module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 584,
"column": 2
} | {
"line": 586,
"column": 50
} | {
"line": 588,
"column": 0
} | [
{
"pp": "α : Type u_3\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite μ\nf : α → E\nhμν : μ ≪ ν\n⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ",
"ppTerm": "?m.... | [] | nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
rw [← integrable_withDensity_iff_integrable_smul' (E := E)
(measurable_rnDeriv μ ν) (rnDeriv_lt_top μ ν)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 584,
"column": 2
} | {
"line": 586,
"column": 50
} | {
"line": 588,
"column": 0
} | [
{
"pp": "α : Type u_3\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite μ\nf : α → E\nhμν : μ ≪ ν\n⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ",
"ppTerm": "?m.... | [] | nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
rw [← integrable_withDensity_iff_integrable_smul' (E := E)
(measurable_rnDeriv μ ν) (rnDeriv_lt_top μ ν)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Basic | {
"line": 667,
"column": 61
} | {
"line": 669,
"column": 69
} | {
"line": 671,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\nP : Type u_1\ninst✝ : L.Structure P\nh : N ≃[L] P\n⊢ Function.Injective h.comp",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.Equiv.ext",
"FirstOrder.Language.E... | [] | by
intro f g hfg
ext x; exact h.injective (congr_fun (congr_arg DFunLike.coe hfg) x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Ultraproducts | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 37
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case refine_2\nα : 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\nn : ℕ\nx : β → (a : α) → M a\nk : ℕ\nφ : L.BoundedFormula β (k + 1)\nih :\n ∀ (v : Fin (k + 1) → (a : α) → M a),\n (φ.Realize (fun i... | [] | · simp only [Fin.snoc_castSucc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.ElementaryMaps | {
"line": 117,
"column": 29
} | {
"line": 117,
"column": 43
} | {
"line": 117,
"column": 44
} | [
{
"pp": "L : Language\nM : Type u_1\nN : Type u_2\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nφ : M ↪ₑ[L] N\nn : ℕ\nf : L.Functions n\nx : Fin n → M\nh : (Formula.graph f).Realize (⇑φ ∘ Fin.cons (funMap f x) x) ↔ funMap f x = funMap f x\n⊢ φ (funMap f x) = funMap f (⇑φ ∘ x)",
"ppTerm": "?m.45",
"ass... | [
"L : Language\nM : Type u_1\nN : Type u_2\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nφ : M ↪ₑ[L] N\nn : ℕ\nf : L.Functions n\nx : Fin n → M\nh : (Formula.graph f).Realize (Fin.cons (φ (funMap f x)) (⇑φ ∘ x)) ↔ funMap f x = funMap f x\n⊢ φ (funMap f x) = funMap f (⇑φ ∘ x)"
] | Fin.comp_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 447,
"column": 2
} | {
"line": 464,
"column": 24
} | {
"line": 466,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nφ : L.BoundedFormula α n\nf : ↥φ.freeVarFinset → β\nv : β → M\nxs : Fin n → M\nv' : α → M\nhv' : ∀ (a : ↥φ.freeVarFinset), v (f a) = v' ↑a\n⊢ (φ.restrictFreeVar f).Realize v xs ↔ φ.Realize v' xs",
... | [] | induction φ with
| falsum => rfl
| equal =>
simp only [Realize, restrictFreeVar]
rw [realize_restrictVarLeft v' (by simp [hv']), realize_restrictVarLeft v' (by simp [hv'])]
simp
| rel =>
simp only [Realize, restrictFreeVar]
congr!
rw [realize_restrictVarLeft v' (by simp [hv'])]
simp
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.ModelTheory.Semantics | {
"line": 447,
"column": 2
} | {
"line": 464,
"column": 24
} | {
"line": 466,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nφ : L.BoundedFormula α n\nf : ↥φ.freeVarFinset → β\nv : β → M\nxs : Fin n → M\nv' : α → M\nhv' : ∀ (a : ↥φ.freeVarFinset), v (f a) = v' ↑a\n⊢ (φ.restrictFreeVar f).Realize v xs ↔ φ.Realize v' xs",
... | [] | induction φ with
| falsum => rfl
| equal =>
simp only [Realize, restrictFreeVar]
rw [realize_restrictVarLeft v' (by simp [hv']), realize_restrictVarLeft v' (by simp [hv'])]
simp
| rel =>
simp only [Realize, restrictFreeVar]
congr!
rw [realize_restrictVarLeft v' (by simp [hv'])]
simp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 447,
"column": 2
} | {
"line": 464,
"column": 24
} | {
"line": 466,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nφ : L.BoundedFormula α n\nf : ↥φ.freeVarFinset → β\nv : β → M\nxs : Fin n → M\nv' : α → M\nhv' : ∀ (a : ↥φ.freeVarFinset), v (f a) = v' ↑a\n⊢ (φ.restrictFreeVar f).Realize v xs ↔ φ.Realize v' xs",
... | [] | induction φ with
| falsum => rfl
| equal =>
simp only [Realize, restrictFreeVar]
rw [realize_restrictVarLeft v' (by simp [hv']), realize_restrictVarLeft v' (by simp [hv'])]
simp
| rel =>
simp only [Realize, restrictFreeVar]
congr!
rw [realize_restrictVarLeft v' (by simp [hv'])]
simp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Substructures | {
"line": 747,
"column": 10
} | {
"line": 747,
"column": 12
} | {
"line": 748,
"column": 2
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nA s : Set M\na : M\n⊢ a ∈ A → a ∈ ↑((closure L[[↑A]]).toFun s)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
],
"usedFVars": [
"M",
"A",
"a"... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nA s : Set M\na : M\nha : a ∈ A\n⊢ a ∈ ↑((closure L[[↑A]]).toFun s)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.ModelTheory.Semantics | {
"line": 869,
"column": 2
} | {
"line": 869,
"column": 41
} | {
"line": 869,
"column": 41
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nγ : Type u_3\ninst✝ : Finite γ\nφ : L.Formula (α ⊕ γ)\nv : α → M\nv' : Fin 0 → M\n⊢ Realize (Formula.iExs γ φ) v v' ↔ ∃ i, φ.Realize (Sum.elim v i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nγ : Type u_3\ninst✝ : Finite γ\nφ : L.Formula (α ⊕ γ)\nv : α → M\nv' : Fin 0 → M\n⊢ Realize (Formula.iExs γ φ) v v' = (Formula.iExs γ φ).Realize v"
] | rw [← Formula.realize_iExs, iff_iff_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Definability | {
"line": 163,
"column": 87
} | {
"line": 167,
"column": 68
} | {
"line": 169,
"column": 0
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\ns : Set (α → M)\nhf : A.Definable L s\n⊢ A.Definable L sᶜ",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"FirstOrder.Language.withConstantsStructure",
"Set.compl_o... | [] | by
rcases hf with ⟨φ, hφ⟩
refine ⟨φ.not, ?_⟩
ext v
rw [hφ, compl_ofPred, mem_ofPred, mem_ofPred, Formula.realize_not] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Skolem | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 68
} | {
"line": 87,
"column": 2
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : L.Structure M\nS : (L.sum L.skolem₁).Substructure M\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥(LHom.sumInl.substructureReduct S)\na : M\nh : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\n⊢ ∃ b, φ.Realize default (Fin.snoc (Subtyp... | [
"L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : L.Structure M\nS : (L.sum L.skolem₁).Substructure M\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥(LHom.sumInl.substructureReduct S)\na : M\nh : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\nφ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFu... | let φ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFunction φ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.ModelTheory.Semantics | {
"line": 1057,
"column": 88
} | {
"line": 1059,
"column": 95
} | {
"line": 1061,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nr : L.Relations 2\n⊢ M ⊨ r.transitive ↔ IsTrans M fun x y ↦ RelMap r ![x, y]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr... | [] | by
rw [isTrans_def]
exact forall₃_congr fun _ _ _ ↦ imp_congr realize_rel₂ <| imp_congr realize_rel₂ realize_rel₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Definability | {
"line": 533,
"column": 41
} | {
"line": 545,
"column": 63
} | {
"line": 547,
"column": 0
} | [
{
"pp": "M : Type u_1\nL : Language\ninst✝¹ : L.Structure M\nα : Type u_2\nβ : Type u_3\nA : Set M\nf : (α → M) → M\ninst✝ : Finite α\ng : (β → M) → α → M\nhg : DefinableMap L A g\nhf : DefinableFun L A f\n⊢ DefinableFun L A fun v ↦ f (g v)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
... | [] | by
let G : (Option β → M) → Option α → M := fun w j =>
match j with
| none => w none
| some i => g (w ∘ some) i
have hG : A.DefinableMap L G := by
intro i
cases i with
| none => fun_prop
| some j =>
simpa [tupleGraph] using!
((hg j).preimage_comp fun | none => none | some i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Satisfiability | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 58
} | {
"line": 244,
"column": 2
} | [
{
"pp": "L : Language\nM : Type w'\ninst✝ : L.Structure M\niM : Infinite M\nκ : Cardinal.{w}\nh1 : ℵ₀ ≤ κ\nh2 : lift.{w, max u v} L.card ≤ lift.{max u v, w} κ\nx✝ : lift.{w', w} κ ≤ lift.{w, w'} #M ∨ lift.{w, w'} #M < lift.{w', w} κ\n⊢ ∃ N, (Nonempty (↑N ↪ₑ[L] M) ∨ Nonempty (M ↪ₑ[L] ↑N)) ∧ #↑N = κ",
"ppTerm... | [] | cases le_or_gt (lift.{w'} κ) (Cardinal.lift.{w} #M) with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 33
} | {
"line": 346,
"column": 2
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.BoundedFormula α n\n⊢ T ⊨ᵇ φ.toFormula ↔ T ⊨ᵇ φ",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.Theory.ModelType",
"FirstOrder.Language.Theory.ModelsBoundedFormula",
"Sum",
"instOfNat... | [
"case refine_1\nL : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.BoundedFormula α n\nh : T ⊨ᵇ φ.toFormula\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ φ.Realize v xs",
"case refine_2\nL : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ : L.BoundedFormula α n\n⊢ T ⊨ᵇ φ → T ⊨ᵇ φ.toFormula"
] | refine ⟨fun h M v xs => ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 74,
"column": 7
} | {
"line": 74,
"column": 25
} | {
"line": 74,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type v\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\nthis : #(Localization S) = #R\n⊢ Cardinal.lift.{u, v} #L = Cardinal.lift.{v, u} #R",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝³ : CommRing R\nL : Type v\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\nthis : Cardinal.lift.{v, u} #(Localization S) = Cardinal.lift.{v, u} #R\n⊢ Cardinal.lift.{u, v} #L = Cardinal.lift.{v, u} #R"
] | ← lift_inj.{u, v}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 78,
"column": 70
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | by
simpa using lift_cardinalMk L S hS | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 50
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case neg\na : ℕ\nh : ¬ℵ₀ ≤ ↑a\np : Cardinal.{u_1}\nk : ℕ\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = ↑a\nkey : p ≤ ↑a\n⊢ ∃ n, ↑a = ↑n ∧ ∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Preorder.toLT",
"Cardinal",
... | [
"case neg\na : ℕ\nh : ¬ℵ₀ ≤ ↑a\nk : ℕ\nhk : 0 < k\np : ℕ\nhp : Prime ↑p\nhpk : ↑p ^ k = ↑a\nkey : ↑p ≤ ↑a\n⊢ ∃ n, ↑a = ↑n ∧ ∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n"
] | lift p to ℕ using key.trans_lt natCast_lt_aleph0 | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.FieldTheory.Finite.Extension | {
"line": 137,
"column": 46
} | {
"line": 137,
"column": 72
} | {
"line": 137,
"column": 72
} | [
{
"pp": "k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nn : ℕ\ninst✝² : NeZero n\nl : Type u_2\ninst✝¹ : Field l\ninst✝ : Algebra k l\nh : Module.finrank k l = n\nthis✝¹ : Module.Finite k l\nthis✝ : Finite l\nthis : Fintype l\n⊢ IsSplittingField k l (X... | [
"k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nn : ℕ\ninst✝² : NeZero n\nl : Type u_2\ninst✝¹ : Field l\ninst✝ : Algebra k l\nh : Module.finrank k l = n\nthis✝¹ : Module.Finite k l\nthis✝ : Finite l\nthis : Fintype l\n⊢ IsSplittingField k l (X ^ Fintype.c... | ← Fintype.card_eq_nat_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 124,
"column": 22
} | {
"line": 124,
"column": 33
} | {
"line": 126,
"column": 0
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝ : CommSemiring K\na : K\np : MvPolynomial σ K\n⊢ (fun e ↦ (eval e) (a • p)) = (RingHom.id K) a • fun e ↦ (eval e) p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Pi.Function.module",
"Nat.instMulZer... | [] | ext e; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 124,
"column": 22
} | {
"line": 124,
"column": 33
} | {
"line": 126,
"column": 0
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝ : CommSemiring K\na : K\np : MvPolynomial σ K\n⊢ (fun e ↦ (eval e) (a • p)) = (RingHom.id K) a • fun e ↦ (eval e) p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Pi.Function.module",
"Nat.instMulZer... | [] | ext e; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Connected | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 35
} | {
"line": 26,
"column": 2
} | [
{
"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}"
] | rw [((t1Space_TFAE X).out 0 1 :)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 92,
"column": 4
} | {
"line": 103,
"column": 13
} | {
"line": 106,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fin... | [] | apply inst.isLiouville a ι c hc (B.val ∘ u) (B.val v)
dsimp only [coe_val, Function.comp_apply]
conv =>
rhs
congr
· rhs
intro x
rhs
apply logDeriv_algebraMap (u x)
· apply (deriv_algebraMap v)
simp_rw [IsScalarTower.algebraMap_apply F B K]
norm_cast | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 92,
"column": 4
} | {
"line": 103,
"column": 13
} | {
"line": 106,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fin... | [] | apply inst.isLiouville a ι c hc (B.val ∘ u) (B.val v)
dsimp only [coe_val, Function.comp_apply]
conv =>
rhs
congr
· rhs
intro x
rhs
apply logDeriv_algebraMap (u x)
· apply (deriv_algebraMap v)
simp_rw [IsScalarTower.algebraMap_apply F B K]
norm_cast | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PerfectClosure | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 90
} | {
"line": 297,
"column": 4
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\na : PerfectClosure K p\nx✝ : ℕ × K\nm : ℕ\nx : K\n⊢ mk K p ((0, 0).1 + (m, x).1, (⇑(frobenius K p))^[(m, x).1] (0, 0).2 * (⇑(frobenius K p))^[(0, 0).1] (m, x).2) =\n mk K p (0, 0)",
"ppTerm": "?m.219",
"a... | [] | simp only [zero_add, iterate_zero, id_eq, iterate_map_zero, zero_mul, mk_zero_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 86
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\n⊢ ∃ r, x = r ^ n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.natDegree_X",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"Field.isDomain",
... | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\nr : R\nhr : (X ^ n - C x).IsRoot r\n⊢ ∃ r, x = r ^ n"
] | rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.FieldTheory.PerfectClosure | {
"line": 378,
"column": 6
} | {
"line": 378,
"column": 22
} | {
"line": 378,
"column": 23
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ\n⊢ ↑x = 0 ↔ ↑x = 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"A... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ\n⊢ ↑x = ↑0 ↔ ↑x = 0"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 400,
"column": 10
} | {
"line": 400,
"column": 80
} | {
"line": 401,
"column": 8
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn : ℕ\nh : mk K p x ^ n = 0\n⊢ mk K p x ^ p ^ n = 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Nat.lt_pow_self",
... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn : ℕ\nh : mk K p x ^ n = 0\n⊢ mk K p x ^ (p ^ n - n + n) = 0"
] | ← Nat.sub_add_cancel ((n.lt_pow_self (Fact.out : p.Prime).one_lt).le), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CosetCover | {
"line": 73,
"column": 2
} | {
"line": 75,
"column": 66
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝¹ : Group G\nD H : Subgroup G\ninst✝ : D.FiniteIndex\nhD_le_H : D ≤ H\nt : Set ↥H\nht : IsComplement t ↑(D.subgroupOf H) ∧ 1 ∈ t\nhf : t.Finite\nx : G\n⊢ x ∈ H → ∃ y ∈ t, ∃ d ∈ D, ↑y * d = x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Subgrou... | [] | · intro hx
exact ⟨_, (ht.1.toLeftFun ⟨x, hx⟩).2, _,
ht.1.inv_toLeftFun_mul_mem ⟨x, hx⟩, mul_inv_cancel_left _ _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Algebraic.LinearIndependent | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 30
} | {
"line": 33,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\nH : Transcendental F x\n⊢ LinearIndependent F fun a ↦ (x - (algebraMap F E) a)⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Al... | [
"F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\nH : ∀ (p : F[X]), (aeval x) p = 0 → p = 0\n⊢ LinearIndependent F fun a ↦ (x - (algebraMap F E) a)⁻¹"
] | rw [transcendental_iff] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.MvRatFunc.Rank | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 69
} | {
"line": 32,
"column": 2
} | [
{
"pp": "case refine_1\nσ : Type u\nF : Type v\ninst✝¹ : Field F\ninst✝ : Nonempty σ\nR : Type (max v u) := MvPolynomial σ F\nK : Type (max u v) := FractionRing R\n⊢ #(FractionRing (MvPolynomial σ F)) ≤ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀",
"ppTerm": "?refine_1",
"assigned": true,
"usedCo... | [
"case refine_2\nσ : Type u\nF : Type v\ninst✝¹ : Field F\ninst✝ : Nonempty σ\nR : Type (max v u) := MvPolynomial σ F\nK : Type (max u v) := FractionRing R\n⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))"
] | · rw [FractionRing.cardinalMk, MvPolynomial.cardinalMk_eq_max_lift] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.CosetCover | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 68
} | {
"line": 383,
"column": 2
} | [
{
"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\nhcovers : ⋃ p ∈ s, ↑p = Set.univ\n⊢ False",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"False",
"AddSubgroup... | [
"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\nhcovers : ⋃ p ∈ s, ↑p = Set.univ\np : Subspace k E\nhp : p ∈ s\nhfi : (Submodule.toAddSubgroup p).FiniteIndex\n⊢ False"
] | have ⟨p, hp, hfi⟩ := Submodule.exists_finiteIndex_of_cover hcovers | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 59
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : ConjRootClass K L\n⊢ x = -y → ∃ a, mk K a = x ∧ ∃ b, mk K b = y ∧ a + b = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddG... | [] | · rintro rfl
induction y with
| h y => exact ⟨-y, mk_neg y, y, rfl, neg_add_cancel _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 65,
"column": 15
} | {
"line": 65,
"column": 22
} | {
"line": 65,
"column": 22
} | [
{
"pp": "case h\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\nx : E\nn : ℕ\nhx : x ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS\n⊢ x⁻¹ ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS",
"ppTerm": "?h",
"assigned": true... | [
"case h\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\nx : E\nn : ℕ\nhx : x ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS\n⊢ (x ^ ringExpChar F ^ n)⁻¹ ∈ (algebraMap F E).rangeS"
] | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 386,
"column": 2
} | {
"line": 399,
"column": 68
} | {
"line": 401,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nq : ℕ\nhF : ExpChar F q\ninst✝ : ExpChar E q\nn : ℕ\na : E\nhsep : IsSeparable F a\n⊢ minpoly F ((iterateFrobenius E q n) a) = Polynomial.map (iterateFrobenius F q n) (minpoly F a)",
"ppTerm": "?m.29",
"assigned":... | [] | have hai : IsIntegral F a := hsep.isIntegral
have hapi : IsIntegral F (iterateFrobenius E q n a) := hai.pow _
symm
refine Polynomial.eq_of_monic_of_dvd_of_natDegree_le
(minpoly.monic hapi)
(minpoly.monic hai |>.map _)
(minpoly.dvd F (a ^ q ^ n) ?haeval)
?hdeg
· simpa using! Eq.symm <|
(min... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 386,
"column": 2
} | {
"line": 399,
"column": 68
} | {
"line": 401,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nq : ℕ\nhF : ExpChar F q\ninst✝ : ExpChar E q\nn : ℕ\na : E\nhsep : IsSeparable F a\n⊢ minpoly F ((iterateFrobenius E q n) a) = Polynomial.map (iterateFrobenius F q n) (minpoly F a)",
"ppTerm": "?m.29",
"assigned":... | [] | have hai : IsIntegral F a := hsep.isIntegral
have hapi : IsIntegral F (iterateFrobenius E q n a) := hai.pow _
symm
refine Polynomial.eq_of_monic_of_dvd_of_natDegree_le
(minpoly.monic hapi)
(minpoly.monic hai |>.map _)
(minpoly.dvd F (a ^ q ^ n) ?haeval)
?hdeg
· simpa using! Eq.symm <|
(min... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 85
} | {
"line": 258,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (g E).natDegree ≤ ((Φ E).coeff (φ E).natDegree).natDegree",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"_private.Mathlib.FieldTheory.RatFunc.Lurot... | [] | rw [Φ_coeff_φ_natDegree' h]
exact natDegree_le_of_dvd (generator_denom_dvd_c_num h) (num_ne_zero (c_ne_zero h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 85
} | {
"line": 258,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (g E).natDegree ≤ ((Φ E).coeff (φ E).natDegree).natDegree",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"_private.Mathlib.FieldTheory.RatFunc.Lurot... | [] | rw [Φ_coeff_φ_natDegree' h]
exact natDegree_le_of_dvd (generator_denom_dvd_c_num h) (num_ne_zero (c_ne_zero h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Face.Basic | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 66
} | {
"line": 164,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nC F : PointedCone R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nhF : F.IsFaceOf C\na : R\nx : M\... | [] | exact ⟨x, hF.mem_of_smul_add_mem hx hy ha (hf hz₂ ▸ hz₁), rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Convex.Cone.Face.Basic | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 59
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nC F : PointedCone R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nsub : map f F ≤ m... | [
"case refine_2\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nC F : PointedCone R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nsub : map f F ≤ map f C\nx✝ y... | simp only [mem_map, forall_exists_index, and_imp] at hF | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Convex.Cone.Face.Basic | {
"line": 268,
"column": 22
} | {
"line": 268,
"column": 24
} | {
"line": 268,
"column": 25
} | [
{
"pp": "case mem_of_smul_add_mem\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : DivisionRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nC₁ : PointedCone R M\nC₂ : PointedCone R N\nF : PointedCone R (M × N)... | [
"case mem_of_smul_add_mem\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : DivisionRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nC₁ : PointedCone R M\nC₂ : PointedCone R N\nF : PointedCone R (M × N)\nhF : F.IsF... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Diffeology.Basic | {
"line": 273,
"column": 48
} | {
"line": 273,
"column": 79
} | {
"line": 273,
"column": 80
} | [
{
"pp": "X : Type u_1\nd : DiffeologicalSpace X\nt : TopologicalSpace X\nh : dTopology = t\nu✝ : Set X\n⊢ TopologicalSpace.IsOpen u✝ ↔ ∀ {n : ℕ}, ∀ p ∈ plots n, IsOpen (p ⁻¹' u✝)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"TopologicalSpace.IsOpen",
... | [
"X : Type u_1\nd : DiffeologicalSpace X\nt : TopologicalSpace X\nh : dTopology = t\nu✝ : Set X\n⊢ TopologicalSpace.IsOpen u✝ ↔ TopologicalSpace.IsOpen u✝"
] | ← d.isOpen_iff_preimages_plots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 53
} | {
"line": 442,
"column": 6
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\nF : Type u_1 := AlgebraicClosure K\nH✝ : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0\nα : F\nhα : (aeval α) (Q₂ h) = 0\neq :\n (Polynomial.mapRingHom (algebraMap K F)) (g E) * Polynomial.C ((aeval α) (f E)) =\n ... | [
"case inr\nK : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\nF : Type u_1 := AlgebraicClosure K\nH✝ : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0\nα : F\nhα : (aeval α) (Q₂ h) = 0\neq :\n (Polynomial.mapRingHom (algebraMap K F)) (g E) * Polynomial.C ((aeval α) (f E)) =\n (Polynomi... | refine ⟨Polynomial.C_injective.ne_iff.mp ?_, H⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Diffeology.Basic | {
"line": 403,
"column": 14
} | {
"line": 403,
"column": 21
} | {
"line": 403,
"column": 21
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : FiniteDimensional ℝ X\nd : DiffeologicalSpace X\nx✝ : IsContDiffCompatible X\n⊢ d = NormedSpace.toDiffeology X",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Real",
"Diffeology.IsPlot",
... | [
"X : Type u_1\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : FiniteDimensional ℝ X\nd : DiffeologicalSpace X\nx✝ : IsContDiffCompatible X\nn : ℕ\np : EuclideanSpace ℝ (Fin n) → X\n⊢ IsPlot p ↔ IsPlot p"
] | ext n p | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Geometry.Diffeology.Basic | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 9
} | {
"line": 478,
"column": 9
} | [
{
"pp": "X : Type u_1\nd d' : DiffeologicalSpace X\nh : d.toPlots = d'.toPlots\n⊢ d = d'",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real",
"Diffeology.IsPlot",
"EuclideanSpace",
"DiffeologicalSpace.ext",
"funext",
"Nat",
"propext",
"Fin"... | [
"X : Type u_1\nd d' : DiffeologicalSpace X\nh : d.toPlots = d'.toPlots\nn : ℕ\np : EuclideanSpace ℝ (Fin n) → X\n⊢ IsPlot p ↔ IsPlot p"
] | ext n p | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 75
} | {
"line": 302,
"column": 0
} | [
{
"pp": "case mpr\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\nhx : x ≠ 0\nθ : Real.Angle\nr : ℝ\nhr : 0 < r\nhy : r • (o.rotation θ) x ≠ 0\n⊢ o.oangle x (r • (o.rotation θ) x) = θ",
"ppTerm": "?mpr",
"a... | [] | rw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 31
} | {
"line": 253,
"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\nh : dist p₃ p₁ = dist p₃ p₂\n⊢ ∠ p₃ (midpoint ℝ p₁ p₂) p₁ = π / 2",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"In... | [
"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\n⊢ ∠ p₃ (midpoint ℝ p₁ p₂) p₁ = π / 2"
] | let m : P := midpoint ℝ p₁ p₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 391,
"column": 2
} | {
"line": 392,
"column": 48
} | {
"line": 394,
"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✝ : n.AtLeastTwo\ns : Simplex ℝ P n\ni j : Fin (n + 1)\n⊢ -1 < ⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫ / (s.height i * s.h... | [] | rw [neg_lt, neg_div', div_lt_one (by simp [height]), neg_lt]
exact neg_mul_lt_inner_vsub_altitudeFoot _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 391,
"column": 2
} | {
"line": 392,
"column": 48
} | {
"line": 394,
"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✝ : n.AtLeastTwo\ns : Simplex ℝ P n\ni j : Fin (n + 1)\n⊢ -1 < ⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫ / (s.height i * s.h... | [] | rw [neg_lt, neg_div', div_lt_one (by simp [height]), neg_lt]
exact neg_mul_lt_inner_vsub_altitudeFoot _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 30
} | {
"line": 55,
"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\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\nx : P × P × P\nhx12 : x.1 ≠ x.2.1\nhx32 : x.2.2 ≠ x.2.1\n⊢ ContinuousAt (fun y ↦ ∡ y.1 y.2.1 y.2.2... | [] | unfold oangle
fun_prop (disch := simp [*]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 30
} | {
"line": 55,
"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\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\nx : P × P × P\nhx12 : x.1 ≠ x.2.1\nhx32 : x.2.2 ≠ x.2.1\n⊢ ContinuousAt (fun y ↦ ∡ y.1 y.2.1 y.2.2... | [] | unfold oangle
fun_prop (disch := simp [*]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 41
} | {
"line": 304,
"column": 4
} | [
{
"pp": "case refine_1\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\nhp₁p₂ : p₁ -ᵥ p₂ ≠ 0\nr : ℝ\nhr : r < 0\nhp₃p₂ : p₃ = r • (p₁ -ᵥ p₂) +ᵥ p₂\n⊢ p₂ -ᵥ p₁ = (1 / (1 - r) * -r + 1 / (1 - r)) • (p₂ -ᵥ... | [
"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\nhp₁p₂ : p₁ -ᵥ p₂ ≠ 0\nr : ℝ\nhr : r < 0\nhp₃p₂ : p₃ = r • (p₁ -ᵥ p₂) +ᵥ p₂\n⊢ 1 / (1 - r) * -r + 1 / (1 - r) = 1"
] | convert! (one_smul ℝ (p₂ -ᵥ p₁)).symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 579,
"column": 4
} | {
"line": 580,
"column": 19
} | {
"line": 582,
"column": 0
} | [
{
"pp": "case inr\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\nhy : y ≠ 0\nh0 : 0 ≤ InnerProductGeometry.angle x y\nhpi : InnerProductGeometry.angle x y ≤ π\nh : o.oangle x y = -↑(InnerProductGeome... | [] | rw [h, eq_comm, Real.Angle.abs_toReal_neg_coe_eq_self_iff]
exact ⟨h0, hpi⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 579,
"column": 4
} | {
"line": 580,
"column": 19
} | {
"line": 582,
"column": 0
} | [
{
"pp": "case inr\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\nhy : y ≠ 0\nh0 : 0 ≤ InnerProductGeometry.angle x y\nhpi : InnerProductGeometry.angle x y ≤ π\nh : o.oangle x y = -↑(InnerProductGeome... | [] | rw [h, eq_comm, Real.Angle.abs_toReal_neg_coe_eq_self_iff]
exact ⟨h0, hpi⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 263,
"column": 8
} | {
"line": 263,
"column": 24
} | {
"line": 263,
"column": 24
} | [
{
"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\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ line[ℝ, p₁,... | [
"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\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ line[ℝ, p₁, p₃]\nh₁₂₄₅ ... | Set.pair_comm p₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 703,
"column": 8
} | {
"line": 703,
"column": 43
} | {
"line": 703,
"column": 44
} | [
{
"pp": "case neg.refine_1\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\nhy : ¬y = 0\nh : o.oangle x y = ↑π\n⊢ InnerProductGeometry.angle x y = π",
"ppTerm": "?neg.refine_1✝",
"assigned": t... | [
"case neg.refine_1\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\nhy : ¬y = 0\nh : o.oangle x y = ↑π\n⊢ |(o.oangle x y).toReal| = π"
] | o.angle_eq_abs_oangle_toReal hx hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 717,
"column": 9
} | {
"line": 717,
"column": 44
} | {
"line": 717,
"column": 45
} | [
{
"pp": "case neg.refine_1\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\nhy : ¬y = 0\nh : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)\n⊢ InnerProductGeometry.angle x y = π / 2",
"ppTerm"... | [
"case neg.refine_1\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\nhy : ¬y = 0\nh : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)\n⊢ |(o.oangle x y).toReal| = π / 2"
] | o.angle_eq_abs_oangle_toReal hx hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 605,
"column": 36
} | {
"line": 606,
"column": 65
} | {
"line": 608,
"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\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₃' : P\nh : Sbtw ℝ p₃ p₂ p₃'\nhp₁p₂ : p₁ ≠ p₂\n⊢ ∡ p₁ p₂ p₃ = ∡ p₁ p₂ p₃' + ↑π",
"pp... | [] | by
rw [← h.oangle₃₂₁_eq_pi, oangle_add hp₁p₂ h.right_ne h.left_ne] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 812,
"column": 2
} | {
"line": 812,
"column": 67
} | {
"line": 813,
"column": 2
} | [
{
"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\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) :... | [
"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\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' ↦ ... | have hc : IsConnected s := isConnected_univ.image _ (by fun_prop) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 940,
"column": 83
} | {
"line": 940,
"column": 98
} | {
"line": 940,
"column": 99
} | [
{
"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\nr₁ r₂ r₃ r₄ : ℝ\nhr₁ : ¬r₁ = 0\n⊢ SignType.sign (-r₃ / r₁ * (r₂ * r₁) + r₄ * r₁) * (o.oangle x y).sign =\n SignType.sign (r₁ * r₄ - r₂ * r₃) * ... | [
"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\nr₁ r₂ r₃ r₄ : ℝ\nhr₁ : ¬r₁ = 0\n⊢ SignType.sign (-r₃ / r₁ * (r₁ * r₂) + r₄ * r₁) * (o.oangle x y).sign =\n SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x ... | mul_comm r₂ r₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 634,
"column": 2
} | {
"line": 637,
"column": 20
} | {
"line": 639,
"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\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⊢ (∡ p₃ p₁ p₂).sin = dist p₃ ... | [] | rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.sin_coe,
sin_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (left_ne_of_oangle_eq_pi_div_two h)),
dist_comm p₁ p₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 720,
"column": 6
} | {
"line": 721,
"column": 27
} | {
"line": 722,
"column": 6
} | [
{
"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 : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\nhr : Same... | [
"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 : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\nhr : SameRay ℝ (p₂ -ᵥ... | simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_ofPred, Set.mem_univ, true_and,
Prod.ext_iff] at hp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 923,
"column": 2
} | {
"line": 923,
"column": 100
} | {
"line": 924,
"column": 2
} | [
{
"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 : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄ + ↑π\nhs : line[ℝ, p₁, p₂].SSameSide ... | [
"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 : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄ + ↑π\nhs : line[ℝ, p₁, p₂].SSameSide p₃ p₄\nh₃₂ :... | obtain h := angle_eq_pi_sub_angle_div_two_of_oangle_eq_of_sOppSide h₁₂ ha' (hs.trans_sOppSide hs') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 38
} | {
"line": 162,
"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\ns : Sphere P\np q : P\n⊢ q ∈ s ∧ dist q s.center ^ 2 = dist p s.center ^ 2 + dist q p ^ 2 ↔\n 0 ≤ s.radius ∧\n dist q p ^ 2 = s.radius ^ 2 - dist p s.c... | [
"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\np q : P\n⊢ dist q s.center ^ 2 = dist p s.center ^ 2 + dist q p ^ 2 →\n (q ∈ s ↔ 0 ≤ s.radius ∧ dist q p ^ 2 = s.radius ^ 2 - dist p s.center ^ 2)"
] | rw [← and_assoc, and_congr_left_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 54
} | {
"line": 338,
"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\ns₁ s₂ : Sphere P\nas : AffineSubspace ℝ P\n⊢ (∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center) ∨ ∀ p ∈ as, ¬Sbtw ℝ s₁.center p s₂.center",
"ppTerm": "?m.69",
"a... | [
"case pos\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\ns₁ s₂ : Sphere P\nas : AffineSubspace ℝ P\nh : ∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center\n⊢ (∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center) ∨ ∀ p ∈ as, ¬Sbtw ℝ s₁.center p... | by_cases! h : ∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 274,
"column": 10
} | {
"line": 277,
"column": 27
} | {
"line": 278,
"column": 10
} | [
{
"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\np : P\nh : dist p s.center < s.radius\nhp : p ≠ s.center\nhb : ℝ ∙ (p -ᵥ s.center) ≠ ⊤\nhn : ¬Subsingleton V\nv : V\nhv0 : v ≠ 0\nhv : ∀ (w : V),... | [
"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\np : P\nh : dist p s.center < s.radius\nhp : p ≠ s.center\nhb : ℝ ∙ (p -ᵥ s.center) ≠ ⊤\nhn : ¬Subsingleton V\nv : V\nhv0 : v ≠ 0\nhv : ∀ (w : V), ∃ c, c • v ... | have hc0 : c ≠ 0 := by
rintro rfl
rw [zero_smul, eq_comm, vsub_eq_zero_iff_eq] at hc
simp [hc] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 14
} | {
"line": 321,
"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 : V\nhv : v ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ\nhv1 : ‖v... | [
"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 : V\nhv : v ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ\nhv1 : ‖v‖ = 1\nhr : ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 21
} | {
"line": 162,
"column": 21
} | [
{
"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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ s.excenterWeights signs i ≠ 0",
"... | [
"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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ ((∑ i, s.excenterWeightsUnnorm signs i)⁻¹ • s.exc... | excenterWeights | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 182,
"column": 27
} | {
"line": 182,
"column": 42
} | {
"line": 182,
"column": 42
} | [
{
"pp": "V : 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\nsigns : Finset (Fin (n + 1))\ninst✝ : Decidable (s.ExcenterExists signs)\n⊢ ∑ i, s.excenterWeights signs i = if ∑... | [
"V : 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\nsigns : Finset (Fin (n + 1))\ninst✝ : Decidable (s.ExcenterExists signs)\n⊢ ∑ i, ((∑ i, s.excenterWeightsUnnorm signs i)⁻¹ • ... | excenterWeights | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 65
} | {
"line": 241,
"column": 4
} | [
{
"pp": "case e'_2.e'_6\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\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\nj : Fin (n + 1)\nhj : j ∈ (Finset.univ.erase 0).erase i\n⊢ s.points i... | [
"case e'_2.e'_6\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\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\nj : Fin (n + 1)\nhj : j ≠ i ∧ j ≠ 0\n⊢ s.points i -ᵥ s.points 0 ∈ vectorSpan ℝ (S... | simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hj | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 31
} | {
"line": 244,
"column": 4
} | [
{
"pp": "case e'_2.e'_6.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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\nj : Fin (n + 1)\nhj : j ≠ i ∧ j ≠ 0\n⊢ ∃ x ∈ {j}ᶜ, s.points ... | [] | exact ⟨i, hj.1.symm, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 31
} | {
"line": 244,
"column": 4
} | [
{
"pp": "case e'_2.e'_6.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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\nj : Fin (n + 1)\nhj : j ≠ i ∧ j ≠ 0\n⊢ ∃ x ∈ {j}ᶜ, s.points ... | [] | exact ⟨i, hj.1.symm, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 31
} | {
"line": 244,
"column": 4
} | [
{
"pp": "case e'_2.e'_6.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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\nj : Fin (n + 1)\nhj : j ≠ i ∧ j ≠ 0\n⊢ ∃ x ∈ {j}ᶜ, s.points ... | [] | exact ⟨i, hj.1.symm, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 247,
"column": 15
} | {
"line": 247,
"column": 22
} | {
"line": 247,
"column": 22
} | [
{
"pp": "case e'_3\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\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\n⊢ 0 = -((s.height 0)⁻¹ ^ 2 * s.height 0 ^ 2) + (s.height i)⁻¹ ^ 2 * s.heig... | [
"case e'_3\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\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhi : i ≠ 0\n⊢ 0 = -((s.height 0 ^ 2)⁻¹ * s.height 0 ^ 2) + (s.height i)⁻¹ ^ 2 * s.height i ^ 2"
] | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 309,
"column": 25
} | {
"line": 309,
"column": 32
} | {
"line": 309,
"column": 32
} | [
{
"pp": "case e'_2.e'_5\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 : Fin (n + 1)\n⊢ 0 < (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹",
"pp... | [
"case e'_2.e'_5\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 : Fin (n + 1)\n⊢ 0 < ∑ i_1, s.excenterWeightsUnnorm {i} i_1"
] | inv_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 317,
"column": 25
} | {
"line": 317,
"column": 32
} | {
"line": 317,
"column": 32
} | [
{
"pp": "case e'_2.e'_5\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 : i ≠ j\n⊢ 0 < (∑ i_1, s.excenterWeightsUnnorm {i} i_1... | [
"case e'_2.e'_5\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 : i ≠ j\n⊢ 0 < ∑ i_1, s.excenterWeightsUnnorm {i} i_1"
] | inv_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 618,
"column": 42
} | {
"line": 618,
"column": 66
} | {
"line": 618,
"column": 66
} | [
{
"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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ n - 1 ≠ n",
"ppTerm": "?m.87",
... | [] | have := NeZero.ne n; lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 618,
"column": 42
} | {
"line": 618,
"column": 66
} | {
"line": 618,
"column": 66
} | [
{
"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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ n - 1 ≠ n",
"ppTerm": "?m.87",
... | [] | have := NeZero.ne n; lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 383,
"column": 56
} | {
"line": 394,
"column": 98
} | {
"line": 396,
"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\nn : ℕ\ns : Simplex ℝ P n\np : P\nhr : ∃ r, ∀ (i : Fin (n + 1)), dist (s.points i) p = r\n⊢ ↑(s.orthogonalProjectionSpan p) = s.circumcenter",
"ppTerm": "?m... | [] | by
change ∃ r : ℝ, ∀ i, (fun x => dist x p = r) (s.points i) at hr
have hr : ∃ (r : ℝ), ∀ (a : P),
a ∈ Set.range (fun (i : Fin (n + 1)) => s.points i) → dist a p = r := by
obtain ⟨r, hr⟩ := hr
use r
refine Set.forall_mem_range.mpr ?_
exact hr
rw [exists_dist_eq_iff_exists_dist_orthogonalProj... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 73
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case inl\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\nc p x : P\nhx : Sbtw ℝ p x c\n⊢ ∠ p p x + ∠ x p c = ∠ p p c",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"InnerProductS... | [
"case 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\na c p x : P\nhx : Sbtw ℝ a x c\nhpa : p ≠ a\n⊢ ∠ a p x + ∠ x p c = ∠ a p c"
] | · simp [(hx.angle_eq_right x).symm.trans (angle_self_of_ne hx.ne_left)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1065,
"column": 10
} | {
"line": 1065,
"column": 51
} | {
"line": 1065,
"column": 51
} | [
{
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\nhne : i ≠ j\nr : ℝ\nhr... | [
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\nhne : i ≠ j\nr : ℝ\nhr : r ∈ Set.I... | ← ContinuousAffineMap.lineMap_toAffineMap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 500,
"column": 26
} | {
"line": 506,
"column": 53
} | {
"line": 508,
"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\np₁ p₂ p₃ : P\nh₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃\nh₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃\nhne : ∠ p₁ p₂ p₃ ≠ ∠ p₂ p₃ p₁ ∨ ∠ p₁ p₂ p₃ ≠ ∠ p₃ p₁ p₂ ∨ ∠ p₂ p₃ p₁ ≠ ∠ p₃ p₁ p₂... | [] | by
by_cases h : p₂ = p₁
· rw [h, angle_self_left]
linarith [Real.pi_pos]
· rcases hne with hne | hne | hne <;>
rcases hne.lt_or_gt with hne | hne <;>
linarith [angle_add_angle_add_angle_eq_pi p₃ h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 528,
"column": 6
} | {
"line": 528,
"column": 15
} | {
"line": 529,
"column": 6
} | [
{
"pp": "case pos\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\np₂ p₃ : P\nh₂₃₁ : ∠ p₂ p₂ p₃ ≤ ∠ p₂ p₃ p₂\nh₃₁₂ : ∠ p₂ p₂ p₃ ≤ ∠ p₃ p₂ p₂\nhne : ∠ p₂ p₂ p₃ ≠ ∠ p₂ p₃ p₂ ∨ ∠ p₂ p₂ p₃ ≠ ∠ p₃ p₂ p₂ ∨ ∠ p₂ p₃ p₂ ≠ ∠ p... | [
"case pos\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\np₂ : P\nh₂₃₁ h₃₁₂ : ∠ p₂ p₂ p₂ ≤ ∠ p₂ p₂ p₂\nhne : ∠ p₂ p₂ p₂ ≠ ∠ p₂ p₂ p₂ ∨ ∠ p₂ p₂ p₂ ≠ ∠ p₂ p₂ p₂ ∨ ∠ p₂ p₂ p₂ ≠ ∠ p₂ p₂ p₂\n⊢ ∠ p₂ p₂ p₂ < π / 3"
] | subst h₂₃ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1233,
"column": 22
} | {
"line": 1233,
"column": 37
} | {
"line": 1233,
"column": 37
} | [
{
"pp": "V : 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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\nhe : Pi.single... | [
"V : 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\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\nhe : Pi.single j 1 = s.tou... | excenterWeights | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 249,
"column": 57
} | {
"line": 249,
"column": 89
} | {
"line": 249,
"column": 89
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\ny : V\nhy left✝ : y ≠ 0\nr : ℝ≥0\nhr : 0 < ↑r\nhxz₁ : ¬angle 0 (↑r • y) = π\nhxz₂ : ¬angle 0 (↑r • y) = 0\nh_sin_xz : Real.sin (angle 0 (↑r • y)) ≠ 0\n⊢ y ∈ ℝ≥0 ∙ r • y",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\ny : V\nhy left✝ : y ≠ 0\nr : ℝ≥0\nhr : 0 < ↑r\nhxz₁ : ¬angle 0 (↑r • y) = π\nhxz₂ : ¬angle 0 (↑r • y) = 0\nh_sin_xz : Real.sin (angle 0 (↑r • y)) ≠ 0\n⊢ y ∈ ℝ≥0 ∙ y",
"case pos.hr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ni... | Submodule.span_singleton_smul_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 113,
"column": 2
} | {
"line": 114,
"column": 74
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case right\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\ninst✝ : NeZero n\ns : Simplex ℝ P n\n⊢ Set.range s.medial.points ⊆ Metric.sphere s.ninePointCircle.center s.ninePointCircle.radius",
"p... | [] | · rw [Set.range_subset_iff]
simpa [medial_points] using s.faceOppositeCentroid_mem_ninePointCircle | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 207,
"column": 62
} | {
"line": 210,
"column": 10
} | {
"line": 212,
"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 : Triangle ℝ P\ni : Fin 3\n⊢ Simplex.eulerPoint s i = midpoint ℝ s.orthocenter (s.points i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants... | [] | by
apply vsub_right_cancel (p := s.points i)
rw [orthocenter_eq_mongePoint, Simplex.points_vsub_eulerPoint, vsub_midpoint]
norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 98
} | {
"line": 65,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : s.AcuteAngled\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i... | [] | convert! h (i₁ := e.symm i₁) (i₂ := e.symm i₂) (i₃ := e.symm i₃) ?_ ?_ ?_ using 1 <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 98
} | {
"line": 65,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : s.AcuteAngled\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i... | [] | convert! h (i₁ := e.symm i₁) (i₂ := e.symm i₂) (i₃ := e.symm i₃) ?_ ?_ ?_ using 1 <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 98
} | {
"line": 65,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : s.AcuteAngled\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i... | [] | convert! h (i₁ := e.symm i₁) (i₂ := e.symm i₂) (i₃ := e.symm i₃) ?_ ?_ ?_ using 1 <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 288,
"column": 4
} | {
"line": 290,
"column": 37
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case e'_3.e'_6\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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ centroidWeights ℝ {e.symm i₁, e.symm i₂}ᶜ = centroidWeigh... | [] | simp_rw [centroidWeights, Function.const_comp, Finset.card_compl]
congr 4
by_cases h : i₁ = i₂ <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 288,
"column": 4
} | {
"line": 290,
"column": 37
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case e'_3.e'_6\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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ centroidWeights ℝ {e.symm i₁, e.symm i₂}ᶜ = centroidWeigh... | [] | simp_rw [centroidWeights, Function.const_comp, Finset.card_compl]
congr 4
by_cases h : i₁ = i₂ <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 18
} | {
"line": 67,
"column": 0
} | [
{
"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\np : P\nv : V\nhv : ¬v = 0\n⊢ -2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫ = 0 ∨ -2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫ = -2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 500,
"column": 60
} | {
"line": 528,
"column": 98
} | {
"line": 530,
"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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\nh₁ : t₂.points... | [] | by
symm
rw [← h₂, t₂.affineSpan_pair_eq_altitude_iff]
rw [h₂]
use t₁.independent.injective.ne hi₁₂
have he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points) := by
refine ext_of_direction_eq ?_
⟨t₁.points i₃, mem_affineSpan ℝ ⟨j₃, h₃⟩, mem_affineSpan ℝ (Set.mem_range_self _)⟩
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 282,
"column": 13
} | {
"line": 282,
"column": 15
} | {
"line": 282,
"column": 16
} | [
{
"pp": "𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne... | [
"𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne✝ e'✝ : Open... | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 692,
"column": 15
} | {
"line": 692,
"column": 20
} | {
"line": 692,
"column": 21
} | [
{
"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 : Set P\nt t₀ : Triangle ℝ P\nht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)\nht₀o : t₀.orthocenter ∉ Set.range t₀.points\nht₀s : 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\ns : Set P\nt t₀ : Triangle ℝ P\nht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)\nht₀o : t₀.orthocenter ∉ Set.range t₀.points\nht₀s : s = insert t₀.or... | ht₀s, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 291,
"column": 48
} | {
"line": 291,
"column": 72
} | {
"line": 291,
"column": 72
} | [
{
"pp": "𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne... | [] | by simp [φ, inter_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 75
} | {
"line": 401,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | simp [isBoundaryPoint_iff_not_isInteriorPoint, hf.isInteriorPoint_iff hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 75
} | {
"line": 401,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | simp [isBoundaryPoint_iff_not_isInteriorPoint, hf.isInteriorPoint_iff hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.