fintype G\nb : B\np : A[X]\nhp1 : map (algebraMap A B) p = charpoly G b\nhp2 : p.Monic\n⊢ eval... | eval_charpoly | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 308,
"column": 14
} | {
"line": 308,
"column": 23
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : NoZeroDivisors B\nb : B\ni j : ℕ\np : A[X]\nh : map (algebraMap A B) p = (X - C b) ^ i * X ^ j\nf : B ≃ₐ[A] B\nhi : i ≠ 0\nha : ¬b = 0\nhf : eval b ((X - C b) ^ i) * eval b (X ^ j) = eval b ((X... | eval_pow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 47
} | [
{
"pp": "k : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nH : Set Gal(K/k)\nL : FiniteGaloisIntermediateField k K\nle : ↑L.fixingSubgroup ⊆ H\n⊢ ∃ t ⊆ (fun a ↦ (mulEquivToLimit k K).toEquiv a) '' H, IsOpen t ∧ 1 ∈ t",
"usedConstants": [
"MulEq... | use mulEquivToLimit k K '' L.1.fixingSubgroup | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.PerfectClosure | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 84
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1,\n (⇑(frobenius K p))^[(n... | rw [← iterate_succ_apply, iterate_succ_apply', iterate_succ_apply', ← map_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 59
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_6\ninst✝²⁴ : CommRing A\ninst✝²³ : CommRing B\ninst✝²² : Algebra A B\nAₘ : Type u_9\nBₘ : Type u_10\ninst✝²¹ : CommRing Aₘ\ninst✝²⁰ : CommRing Bₘ\ninst✝¹⁹ : Algebra Aₘ Bₘ\ninst✝¹⁸ : Algebra A Aₘ\ninst✝¹⁷ : Algebra B Bₘ\ninst✝¹⁶ : Algebra A Bₘ\ninst✝¹⁵ : IsScalarTower ... | have : IsLocalization (algebraMapSubmonoid B A⁰) L :=
IsIntegralClosure.isLocalization _ (FractionRing A) _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 49
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_6\ninst✝²⁴ : CommRing A\ninst✝²³ : CommRing B\ninst✝²² : Algebra A B\nAₘ : Type u_9\nBₘ : Type u_10\ninst✝²¹ : CommRing Aₘ\ninst✝²⁰ : CommRing Bₘ\ninst✝¹⁹ : Algebra Aₘ Bₘ\ninst✝¹⁸ : Algebra A Aₘ\ninst✝¹⁷ : Algebra B Bₘ\ninst✝¹⁶ : Algebra A Bₘ\ninst✝¹⁵ : IsScalarTower ... | have : IsIntegralClosure Bₘ Aₘ L :=
IsIntegralClosure.of_isIntegrallyClosed _ _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 465,
"column": 2
} | {
"line": 467,
"column": 73
} | [
{
"pp": "A : Type u_1\nB : Type u_6\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsIntegrallyClosed A\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDomain B\ninst✝³ : IsIntegrallyClosed B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : IsTorsionFree A B\ninst✝ : FiniteDimensional (FractionRing A) (Fr... | rw [← (IsFractionRing.injective A (FractionRing A)).eq_iff,
← (IsFractionRing.injective B (FractionRing B)).eq_iff]
simp only [algebraMap_intNorm_fractionRing, map_zero, norm_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 465,
"column": 2
} | {
"line": 467,
"column": 73
} | [
{
"pp": "A : Type u_1\nB : Type u_6\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsIntegrallyClosed A\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDomain B\ninst✝³ : IsIntegrallyClosed B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : IsTorsionFree A B\ninst✝ : FiniteDimensional (FractionRing A) (Fr... | rw [← (IsFractionRing.injective A (FractionRing A)).eq_iff,
← (IsFractionRing.injective B (FractionRing B)).eq_iff]
simp only [algebraMap_intNorm_fractionRing, map_zero, norm_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 506,
"column": 2
} | {
"line": 507,
"column": 49
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_6\ninst✝²⁵ : CommRing A\ninst✝²⁴ : CommRing B\ninst✝²³ : Algebra A B\nAₘ : Type u_9\nBₘ : Type u_10\ninst✝²² : CommRing Aₘ\ninst✝²¹ : CommRing Bₘ\ninst✝²⁰ : Algebra Aₘ Bₘ\ninst✝¹⁹ : Algebra A Aₘ\ninst✝¹⁸ : Algebra B Bₘ\ninst✝¹⁷ : Algebra A Bₘ\ninst✝¹⁶ : IsScalarTower ... | have : IsIntegralClosure Bₘ Aₘ L :=
IsIntegralClosure.of_isIntegrallyClosed _ _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 36
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\n⊢ ∃ x ∉ k, IsSeparable (↥k) x",
"usedConstants": [
"DivisionRing.toRing",
"DivisionRing.isDomain",
"ExpChar.exists"
]
}
] | obtain ⟨p, hp⟩ := ExpChar.exists D | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 128,
"column": 77
} | {
"line": 128,
"column": 90
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : a ∉ k\nha₀ : a ≠ 0\nb : D\nhb1 : ((ad (↥k) D) a) b ≠ 0\nm : ℕ\nhm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0\n⊢ 0 b = 0",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CosetCover | {
"line": 206,
"column": 2
} | {
"line": 207,
"column": 64
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\nj : ι\nhj : j ∈ s\nhcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) = Set.univ\n⊢ ∃ k ∈ s, (H k).FiniteIndex",
"usedConstants": [
"Eq.mpr",
"... | · rw [Set.iUnion₂_congr fun i hi => by rw [(Finset.mem_filter.mp hi).right]] at hcovers'
exact ⟨j, hj, finiteIndex_of_leftCoset_cover_const hcovers'⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.CosetCover | {
"line": 201,
"column": 2
} | {
"line": 210,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\n⊢ ∃ k ∈ s, (H k).FiniteIndex",
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"Finset.coe_empty",
"instSMulOfMul",
"InvOneCla... | have ⟨j, hj⟩ : s.Nonempty := by
by_contra! rfl
rw [← Finset.set_biUnion_coe, Finset.coe_empty, Set.biUnion_empty] at hcovers
exact Set.empty_ne_univ hcovers
by_cases hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) = Set.univ
· rw [Set.iUnion₂_congr fun i hi => by rw [(Finset.mem_filter.mp hi)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CosetCover | {
"line": 201,
"column": 2
} | {
"line": 210,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\n⊢ ∃ k ∈ s, (H k).FiniteIndex",
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"Finset.coe_empty",
"instSMulOfMul",
"InvOneCla... | have ⟨j, hj⟩ : s.Nonempty := by
by_contra! rfl
rw [← Finset.set_biUnion_coe, Finset.coe_empty, Set.biUnion_empty] at hcovers
exact Set.empty_ne_univ hcovers
by_cases hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) = Set.univ
· rw [Set.iUnion₂_congr fun i hi => by rw [(Finset.mem_filter.mp hi)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.KummerExtension | {
"line": 365,
"column": 7
} | {
"line": 365,
"column": 71
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na✝ : K\nH : Irreducible (X ^ n - C a✝)\nL✝ : Type u_1\ninst✝⁵ : Field L✝\ninst✝⁴ : Algebra K L✝\ninst✝³ : IsSplittingField K L✝ (X ^ n - C a✝)\nα : L✝\nhα : α ^ n = (algebraMap K L✝) a✝\nhn : 0 < n\na : K\nL : Type ?u.229271\ninst... | by simpa [degree_X_pow_sub_C hn] using Nat.pos_iff_ne_zero.mp hn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 77,
"column": 11
} | {
"line": 77,
"column": 23
} | [
{
"pp": "case h.h\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nH : (mk K x).carrier = (mk K y).carrier\n⊢ mk K x = mk K y",
"usedConstants": [
"Membership.mem",
"Eq.mp",
"Iff",
"ConjRootClass.mk",
"_private.Mathlib.FieldTheory.M... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.PurelyInseparable.Exponent | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 52
} | [
{
"pp": "case a\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra K L\ninst✝⁶ : HasExponent K L\np : ℕ\ninst✝⁵ : ExpChar K p\ninst✝⁴ : Field F\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\ninst✝ : ExpChar F p\nn : ℕ\nhn : exponent K L ≤ ... | algebraMap_iterateFrobeniusAux K p hn (r • a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 375,
"column": 59
} | {
"line": 377,
"column": 59
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nA B : Subalgebra R S\nH : A.LinearDisjoint B\ninst✝ : Flat R ↥B\nι : Type u_1\na : ι → ↥A\nha : LinearIndependent R a\n⊢ LinearIndependent (↥B.op) (MulOpposite.op ∘ ⇑A.val ∘ a)",
"usedConstants": [
"Subalgebra... | by
have h := Submodule.LinearDisjoint.linearIndependent_left_of_flat H ha
rwa [mulLeftMap_ker_eq_bot_iff_linearIndependent_op] at h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 34
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝¹³ : Field F\ninst✝¹² : Field E\ninst✝¹¹ : Algebra F E\nA : IntermediateField F E\nL : Type w\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra F L\ninst✝⁸ : Algebra L E\ninst✝⁷ : IsScalarTower F L E\nH : A.LinearDisjoint (IsScalarTower.toAlgHom F L E).range\nK : Type u_1\ninst✝⁶ : Fiel... | rw [← AlgHom.range_comp] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 428,
"column": 7
} | {
"line": 428,
"column": 10
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nA B : IntermediateField F E\ninst✝ : Module.Finite F ↥A\nh₁ : A.LinearDisjoint ↥B\nh₂ : A ⊔ B = ⊤\nthis : finrank F ↥(A ⊔ B) = finrank F ↥A * finrank F ↥B\n⊢ finrank (↥A) E = finrank F ↥B",
"usedConstants": [
"L... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 674,
"column": 2
} | {
"line": 690,
"column": 59
} | [
{
"pp": "F : Type u\ninst✝⁴ : Field F\nA : Type v\ninst✝³ : Field A\nB : Type w\ninst✝² : Field B\ninst✝¹ : Algebra F A\ninst✝ : Algebra F B\nH :\n ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),\n fa.fieldRange.LinearDisjoint ↥fb.fieldRange\n⊢ IsField (A ⊗[... | obtain ⟨M, hM⟩ := Ideal.exists_maximal (A ⊗[F] B)
apply not_imp_not.1 (Ring.ne_bot_of_isMaximal_of_not_isField hM)
let K : Type (max v w) := A ⊗[F] B ⧸ M
letI : Field K := Ideal.Quotient.field _
let i := IsScalarTower.toAlgHom F (A ⊗[F] B) K
let fa := i.comp (Algebra.TensorProduct.includeLeft : A →ₐ[F] _)
l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 674,
"column": 2
} | {
"line": 690,
"column": 59
} | [
{
"pp": "F : Type u\ninst✝⁴ : Field F\nA : Type v\ninst✝³ : Field A\nB : Type w\ninst✝² : Field B\ninst✝¹ : Algebra F A\ninst✝ : Algebra F B\nH :\n ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),\n fa.fieldRange.LinearDisjoint ↥fb.fieldRange\n⊢ IsField (A ⊗[... | obtain ⟨M, hM⟩ := Ideal.exists_maximal (A ⊗[F] B)
apply not_imp_not.1 (Ring.ne_bot_of_isMaximal_of_not_isField hM)
let K : Type (max v w) := A ⊗[F] B ⧸ M
letI : Field K := Ideal.Quotient.field _
let i := IsScalarTower.toAlgHom F (A ⊗[F] B) K
let fa := i.comp (Algebra.TensorProduct.includeLeft : A →ₐ[F] _)
l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LinearDisjoint | {
"line": 798,
"column": 14
} | {
"line": 798,
"column": 36
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nA B : Subalgebra R S\ninst✝ : Algebra.IsIntegral R ↥A\nH : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B\nx y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)\nhxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((t... | LinearMap.rTensor_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 85
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (g E).natDegree ≤ (c E).num.natDegree",
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"_private.Mathlib.FieldTheory.RatFunc.Luroth.0.RatFunc.Luroth.generator_denom_dvd_c_num",
"_private.Mathlib.FieldTheor... | exact natDegree_le_of_dvd (generator_denom_dvd_c_num h) (num_ne_zero (c_ne_zero h)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 264,
"column": 31
} | {
"line": 264,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ m E ≤ (Bivariate.swap ((Φ E).sum fun n a ↦ (monomial n) a)).natDegree",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.algebraOfAlgebra",
... | sum_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 146,
"column": 2
} | {
"line": 170,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\np : P\n⊢ affineSpan ℝ {p, s.points i} = s.altitude i ↔\n p ≠ s.points i ∧\n p ∈ affineSpa... | rw [eq_iff_direction_eq_of_mem (mem_affineSpan ℝ (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
(s.mem_altitude _),
← vsub_right_mem_direction_iff_mem (mem_affineSpan ℝ (Set.mem_range_self i)) p,
direction_affineSpan, direction_affineSpan, direction_affineSpan]
constructor
· intro h
constructor
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 146,
"column": 2
} | {
"line": 170,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\np : P\n⊢ affineSpan ℝ {p, s.points i} = s.altitude i ↔\n p ≠ s.points i ∧\n p ∈ affineSpa... | rw [eq_iff_direction_eq_of_mem (mem_affineSpan ℝ (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
(s.mem_altitude _),
← vsub_right_mem_direction_iff_mem (mem_affineSpan ℝ (Set.mem_range_self i)) p,
direction_affineSpan, direction_affineSpan, direction_affineSpan]
constructor
· intro h
constructor
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 243,
"column": 8
} | {
"line": 243,
"column": 27
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₃ p₁ = dist p₃ p₂\nm : P := midpoint ℝ p₁ p₂\nh1 : p₃ -ᵥ p₁ = p₃ -ᵥ m - (p₁ -ᵥ m)\n⊢ p₃ -ᵥ p₂ = p₃ -ᵥ m + (p₁ -ᵥ m)",
"usedConstant... | left_vsub_midpoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 432,
"column": 12
} | {
"line": 432,
"column": 28
} | [
{
"pp": "case refine_2.inr.inr.inl.inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : Wbtw ℝ p₂ p₃ p₁\n⊢ Collinear ℝ {p₁, p₂, p₃}",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.... | Set.insert_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\nx : ↑↑s\n⊢ dist p ↑((or... | simp_rw [pow_two,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p x.property] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 14
} | [
{
"pp": "case neg.mpr.inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nH : x ≠ 0 ∧ 0 • (o.rotation ↑(π / 2)) x ≠ 0\n⊢ (∃ r, 0 < r ∧ 0 • (o.rotation ↑(π / 2)) x = r • (o.rotation ↑(π / 2)) x) ∨\n ∃ r, 0 < ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 14
} | [
{
"pp": "case neg.mpr.inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nH : x ≠ 0 ∧ 0 • (o.rotation ↑(π / 2)) x ≠ 0\n⊢ (∃ r, 0 < r ∧ 0 • (o.rotation ↑(π / 2)) x = r • (o.rotation ↑(π / 2)) x) ∨\n ∃ r, 0 < ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 14
} | [
{
"pp": "case neg.mpr.inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nH : x ≠ 0 ∧ 0 • (o.rotation ↑(π / 2)) x ≠ 0\n⊢ (∃ r, 0 < r ∧ 0 • (o.rotation ↑(π / 2)) x = r • (o.rotation ↑(π / 2)) x) ∨\n ∃ r, 0 < ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 436,
"column": 6
} | {
"line": 436,
"column": 14
} | [
{
"pp": "case h\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nr : ℝ\nH : x ≠ 0 ∧ r • (o.rotation ↑(π / 2)) x ≠ 0\nhr0 : r < 0\n⊢ 0 < -r ∧ r • (o.rotation ↑(π / 2)) x = - -r • (o.rotation ↑(π / 2)) x",
"usedCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 232,
"column": 11
} | {
"line": 232,
"column": 51
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh : 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₄ p₅ p₆\n⊢ Collinear ℝ {p₁, p₂, p₃} ↔ Collinear... | ← oangle_eq_zero_or_eq_pi_iff_collinear, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 194,
"column": 4
} | {
"line": 196,
"column": 57
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : o.oangle x y = ↑(π / 2)\nhs : (o.oangle x (x + y)).sign = 1\n⊢ ‖x‖ / Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x + y‖",
"usedConstants": [
... | InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 383,
"column": 12
} | {
"line": 383,
"column": 15
} | [
{
"pp": "case inl.inl.h\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆\nh1 : p₂ ≠ p₁\nh2 : p₂ ≠ p₃\nh3... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 385,
"column": 12
} | {
"line": 385,
"column": 15
} | [
{
"pp": "case inl.inr.h\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆\nh1 : p₂ ≠ p₁\nh2 : p₂ ≠ p₃\nh3... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 387,
"column": 12
} | {
"line": 387,
"column": 15
} | [
{
"pp": "case inr.inl.h\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆\nh1 : p₂ ≠ p₁\nh2 : p₂ ≠ p₃\nh3... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 389,
"column": 12
} | {
"line": 389,
"column": 15
} | [
{
"pp": "case inr.inr.h\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆\nh1 : p₂ ≠ p₁\nh2 : p₂ ≠ p₃\nh3... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 349,
"column": 4
} | {
"line": 350,
"column": 53
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : o.oangle x y = ↑(π / 2)\nhs : (o.oangle y (y - x)).sign = 1\n⊢ Real.cos (InnerProductGeometry.angle y (y - x)) * ‖y - x‖ = ‖y‖",
"usedConstants": [
... | InnerProductGeometry.cos_angle_sub_mul_norm_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 666,
"column": 58
} | {
"line": 666,
"column": 77
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ : P\nh : p₁ ≠ p₂\nr : ℝ\n⊢ ‖p₁ -ᵥ midpoint ℝ p₁ p₂ - r • (o.rotation ↑(π / 2)... | left_vsub_midpoint, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 542,
"column": 4
} | {
"line": 542,
"column": 19
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₃ p₁ p₂).sign = 1\n⊢ ↑(Real.arccos (dist p₁ p₂ /... | dist_comm p₁ p₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 704,
"column": 17
} | {
"line": 704,
"column": 19
} | [
{
"pp": "case refine_1\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃,... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 19
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₃ p₁ p₂).sign = 1\n⊢ ↑(Real.arcsin (dist p₃ p₂ /... | dist_comm p₁ p₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 704,
"column": 17
} | {
"line": 704,
"column": 19
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃,... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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