module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 39
} | {
"line": 541,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\nF : Set (Pattern A G)\nh_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}\np : Pattern A G\n⊢ IsClosed[... | [
"A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\nF : Set (Pattern A G)\nh_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}\np : Pattern A G\nhp : p ∈ F\n⊢ IsClosed[... | refine isClosed_iInter (fun hp => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 367,
"column": 7
} | {
"line": 377,
"column": 82
} | {
"line": 379,
"column": 0
} | [] | [] | ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν
≤ ∫ x in t, (μ.rnDeriv ν x).toReal ∂ν := by
refine setIntegral_mono_set ?_ ?_ (LE.le.eventuallyLE (subset_toMeasurable _ _))
· exact integrableOn_toReal_rnDeriv hμt
· exact ae_of_all _ (by simp)
_ = (withDensity ν (rnDeriv μ ν)).real t := setIntegral_toR... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Dynamics.TopologicalEntropy.NetEntropy | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 37
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case inr\nX : Type u_1\nU : SetRel X X\nT : X → X\nF : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nn : ℕ\nh✝ : netMaxcard T F U n < ⊤\ns : Finset X\ns_net : IsDynNetIn T F U n ↑s\ns_card : ↑s.card = netMaxcard T F U n\nh : ¬F ⊆ ⋃ y ∈ ↑s, ball y (dynEntourage T (U ○ U) n)\nx : X\nx_F : x ∈ F\nx_uncov : ... | [
"case inr\nX : Type u_1\nU : SetRel X X\nT : X → X\nF : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nn : ℕ\nh✝ : netMaxcard T F U n < ⊤\ns : Finset X\ns_net : IsDynNetIn T F U n ↑s\ns_card : ↑s.card = netMaxcard T F U n\nh : ¬F ⊆ ⋃ y ∈ ↑s, ball y (dynEntourage T (U ○ U) n)\nx : X\nx_F : x ∈ F\nx_uncov : ∀ x_1 ∈ s, x... | rw [← s.coe_insert x] at larger_net | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 541,
"column": 2
} | {
"line": 542,
"column": 71
} | {
"line": 543,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\ns : Set β\nhs : MeasurableSet s\n⊢ (map f (ν.withDensity (μ.rnDeriv ν))) s = ((map f ν).withDensity ((map f μ).rnDeriv (map f... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\ns : Set β\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in f ⁻¹' s, μ.rnDeriv ν a ∂ν = ∫⁻ (x : α) in f ⁻¹' s, (map f μ).rnDeriv (map f ν) (f x) ∂ν"... | rw [hf.map_apply, withDensity_apply _ (hf.measurable hs), withDensity_apply _ hs,
setLIntegral_map hs (Measure.measurable_rnDeriv _ _) hf.measurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Cardinality | {
"line": 56,
"column": 33
} | {
"line": 56,
"column": 63
} | {
"line": 57,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type v\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\n⊢ (sum fun x ↦ ℵ₀) = lift.{v, u} #R[X] * ℵ₀",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": ... | [] | by rw [sum_const, lift_aleph0] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Basic | {
"line": 462,
"column": 14
} | {
"line": 462,
"column": 74
} | {
"line": 464,
"column": 2
} | [
{
"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\nhnp : N ↪[L] P\nhmn : M ↪[L] N\n⊢ ∀ {n : ℕ} (f : L.Functions n) (x : Fin n → M), (⇑hnp ∘ ⇑hmn) (funMap f x) = funMap f ((⇑hnp ... | [] | by intros; simp only [Function.comp_apply, map_fun]; trivial | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Basic | {
"line": 636,
"column": 16
} | {
"line": 636,
"column": 76
} | {
"line": 638,
"column": 4
} | [
{
"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\nhnp : N ≃[L] P\nhmn : M ≃[L] N\n⊢ ∀ {n : ℕ} (f : L.Functions n) (x : Fin n → M), (⇑hnp ∘ ⇑hmn) (funMap f x) = funMap f ((⇑hnp ... | [] | by intros; simp only [Function.comp_apply, map_fun]; trivial | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Algebra.Field.CharP | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 12
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case inr\np : ℕ\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : CharP K p\nhp : p = 0\n⊢ K ⊨ if p = 0 then (fun q ↦ ∼(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {⊥}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NegZeroClass... | [
"case inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : CharP K 0\n⊢ K ⊨ if 0 = 0 then (fun q ↦ ∼(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime 0 then {eqZero 0} else {⊥}"
] | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.ModelTheory.Semantics | {
"line": 133,
"column": 47
} | {
"line": 136,
"column": 37
} | {
"line": 138,
"column": 0
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\ninst✝¹ : L.Structure M\nβ : Type v'\ninst✝ : L'.Structure M\nc : {n : ℕ} → L.Functions n → L'.Term (Fin n)\nhc : ∀ {n : ℕ} (g : L.Functions n) (y : Fin n → M), realize y g.term = realize y (c g)\nv : β → M\nx : L.Term β\n⊢ realize v (x.substFunc fun {n} ↦ c) = r... | [] | by
induction x with
| var => simp
| func f ts ih => simp [← ih, ← hc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Syntax | {
"line": 665,
"column": 13
} | {
"line": 665,
"column": 32
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case equal\nL : Language\nL' : Language\nα : Type u'\nn : ℕ\nL'' : Language\nφ : L' →ᴸ L''\nψ : L →ᴸ L'\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ (φ.comp ψ).onBoundedFormula (equal t₁✝ t₂✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (equal t₁✝ t₂✝)",
"ppTerm": "?equal",
"assigned": true,
"... | [] | simp [Term.bdEqual] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Syntax | {
"line": 665,
"column": 13
} | {
"line": 665,
"column": 32
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case equal\nL : Language\nL' : Language\nα : Type u'\nn : ℕ\nL'' : Language\nφ : L' →ᴸ L''\nψ : L →ᴸ L'\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ (φ.comp ψ).onBoundedFormula (equal t₁✝ t₂✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (equal t₁✝ t₂✝)",
"ppTerm": "?equal",
"assigned": true,
"... | [] | simp [Term.bdEqual] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Syntax | {
"line": 665,
"column": 13
} | {
"line": 665,
"column": 32
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case equal\nL : Language\nL' : Language\nα : Type u'\nn : ℕ\nL'' : Language\nφ : L' →ᴸ L''\nψ : L →ᴸ L'\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\n⊢ (φ.comp ψ).onBoundedFormula (equal t₁✝ t₂✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (equal t₁✝ t₂✝)",
"ppTerm": "?equal",
"assigned": true,
"... | [] | simp [Term.bdEqual] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 51
} | {
"line": 454,
"column": 4
} | [
{
"pp": "case rel\nL : 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),... | [
"case rel\nL : 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) = v... | rw [realize_restrictVarLeft v' (by simp [hv'])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Substructures | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 40
} | {
"line": 280,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nS : L.Substructure M := { carrier := range (Term.realize Subtype.val), fun_mem := ⋯ }\nS' : L.Substructure M\nhS' : S' ∈ {S | s ⊆ ↑S}\nt : L.Term { x // x ∈ s }\n⊢ Term.realize Subtype.val t ∈ S'",
"ppTerm": "?m.94",
"assigned": true,
... | [] | exact t.realize_mem _ fun i => hS' i.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.Definability | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 34
} | {
"line": 79,
"column": 2
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\ns : Set (α → M)\nφ : L.BoundedFormula (↑A ⊕ α) 0\nx✝ : α → M\nn✝ : ℕ\nt✝ : L.Term ((↑A ⊕ α) ⊕ Fin n✝)\nxs✝ : Fin n✝ → M\n⊢ Term.realize (Sum.elim x✝ xs✝) (Term.constantsVarsEquivLeft.symm t✝) =\n Term.realize (Sum.elim (Sum.eli... | [
"M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\ns : Set (α → M)\nφ : L.BoundedFormula (↑A ⊕ α) 0\nx✝ : α → M\nn✝ : ℕ\nt✝ : L.Term ((↑A ⊕ α) ⊕ Fin n✝)\nxs✝ : Fin n✝ → M\n⊢ Term.realize (Sum.elim (fun a ↦ ↑a) (Sum.elim x✝ xs✝) ∘ ⇑(Equiv.sumAssoc (↑A) α (Fin n✝))) t✝ =\n Term.realize (Sum.... | simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Definability | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 47
} | {
"line": 93,
"column": 47
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\nB : Set M\ns : Set (α → M)\nhAs : A.Definable L s\nhAB : A ⊆ B\n⊢ B.Definable L s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.definable_iff_empty_definable_with_params",
"F... | [
"M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\nB : Set M\ns : Set (α → M)\nhAs : ∅.Definable L[[↑A]] s\nhAB : A ⊆ B\n⊢ ∅.Definable L[[↑B]] s"
] | definable_iff_empty_definable_with_params | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Skolem | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 55
} | {
"line": 140,
"column": 2
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\n⊢ lift.{... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : lift.{max w w', ... | have h := mk_image_eq_lift _ s' Equiv.ulift.injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.Definability | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 24
} | {
"line": 267,
"column": 6
} | [
{
"pp": "case intro.intro.mp\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 (... | [
"case intro.intro.mp\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 (fun g ↦ g ∘ ... | refine ⟨y, ys, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Semantics | {
"line": 1007,
"column": 4
} | {
"line": 1007,
"column": 35
} | {
"line": 1007,
"column": 35
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nα : Type u'\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nφ : L.Formula α\nv : α → M\n⊢ BoundedFormula.Realize φ (⇑g ∘ v) default = BoundedFormula.Realize φ (⇑g ∘ v) (⇑g ∘ default)"... | [
"L : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nα : Type u'\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nφ : L.Formula α\nv : α → M\n⊢ BoundedFormula.Realize φ (⇑g ∘ v) default = BoundedFormula.Realize φ (⇑g ∘ v) default"
] | Unique.eq_default (g ∘ default) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 1013,
"column": 4
} | {
"line": 1013,
"column": 35
} | {
"line": 1013,
"column": 35
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nφ : L.Sentence\n⊢ Formula.Realize φ (⇑g ∘ default) ↔ Formula.Realize φ default",
"ppTerm": "?m.30",
"assigned": true,
"usedC... | [
"L : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nφ : L.Sentence\n⊢ Formula.Realize φ default ↔ Formula.Realize φ default"
] | Unique.eq_default (g ∘ default) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 1132,
"column": 6
} | {
"line": 1132,
"column": 45
} | {
"line": 1132,
"column": 46
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nT : L.Theory\nh : M ≅[L] N\n⊢ M ⊨ T ↔ N ⊨ T",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.completeTheory",
"congrArg",
"FirstOrder.Languag... | [
"L : Language\nM : Type w\nN : Type u_1\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nT : L.Theory\nh : M ≅[L] N\n⊢ T ⊆ L.completeTheory M ↔ N ⊨ T"
] | Theory.model_iff_subset_completeTheory, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 1132,
"column": 46
} | {
"line": 1132,
"column": 85
} | {
"line": 1133,
"column": 4
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nT : L.Theory\nh : M ≅[L] N\n⊢ T ⊆ L.completeTheory M ↔ N ⊨ T",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.completeTheory",
"congrArg",
"... | [
"L : Language\nM : Type w\nN : Type u_1\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nT : L.Theory\nh : M ≅[L] N\n⊢ T ⊆ L.completeTheory M ↔ T ⊆ L.completeTheory N"
] | Theory.model_iff_subset_completeTheory, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 103,
"column": 4
} | {
"line": 119,
"column": 31
} | {
"line": 119,
"column": 31
} | [
{
"pp": "L : Language\nT : L.Theory\nh : T.IsFinitelySatisfiable\n⊢ T.IsSatisfiable",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",
"Filter.Eventually.filter_mono",
"_private.Mathlib.ModelTheory.Satis... | [] | classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_subtype_subset).some.Carrier
let M' := Filter.Product (Ultrafilter.of (Filter.atTop : Filter (Finset T))) M
have h' : M' ⊨ T := by
refine ⟨fun φ hφ => ?_⟩
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.ModelTheory.Satisfiability | {
"line": 103,
"column": 4
} | {
"line": 119,
"column": 31
} | {
"line": 119,
"column": 31
} | [
{
"pp": "L : Language\nT : L.Theory\nh : T.IsFinitelySatisfiable\n⊢ T.IsSatisfiable",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",
"Filter.Eventually.filter_mono",
"_private.Mathlib.ModelTheory.Satis... | [] | classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_subtype_subset).some.Carrier
let M' := Filter.Product (Ultrafilter.of (Filter.atTop : Filter (Finset T))) M
have h' : M' ⊨ T := by
refine ⟨fun φ hφ => ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Satisfiability | {
"line": 103,
"column": 4
} | {
"line": 119,
"column": 31
} | {
"line": 119,
"column": 31
} | [
{
"pp": "L : Language\nT : L.Theory\nh : T.IsFinitelySatisfiable\n⊢ T.IsSatisfiable",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",
"Filter.Eventually.filter_mono",
"_private.Mathlib.ModelTheory.Satis... | [] | classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_subtype_subset).some.Carrier
let M' := Filter.Product (Ultrafilter.of (Filter.atTop : Filter (Finset T))) M
have h' : M' ⊨ T := by
refine ⟨fun φ hφ => ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 12
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case inr\np : ℕ\nhp : p = 0\n⊢ (Theory.ACF p).IsSatisfiable",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.ring",
"FirstOrder.Language.Theory.IsSatisfiable",
"FirstOrder.Language.Theory.ACF",
"instOfNatNat",
"Nat",
"Eq.ndrec... | [
"case inr\n⊢ (Theory.ACF 0).IsSatisfiable"
] | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.ModelTheory.Satisfiability | {
"line": 341,
"column": 70
} | {
"line": 345,
"column": 68
} | {
"line": 347,
"column": 0
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nφ : L.Formula α\nh : T ⊨ᵇ φ\nM : Type u_1\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Nonempty M\nv : α → M\n⊢ φ.Realize v",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"FirstOrder.Language.Sentence.Realize",
... | [] | by
rw [models_formula_iff_onTheory_models_equivSentence] at h
letI : (constantsOn α).Structure M := constantsOn.structure v
have : M ⊨ (L.lhomWithConstants α).onTheory T := (LHom.onTheory_model _ _).2 inferInstance
exact (Formula.realize_equivSentence _ _).1 (h.realize_sentence M) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeAlgebra | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 72
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : Set S\nh✝ : Subsingleton R\n⊢ Module.rank R ↥((FreeAlgebra.lift R) Subtype.val).range ≤ max #↑s ℵ₀",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
... | [
"case inr\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : Set S\nh✝ : Nontrivial R\n⊢ Module.rank R ↥((FreeAlgebra.lift R) Subtype.val).range ≤ max #↑s ℵ₀"
] | · rw [rank_subsingleton]; exact one_le_aleph0.trans (le_max_right _ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 13
} | {
"line": 123,
"column": 14
} | [
{
"pp": "n : ℕ\nh✝ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : b = 0 ∨ c = 0\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nh : ↑n = 0\n⊢ False",
"ppTerm": "?m.204",
"assigned": true,
"usedConstants": [
"False",
"Cardinal",
"CommSemiring.... | [
"case inl\nn : ℕ\nh✝¹ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nh : ↑n = 0\nh✝ : b = 0\n⊢ False",
"case inr\nn : ℕ\nh✝¹ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀... | cases hbc | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.FieldTheory.Differential.Basic | {
"line": 49,
"column": 74
} | {
"line": 56,
"column": 15
} | {
"line": 58,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\nn : ℕ\na : R\n⊢ logDeriv (a ^ n) = ↑n * logDeriv a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"Nat.recAux",
... | [] | by
induction n with
| zero => simp
| succ n h2 =>
obtain rfl | hb := eq_or_ne a 0
· simp
· rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 25
} | {
"line": 117,
"column": 4
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgeb... | [
"F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgebra F K'\nins... | apply_fun e.symm at h | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 52
} | {
"line": 153,
"column": 6
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [
"F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x : ι), (c x)′... | simp only [v₁, map_div₀, map_sum, map_natCast] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 65
} | {
"line": 38,
"column": 2
} | [
{
"pp": "case C\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (a : ZMod p), (frobenius (MvPolynomial σ (ZMod p)) p) (C a) = (expand p) (C a)",
"ppTerm": "?C",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSe... | [
"case add\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 q : MvPolynomial σ (ZMod p)),\n (frobenius (MvPolynomial σ (ZMod p)) p) p_1 = (expand p) p_1 →\n (frobenius (MvPolynomial σ (ZMod p)) p) q = (expand p) q →\n (frobenius (MvPolynomial σ (ZMod p)) p) (p_1 ... | · intro a; rw [expand_C, frobenius_def, ← C_pow, ZMod.pow_card] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 215,
"column": 54
} | {
"line": 219,
"column": 52
} | {
"line": 221,
"column": 0
} | [
{
"pp": "σ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\n⊢ (evalᵢ σ K).ker = ⊥",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids",
"Semiring.toModule",
... | [] | by
cases nonempty_fintype σ
refine (ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank ?_).mpr (range_evalᵢ σ K)
classical
rw [Module.finrank_fintype_fun_eq_card, finrank_R] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.Connected | {
"line": 25,
"column": 6
} | {
"line": 25,
"column": 34
} | {
"line": 25,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\n⊢ T1Space X",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Continuous",
"Equiv.instEquivLike",
"Specializes",
"Topol... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\n⊢ ∀ (x : X), IsClosed[inst✝] {x}"
] | ((t1Space_TFAE X).out 0 1 :) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.ExtendRight | {
"line": 91,
"column": 29
} | {
"line": 91,
"column": 55
} | {
"line": 91,
"column": 56
} | [
{
"pp": "case convert_2\nK : Type u_1\nL : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra K L\nF : IntermediateField K L\nM : Type u_3\ninst✝⁸ : Field M\ninst✝⁷ : Algebra K M\ninst✝⁶ : Algebra L M\ninst✝⁵ : IsScalarTower K L M\nR : Type u_4\nS : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : CommRing... | [
"case convert_2\nK : Type u_1\nL : Type u_2\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra K L\nF : IntermediateField K L\nM : Type u_3\ninst✝⁸ : Field M\ninst✝⁷ : Algebra K M\ninst✝⁶ : Algebra L M\ninst✝⁵ : IsScalarTower K L M\nR : Type u_4\nS : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² :... | RingHom.codRestrict_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 34
} | {
"line": 370,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℤ\n⊢ ↑x = mk K p (0, ↑x)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"NegZeroClass.toNeg",
"Int.... | [] | cases x <;> simp [natCast K p 0] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.PerfectClosure | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 34
} | {
"line": 370,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℤ\n⊢ ↑x = mk K p (0, ↑x)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"NegZeroClass.toNeg",
"Int.... | [] | cases x <;> simp [natCast K p 0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PerfectClosure | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 34
} | {
"line": 370,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℤ\n⊢ ↑x = mk K p (0, ↑x)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"NegZeroClass.toNeg",
"Int.... | [] | cases x <;> simp [natCast K p 0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PerfectClosure | {
"line": 432,
"column": 8
} | {
"line": 432,
"column": 25
} | {
"line": 433,
"column": 8
} | [
{
"pp": "K : Type u\ninst✝⁵ : CommRing K\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP K p\nL : Type v\ninst✝² : CommSemiring L\ninst✝¹ : CharP L p\ninst✝ : PerfectRing L p\nf : K →+* L\ne : PerfectClosure K p\n⊢ ∀ (x y : ℕ × K), R K p x y → (⇑(frobeniusEquiv L p).symm)^[x.1] (f x.2) = (⇑(frobeniusEquiv L... | [
"K : Type u\ninst✝⁵ : CommRing K\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP K p\nL : Type v\ninst✝² : CommSemiring L\ninst✝¹ : CharP L p\ninst✝ : PerfectRing L p\nf : K →+* L\ne : PerfectClosure K p\nn : ℕ\nx : K\n⊢ (⇑(frobeniusEquiv L p).symm)^[(n, x).1] (f (n, x).2) =\n (⇑(frobeniusEquiv L p).symm)^[(... | rintro - - ⟨n, x⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 228,
"column": 15
} | {
"line": 228,
"column": 54
} | {
"line": 228,
"column": 55
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝² : IsPRadical i p\ninst✝¹ : ExpChar L p\ninst✝ : PerfectRing L p\nx : L\n⊢ (iterateFrobeniusEquiv L p (Classical.choose ⋯).1).symm (i (Classical.choose ⋯).2) = x",
"ppTerm": "?m.39",
"assigne... | [
"K : Type u_1\nL : Type u_2\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝² : IsPRadical i p\ninst✝¹ : ExpChar L p\ninst✝ : PerfectRing L p\nx : L\n⊢ (iterateFrobeniusEquiv L p (Classical.choose ⋯).1).symm (x ^ p ^ (Classical.choose ⋯).1) = x"
] | Classical.choose_spec (lift_aux i p x), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.CardinalEmb | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 57
} | {
"line": 282,
"column": 2
} | [
{
"pp": "case top\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\nhi✝ : IsSuccPrelimit ⊤\nhi : ¬Nonempty ↑(Iio ⊤)\n⊢ filtration ⊤ = ⊥",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
... | [] | have := mk_ne_zero_iff.mp (rank_pos.trans_eq (mk_ord_toType <| Module.rank F E).symm).ne'
rw [← range_coe] at hi; exact (hi inferInstance).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.CardinalEmb | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 57
} | {
"line": 282,
"column": 2
} | [
{
"pp": "case top\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\nhi✝ : IsSuccPrelimit ⊤\nhi : ¬Nonempty ↑(Iio ⊤)\n⊢ filtration ⊤ = ⊥",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
... | [] | have := mk_ne_zero_iff.mp (rank_pos.trans_eq (mk_ord_toType <| Module.rank F E).symm).ne'
rw [← range_coe] at hi; exact (hi inferInstance).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CosetCover | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 39
} | {
"line": 109,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\ns : Finset ι\nH : Subgroup G\ng : ι → G\nhcovers : ⋃ i ∈ s, g i • ↑H = Set.univ\nhind : H.index = s.card\nh : H.index = 0\n⊢ False",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Eq.mp",
"i... | [
"G : Type u_1\ninst✝ : Group G\nι : Type u_2\ns : Finset ι\nH : Subgroup G\ng : ι → G\nhcovers : ⋃ i ∈ s, g i • ↑H = Set.univ\nhind : H.index = s.card\nh : s = ∅\n⊢ False"
] | rw [hind, Finset.card_eq_zero] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 185,
"column": 59
} | {
"line": 200,
"column": 99
} | {
"line": 202,
"column": 0
} | [
{
"pp": "L : Type u_2\nD : Type u_3\ninst✝⁴ : Field L\ninst✝³ : DivisionRing D\ninst✝² : Algebra L D\ninst✝¹ : Algebra.IsAlgebraic L D\ninst✝ : IsCentral L D\nhneq : ⊥ ≠ ⊤\n⊢ ∃ x ∉ ⊥, IsSeparable L x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"Subalgebra.inst... | [] | by
have hcenter : Subalgebra.center L D = ⊥ := le_bot_iff.mp IsCentral.out
have ntrivial : Subring.center D ≠ ⊤ :=
congr(Subalgebra.toSubring $hcenter).trans_ne (Subalgebra.toSubring_injective.ne hneq)
set φ := Subalgebra.equivOfEq (⊥ : Subalgebra L D) (.center L D) hcenter.symm
set equiv : L ≃+* (center D)... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CosetCover | {
"line": 380,
"column": 97
} | {
"line": 386,
"column": 26
} | {
"line": 388,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : Infinite k\ninst✝¹ : AddCommGroup E\ninst✝ : Module k E\ns : Finset (Subspace k E)\nhs : ⊤ ∉ s\n⊢ ⋃ p ∈ s, ↑p ≠ Set.univ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"Submod... | [] | by
intro hcovers
have ⟨p, hp, hfi⟩ := Submodule.exists_finiteIndex_of_cover hcovers
have : Finite (E ⧸ p) := AddSubgroup.finite_quotient_of_finiteIndex
have : Nontrivial (E ⧸ p) := Submodule.Quotient.nontrivial_iff.mpr (ne_of_mem_of_not_mem hp hs)
have : Infinite (E ⧸ p) := Module.Free.infinite k (E ⧸ p)
ex... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.KummerExtension | {
"line": 326,
"column": 73
} | {
"line": 326,
"column": 94
} | {
"line": 326,
"column": 94
} | [
{
"pp": "case adjoin_rootSet'\nK : Type u\ninst✝ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nthis✝¹ : Fact (Irreducible (X ^ n - C a))\nthis✝ : Algebra K K[n√a] := inferInstance\nthis : n ≠ 0\n⊢ eval₂ (of (X ^ n - C a)) (root (X ^ n - C a)) (X ^ n - C a) = 0",
... | [
"case adjoin_rootSet'\nK : Type u\ninst✝ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nthis✝¹ : Fact (Irreducible (X ^ n - C a))\nthis✝ : Algebra K K[n√a] := inferInstance\nthis : n ≠ 0\n⊢ 0 = 0"
] | AdjoinRoot.eval₂_root | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 396,
"column": 83
} | {
"line": 403,
"column": 5
} | {
"line": 405,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\ninst✝ : NeZero n\nσ : Gal(L/K)\n⊢ σ (rootOfSplitsXPowSubC ⋯ a L) = (autEquivRootsOfUnity hζ H... | [] | by
obtain ⟨η, rfl⟩ := (autEquivRootsOfUnity hζ H L).symm.surjective σ
rw [MulEquiv.apply_symm_apply, autEquivRootsOfUnity]
simp only [MulEquiv.symm_trans_apply, AlgEquiv.autCongr_symm, AlgEquiv.symm_symm,
MulEquiv.symm_symm, AlgEquiv.autCongr_apply, AlgEquiv.trans_apply,
adjoinRootXPowSubCEquiv_symm_eq_ro... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Relrank | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 60
} | {
"line": 252,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.FieldTheory.Relrank | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 60
} | {
"line": 252,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 60
} | {
"line": 252,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 147,
"column": 34
} | {
"line": 147,
"column": 41
} | {
"line": 147,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ¬∃ c, f = C c\ne : K[X] ≃ₐ[K] ↥K[f] := Polynomial.algEquivOfTranscendental K f ⋯\nφ : K[X][X] :=\n Polynomial.map (algebraMap K K[X]) f.num - Polynomial.C Polynomial.X * Polynomial.map (algebraMap K K[X]) f.denom\nφ_map : (mapEquiv e.toRingEquiv) φ = f.min... | [
"K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ¬∃ c, f = C c\ne : K[X] ≃ₐ[K] ↥K[f] := Polynomial.algEquivOfTranscendental K f ⋯\nφ : K[X][X] :=\n Polynomial.map (algebraMap K K[X]) f.num - Polynomial.C Polynomial.X * Polynomial.map (algebraMap K K[X]) f.denom\nφ_map : (mapEquiv e.toRingEquiv) φ = f.minpolyX ↥K[f]\... | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Relrank | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 60
} | {
"line": 477,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B C : IntermediateField F E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.FieldTheory.Relrank | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 60
} | {
"line": 477,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B C : IntermediateField F E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 60
} | {
"line": 477,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B C : IntermediateField F E\n⊢ A.relfinrank (B ⊓ C) * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_inf_mul_relrank A B C)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 74
} | {
"line": 184,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nf : K⟮X⟯\nhf₁ : f ∈ E\nhf₂ : f ∉ ⊥\n⊢ IsAlgebraic (↥E) X",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"IntermediateField.instPartialOrder",
"False",
"CommSemiring.toSemiring",
... | [] | exact IsAlgebraic.tower_top_of_subalgebra_le (adjoin_simple_le_iff.mpr hf₁) <|
f.isAlgebraic_adjoin_simple_X (by rintro ⟨c, rfl⟩; exact hf₂ ⟨c, rfl⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 39
} | {
"line": 333,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (θ E).natDegree ≤ m E",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"HEq.refl",
"HSub.hSub",
"_private.Mathlib.FieldTheory.RatFunc... | [
"case e'_4.e'_3\nK : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (f E).natDegree = (Polynomial.C (g E) * Polynomial.map Polynomial.C (f E)).natDegree",
"case e'_4.e'_4\nK : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (g E).natDegree = (Polynomial.C (f E) * Polyn... | convert! natDegree_sub_le _ _ using 3 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 350,
"column": 51
} | {
"line": 350,
"column": 68
} | {
"line": 350,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ Polynomial.C ((algebraMap K[X] K⟮X⟯) (g E)) *\n (Polynomial.map ((algebraMap (↥E) K⟮X⟯).comp (algebraMap ↥K⟮generator E⟯ ↥E))\n (Polynomial.map (algebraMap K ↥K⟮generator E⟯) (generator E).num) -\n Polynomial... | [
"K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ Polynomial.C ((algebraMap K[X] K⟮X⟯) (g E)) *\n (Polynomial.map ((algebraMap (↥E) K⟮X⟯).comp (algebraMap ↥K⟮generator E⟯ ↥E))\n (Polynomial.map (algebraMap K ↥K⟮generator E⟯) (generator E).num) -\n Polynomial.C\n ... | RingHom.coe_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 131,
"column": 36
} | {
"line": 131,
"column": 53
} | {
"line": 131,
"column": 54
} | [
{
"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)\nh :\n finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +\n finrank ℝ ↥((vectorSpan ℝ (s.points ... | [
"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)\nh :\n finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +\n finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 50
} | {
"line": 229,
"column": 4
} | [
{
"pp": "case mpr\n𝕜 : 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 q : P\nhqs : q ∈ s... | [
"case mpr\n𝕜 : 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 q : P\nhqs : q ∈ s\nhpq : p -ᵥ... | simp only [Set.mem_inter_iff, SetLike.mem_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 332,
"column": 4
} | {
"line": 358,
"column": 68
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case a.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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\nr : ℝ\nhr : r ≠ 0\nh : s.points j -ᵥ s.altitudeFoot j = r • (... | [] | · rw [SetLike.mem_coe]
have hk : ∃ k, k ≠ i ∧ k ≠ j := Fin.exists_ne_and_ne_of_two_lt i j
(by linarith only [Nat.AtLeastTwo.one_lt (n := n)])
have hs : vectorSpan ℝ (Set.range s.points) =
vectorSpan ℝ (Set.range (s.faceOpposite i).points) ⊔
vectorSpan ℝ (Set.range (s.faceOpposi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 311,
"column": 60
} | {
"line": 311,
"column": 97
} | {
"line": 312,
"column": 4
} | [
{
"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 -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ o.oangle (p₁ -ᵥ p₂) (p₁ -ᵥ p₂ - (p₁ -ᵥ p₃)) = o.oangl... | [
"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 -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ o.oangle (p₁ -ᵥ p₂) (p₁ -ᵥ p₂ - (p₁ -ᵥ p₃)) = o.oangle (p₁ -ᵥ p₃ ... | ← vsub_sub_vsub_cancel_left p₂ p₃ p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 33
} | {
"line": 304,
"column": 34
} | [
{
"pp": "V : 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 y : V\nr : ℝ\nhr : r ≠ 0\n⊢ 2 • o.oangle (r • x) y = 2 • o.oangle x y",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"InnerProductSpac... | [
"case 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 y : V\nr : ℝ\nhr : r ≠ 0\nh : r < 0\n⊢ 2 • o.oangle (r • x) y = 2 • o.oangle x y",
"case inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpa... | rcases hr.lt_or_gt with (h | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 33
} | {
"line": 311,
"column": 34
} | [
{
"pp": "V : 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 y : V\nr : ℝ\nhr : r ≠ 0\n⊢ 2 • o.oangle x (r • y) = 2 • o.oangle x y",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"InnerProductSpac... | [
"case 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 y : V\nr : ℝ\nhr : r ≠ 0\nh : r < 0\n⊢ 2 • o.oangle x (r • y) = 2 • o.oangle x y",
"case inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpa... | rcases hr.lt_or_gt with (h | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 668,
"column": 2
} | {
"line": 669,
"column": 43
} | {
"line": 671,
"column": 0
} | [
{
"pp": "case neg\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 y : V\nh : ¬0 ≤ (↑(InnerProductGeometry.angle x y)).sign\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : o.oangle x y = -↑(InnerProductGeometry.angle x y)\n⊢ False",
... | [] | exact h (Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi (InnerProductGeometry.angle_nonneg _ _)
(InnerProductGeometry.angle_le_pi _ _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 680,
"column": 2
} | {
"line": 681,
"column": 43
} | {
"line": 683,
"column": 0
} | [
{
"pp": "case neg\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 y : V\nh : ¬0 ≤ (↑(InnerProductGeometry.angle x y)).sign\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : o.oangle x y = ↑(InnerProductGeometry.angle x y)\n⊢ False",
... | [] | exact h (Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi (InnerProductGeometry.angle_nonneg _ _)
(InnerProductGeometry.angle_le_pi _ _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 697,
"column": 2
} | {
"line": 698,
"column": 22
} | {
"line": 699,
"column": 2
} | [
{
"pp": "case pos\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 y : V\nhx : x = 0\n⊢ o.oangle x y = ↑π ↔ InnerProductGeometry.angle x y = π",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [
"case neg\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 y : V\nhx : ¬x = 0\n⊢ o.oangle x y = ↑π ↔ InnerProductGeometry.angle x y = π"
] | · simp [hx, Real.Angle.pi_ne_zero.symm, div_eq_mul_inv,
Real.pi_ne_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 48
} | {
"line": 109,
"column": 0
} | [
{
"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\nv w : V\nr : ℝ\nleft✝ : r > 0\nright✝ : r • v = w\np q : P\n⊢ ((signedDist v) p) q = ((signedDist w) p) q",
"ppTerm": "?m.78",
"assigned": true,
"u... | [] | simpa [*] using (signedDist_smul v p q r).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 934,
"column": 4
} | {
"line": 936,
"column": 74
} | {
"line": 937,
"column": 2
} | [
{
"pp": "case pos\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 y : V\nr₁ r₂ r₃ r₄ : ℝ\nhr₁ : r₁ = 0\n⊢ (o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign = SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign",
... | [] | rw [hr₁, zero_smul, zero_mul, zero_add, zero_sub, Left.sign_neg,
oangle_sign_smul_left, add_comm, oangle_sign_smul_add_smul_right, oangle_rev,
Real.Angle.sign_neg, sign_mul, mul_neg, mul_neg, neg_mul, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 934,
"column": 4
} | {
"line": 936,
"column": 74
} | {
"line": 937,
"column": 2
} | [
{
"pp": "case pos\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 y : V\nr₁ r₂ r₃ r₄ : ℝ\nhr₁ : r₁ = 0\n⊢ (o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign = SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign",
... | [] | rw [hr₁, zero_smul, zero_mul, zero_add, zero_sub, Left.sign_neg,
oangle_sign_smul_left, add_comm, oangle_sign_smul_add_smul_right, oangle_rev,
Real.Angle.sign_neg, sign_mul, mul_neg, mul_neg, neg_mul, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 934,
"column": 4
} | {
"line": 936,
"column": 74
} | {
"line": 937,
"column": 2
} | [
{
"pp": "case pos\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 y : V\nr₁ r₂ r₃ r₄ : ℝ\nhr₁ : r₁ = 0\n⊢ (o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign = SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign",
... | [] | rw [hr₁, zero_smul, zero_mul, zero_add, zero_sub, Left.sign_neg,
oangle_sign_smul_left, add_comm, oangle_sign_smul_add_smul_right, oangle_rev,
Real.Angle.sign_neg, sign_mul, mul_neg, mul_neg, neg_mul, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 28
} | {
"line": 208,
"column": 0
} | [
{
"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\np q : P\n⊢ ((signedDist (q -ᵥ p)) p) q = dist p q",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"signedDist_vsub_self"
],
... | [] | apply signedDist_vsub_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Euclidean.Basic | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 6
} | {
"line": 99,
"column": 0
} | [
{
"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\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ = ⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫",
"ppTerm": "?m.71... | [] | rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Basic | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 6
} | {
"line": 99,
"column": 0
} | [
{
"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\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ = ⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫",
"ppTerm": "?m.71... | [] | rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 91
} | {
"line": 137,
"column": 2
} | [
{
"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\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (... | [
"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\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (c -ᵥ a)\nhpc... | have hpc : ⟪p -ᵥ a, c -ᵥ a⟫ = 0 := by simpa [ht0.ne', hb, inner_smul_right] using h_inner | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 87
} | {
"line": 144,
"column": 0
} | [
{
"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\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (... | [] | simpa only [Real.sqrt_sq dist_nonneg] using Real.sqrt_lt_sqrt (sq_nonneg _) h_sq_ineq | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.Basic | {
"line": 148,
"column": 35
} | {
"line": 148,
"column": 52
} | {
"line": 148,
"column": 53
} | [
{
"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\ns : AffineSubspace ℝ P\ninst✝ : FiniteDimensional ℝ ↥s.direction\nhd : finrank ℝ ↥s.direction = 2\nc₁ c₂ p₁ p₂ p : P\nhc₁s : c₁ ∈ s\nhc₂s : c₂ ... | [
"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\ns : AffineSubspace ℝ P\ninst✝ : FiniteDimensional ℝ ↥s.direction\nhd : finrank ℝ ↥s.direction = 2\nc₁ c₂ p₁ p₂ p : P\nhc₁s : c₁ ∈ s\nhc₂s : c₂ ∈ s\nhp₁s : ... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 521,
"column": 6
} | {
"line": 521,
"column": 50
} | {
"line": 521,
"column": 51
} | [
{
"pp": "case neg\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\ns : Sphere P\np₁ p₂ : P\nhp₁ : dist p₁ s.center = s.radius\nhp₂ : dist p₂ s.center ≤ s.radius\nh : ¬p₁ = p₂\n⊢ 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫",
"... | [
"case neg\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\ns : Sphere P\np₁ p₂ : P\nhp₁ : dist p₁ s.center = s.radius\nhp₂ : dist p₂ s.center ≤ s.radius\nh : ¬p₁ = p₂\n⊢ 0 < ⟪p₁ -ᵥ s.center - (p₂ -ᵥ s.center), p₁ -ᵥ s.ce... | ← vsub_sub_vsub_cancel_right p₁ p₂ s.center, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 95,
"column": 2
} | {
"line": 98,
"column": 10
} | {
"line": 100,
"column": 0
} | [
{
"pp": "case refine_2\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\ns : Sphere P\nas : AffineSubspace ℝ P\n⊢ s.radius = 0 ∧ s.center ∈ as → s.IsTangentAt s.center as",
"ppTerm": "?refine_2",
"assigned": t... | [] | · rintro ⟨hr, hm⟩
refine ⟨?_, hm, ?_⟩
· rw [center_mem_iff, hr]
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 179,
"column": 2
} | {
"line": 187,
"column": 54
} | {
"line": 189,
"column": 0
} | [
{
"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\ns : Sphere P\nas : AffineSubspace ℝ P\nh : s.IsTangent as\n⊢ Metric.infDist s.center ↑as = s.radius",
"ppTerm": "?m.24",
"assigned": true,
"usedCon... | [] | obtain ⟨p, h⟩ := h
refine le_antisymm ?_ ?_
· convert! Metric.infDist_le_dist_of_mem h.mem_space
rw [mem_sphere'.1 h.mem_sphere]
· rw [Metric.infDist_eq_iInf]
have : Nonempty as := ⟨⟨p, h.mem_space⟩⟩
refine le_ciInf fun x ↦ ?_
rw [dist_comm]
exact h.isTangent.radius_le_dist_center x.property | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 179,
"column": 2
} | {
"line": 187,
"column": 54
} | {
"line": 189,
"column": 0
} | [
{
"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\ns : Sphere P\nas : AffineSubspace ℝ P\nh : s.IsTangent as\n⊢ Metric.infDist s.center ↑as = s.radius",
"ppTerm": "?m.24",
"assigned": true,
"usedCon... | [] | obtain ⟨p, h⟩ := h
refine le_antisymm ?_ ?_
· convert! Metric.infDist_le_dist_of_mem h.mem_space
rw [mem_sphere'.1 h.mem_sphere]
· rw [Metric.infDist_eq_iInf]
have : Nonempty as := ⟨⟨p, h.mem_space⟩⟩
refine le_ciInf fun x ↦ ?_
rw [dist_comm]
exact h.isTangent.radius_le_dist_center x.property | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 373,
"column": 4
} | {
"line": 375,
"column": 30
} | {
"line": 376,
"column": 4
} | [
{
"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\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center < s.radius\nhpc : p ≠ s.center\nv : ↥(s.orthRadius p).direction\nhv : ∀ (w : ↥(s... | [
"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\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center < s.radius\nhpc : p ≠ s.center\nv : ↥(s.orthRadius p).direction\nhv : ∀ (w : ↥(s.orthRadius ... | have hvp : 0 < √(s.radius ^ 2 - dist p s.center ^ 2) := by
rw [Real.sqrt_pos, sub_pos, sq_lt_sq, abs_of_nonneg dist_nonneg]
exact lt_abs.2 (.inl hp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 51,
"column": 49
} | {
"line": 51,
"column": 88
} | {
"line": 53,
"column": 0
} | [
{
"pp": "V : 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 y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ 2 • o.oangle (x - y) (x - z) = 2 • o.oangle (y - x) ... | [] | rw [← oangle_neg_neg, neg_sub, neg_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 51,
"column": 49
} | {
"line": 51,
"column": 88
} | {
"line": 53,
"column": 0
} | [
{
"pp": "V : 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 y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ 2 • o.oangle (x - y) (x - z) = 2 • o.oangle (y - x) ... | [] | rw [← oangle_neg_neg, neg_sub, neg_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 51,
"column": 49
} | {
"line": 51,
"column": 88
} | {
"line": 53,
"column": 0
} | [
{
"pp": "V : 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 y z : V\nhxyne : x ≠ y\nhxzne : x ≠ z\nhxy : ‖x‖ = ‖y‖\nhxz : ‖x‖ = ‖z‖\nhy : y ≠ 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ 2 • o.oangle (x - y) (x - z) = 2 • o.oangle (y - x) ... | [] | rw [← oangle_neg_neg, neg_sub, neg_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 9
} | {
"line": 95,
"column": 2
} | [
{
"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\ns : Sphere P\nhd : s.IsDiameter p₁ p₃\no : P := s.center\nh_center : o = midpoint ℝ p₁ p₃\nh_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)\n⊢ -⟪p₃ -ᵥ o, p₃ -ᵥ o⟫_ℝ ... | [
"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\ns : Sphere P\nhd : s.IsDiameter p₁ p₃\no : P := s.center\nh_center : o = midpoint ℝ p₁ p₃\nh_opp : p₁ -ᵥ o = -(p₃ -ᵥ o)\n⊢ -⟪p₃ -ᵥ o, p₃ -ᵥ o⟫_ℝ + ⟪o -ᵥ p₂, ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 189,
"column": 22
} | {
"line": 189,
"column": 66
} | {
"line": 189,
"column": 67
} | [
{
"pp": "V : Type u_3\nP : Type u_4\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)\ns : Sphere P\np₁ p₂ p₃ p₄ : P\nhp₁ : ‖p₁ -ᵥ s.center‖ = s.radius\nhp₂ : ‖p₂ -ᵥ s.center‖ = s.radiu... | [
"V : Type u_3\nP : Type u_4\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)\ns : Sphere P\np₁ p₂ p₃ p₄ : P\nhp₁ : ‖p₁ -ᵥ s.center‖ = s.radius\nhp₂ : ‖p₂ -ᵥ s.center‖ = s.radius\nhp₃ : ‖p₃... | ← vsub_sub_vsub_cancel_right p₁ p₂ s.center, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 263,
"column": 11
} | {
"line": 263,
"column": 18
} | {
"line": 263,
"column": 18
} | [
{
"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)\n⊢ (s.height i)⁻¹ -\n ∑ j with j ≠ i,\n -(⟪s.points i -ᵥ s.altitudeFoot i, s.points j ... | [
"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)\n⊢ (s.height i)⁻¹ -\n ∑ x with x ≠ i,\n -(⟪s.points i -ᵥ s.altitudeFoot i, s.points x -ᵥ s.altitud... | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 302,
"column": 8
} | {
"line": 302,
"column": 30
} | {
"line": 302,
"column": 30
} | [
{
"pp": "case a\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nh : j ∈ {k | k ≠ i}\n⊢ s.excenterWeightsUnnorm {i} j = (s.height... | [
"case a\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nh : j ≠ i\n⊢ s.excenterWeightsUnnorm {i} j = (s.height j)⁻¹"
] | Finset.mem_filter_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 9
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case inr.inr.inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhy : y ≠ 0\nhx : x ≠ 0\nhxy : x ≠ y\nh_sin : ∀ (x y : V), x ≠ 0 → y ≠ 0 → Real.sin (angle x y) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) / (‖x‖ * ‖y‖)\nhsub : x - y ≠ 0\n⊢ √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ ^ 2)... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 532,
"column": 57
} | {
"line": 546,
"column": 68
} | {
"line": 548,
"column": 0
} | [
{
"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\n⊢ s.incenter ∈ s.interior",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
have h := s.excenterExists_empty.sum_excenterWeights_eq_one
rw [incenter_eq_affineCombination, s.affineCombination_mem_interior_iff h]
intro i
refine ⟨s.excenterWeights_empty_pos i, ?_⟩
by_contra! hp
obtain ⟨j, hj⟩ := exists_ne i
rw [← Finset.sum_add_sum_compl {j, i}, Finset.sum_pair hj] at h
revert ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 23
} | {
"line": 305,
"column": 4
} | [
{
"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\ns : Simplex ℝ P 1\n⊢ Set.univ.Pairwise fun i j ↦\n dist (s.points i) (Finset.centroid ℝ univ s.points) = dist (s.points j) (Finset.centroid ℝ univ s.points)... | [
"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\ns : Simplex ℝ P 1\ni : Fin 2\nhi : i ∈ Set.univ\nj : Fin 2\nhj : j ∈ Set.univ\nhij : i ≠ j\n⊢ dist (s.points i) (Finset.centroid ℝ univ s.points) = dist (s.points j) (Fins... | intro i hi j hj hij | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 11
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nhxy : x ≠ -y\nthis : x + y ≠ 0\n⊢ √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) * (⟪x, x⟫ + ⟪x, y⟫ + (⟪x, y⟫ + ⟪y, y⟫)) =\n √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) * (⟪x, y⟫ + ⟪y, y⟫) + (⟪x, x⟫... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 491,
"column": 2
} | {
"line": 493,
"column": 8
} | {
"line": 495,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ ∑ j, pointWeightsWithCircumcenter i j = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Finset.univ",
"Finset.sum_ite_eq'",
"Real.instZero",
"congrArg",
"HEq.refl",
"Finse... | [] | convert! sum_ite_eq' univ (pointIndex i) (Function.const _ (1 : ℝ)) with j
· cases j <;> simp [pointWeightsWithCircumcenter]
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 491,
"column": 2
} | {
"line": 493,
"column": 8
} | {
"line": 495,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ ∑ j, pointWeightsWithCircumcenter i j = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Finset.univ",
"Finset.sum_ite_eq'",
"Real.instZero",
"congrArg",
"HEq.refl",
"Finse... | [] | convert! sum_ite_eq' univ (pointIndex i) (Function.const _ (1 : ℝ)) with j
· cases j <;> simp [pointWeightsWithCircumcenter]
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 34
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhz : ‖z‖ = 1\nH : angle x y + angle y z ≤ π\n⊢ Real.cos (angle x y + angle y z) ≤ Real.cos (angle x z)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhz : ‖z‖ = 1\nH : angle x y + angle y z ≤ π\nH1 : ⟪x, z⟫ = ⟪x, z⟫\n⊢ Real.cos (angle x y + angle y z) ≤ Real.cos (angle x z)"
] | have H1 : ⟪x, z⟫ = ⟪x, z⟫ := rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 598,
"column": 34
} | {
"line": 598,
"column": 73
} | {
"line": 599,
"column": 4
} | [
{
"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\nn : ℕ\ns : Simplex ℝ P n\ni₁ i₂ : Fin (n + 1)\nh : i₁ ≠ i₂\nhc : #{i₁, i₂} = 2\nW : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})\nh_faces : ↑((ort... | [
"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\nn : ℕ\ns : Simplex ℝ P n\ni₁ i₂ : Fin (n + 1)\nh : i₁ ≠ i₂\nhc : #{i₁, i₂} = 2\nW : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})\nh_faces : ↑((orthogonalProje... | s.orthogonalProjection_circumcenter hc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 921,
"column": 4
} | {
"line": 921,
"column": 21
} | {
"line": 921,
"column": 22
} | [
{
"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\nhf : Fact (Module.finrank ℝ V = n)\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ F... | [
"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\nhf : Fact (Module.finrank ℝ V = n)\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ n - 1 + 1 = M... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 49
} | {
"line": 401,
"column": 4
} | [
{
"pp": "case neg\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\na b c : P\nhbc : ¬b = c\nm : P := midpoint ℝ b c\n⊢ dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2)",
"ppTer... | [
"case neg\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\na b c : P\nhbc : ¬b = c\nm : P := midpoint ℝ b c\nthis : dist b c ≠ 0\n⊢ dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2)"
] | have : dist b c ≠ 0 := (dist_pos.mpr hbc).ne' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 423,
"column": 54
} | {
"line": 423,
"column": 64
} | {
"line": 423,
"column": 64
} | [
{
"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\na b c a' b' c' : P\nr : ℝ\nh : ∠ a' b' c' = ∠ a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2... | [
"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\na b c a' b' c' : P\nr : ℝ\nh : ∠ a' b' c' = ∠ a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab₁ : a =... | mul_zero r | Lean.Elab.Tactic.evalRewriteSeq | null |
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