module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 584,
"column": 12
} | {
"line": 584,
"column": 44
} | {
"line": 584,
"column": 44
} | [
{
"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.... | [
"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 (ν.withDensity (μ.rnDeriv ν))"
] | ← withDensity_rnDeriv_eq μ ν hμν | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 666,
"column": 71
} | {
"line": 669,
"column": 64
} | {
"line": 671,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\ninst✝² : SigmaFinite μ\nν₁ ν₂ : Measure G\ninst✝¹ : SigmaFinite ν₁\ninst✝ : SigmaFinite ν₂\nhν₁ : ν₁ ≪ μ\nhν₂ : ν₂ ≪ μ\n⊢ (ν₁ ∗ₘ ν₂).rnDeriv μ =ᵐ[μ]... | [] | by
rw [← withDensity_eq_iff_of_sigmaFinite (by fun_prop) (by fun_prop),
← mconv_eq_withDensity_mlconvolution_rnDeriv hν₁ hν₂,
withDensity_rnDeriv_eq _ _ (mconv_absolutelyContinuous hν₂)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.LanguageMap | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 15
} | {
"line": 528,
"column": 4
} | [
{
"pp": "case refine_2\nL : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\n⊢ ∀ {n : ℕ} (r : L[[↑A]].Relations n) (x : Fin n → M), RelMap r (f.toFun ∘ x) ↔ RelMap r x",
... | [
"case refine_2\nL : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\nn : ℕ\nR : L[[↑A]].Relations n\nx : Fin n → M\n⊢ RelMap R (f.toFun ∘ x) ↔ RelMap R x"
] | intro n R x | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 39
} | {
"line": 141,
"column": 39
} | [
{
"pp": "R : Type u\nK : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsAlgClosed K\nι : Type w\nv : ι → K\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K\nthis : ℵ₀ ≤ Cardinal.lift.{max u v, w} #ι\n⊢ ℵ₀ ≤ max (Cardinal.lift.{max v w, u} #R) (Car... | [] | · exact le_max_of_le_right this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.Syntax | {
"line": 124,
"column": 19
} | {
"line": 124,
"column": 28
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel id (_ts✝ a) = _ts✝ a\n⊢ relabel id (func _f✝ _ts✝) = func _f✝ _ts✝",
"ppTerm": "?func",
"assigned": true,
"usedConstants": [
"congrArg",
"FirstOrder.Languag... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Syntax | {
"line": 124,
"column": 19
} | {
"line": 124,
"column": 28
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel id (_ts✝ a) = _ts✝ a\n⊢ relabel id (func _f✝ _ts✝) = func _f✝ _ts✝",
"ppTerm": "?func",
"assigned": true,
"usedConstants": [
"congrArg",
"FirstOrder.Languag... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Syntax | {
"line": 124,
"column": 19
} | {
"line": 124,
"column": 28
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel id (_ts✝ a) = _ts✝ a\n⊢ relabel id (func _f✝ _ts✝) = func _f✝ _ts✝",
"ppTerm": "?func",
"assigned": true,
"usedConstants": [
"congrArg",
"FirstOrder.Languag... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Syntax | {
"line": 135,
"column": 19
} | {
"line": 135,
"column": 28
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nβ : Type v'\nγ : Type u_1\nf : α → β\ng : β → γ\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel g (relabel f (_ts✝ a)) = relabel (g ∘ f) (_ts✝ a)\n⊢ relabel g (relabel f (func _f✝ _ts✝)) = relabel (g ∘ f) (func _f✝ _ts✝)",
... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Syntax | {
"line": 135,
"column": 19
} | {
"line": 135,
"column": 28
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nβ : Type v'\nγ : Type u_1\nf : α → β\ng : β → γ\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel g (relabel f (_ts✝ a)) = relabel (g ∘ f) (_ts✝ a)\n⊢ relabel g (relabel f (func _f✝ _ts✝)) = relabel (g ∘ f) (func _f✝ _ts✝)",
... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Syntax | {
"line": 135,
"column": 19
} | {
"line": 135,
"column": 28
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case func\nL : Language\nα : Type u'\nβ : Type v'\nγ : Type u_1\nf : α → β\ng : β → γ\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), relabel g (relabel f (_ts✝ a)) = relabel (g ∘ f) (_ts✝ a)\n⊢ relabel g (relabel f (func _f✝ _ts✝)) = relabel (g ∘ f) (func _f✝ _ts✝)",
... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 13
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case func.add\nα : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CompatibleRing R\np : FreeCommRing α\nv : α → R\na : Fin 2 → ring.Term α\nih : ∀ (a_1 : Fin 2), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n⊢ Term.realize v (func ringFunc.add a) = (FreeCo... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 13
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case func.mul\nα : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CompatibleRing R\np : FreeCommRing α\nv : α → R\na : Fin 2 → ring.Term α\nih : ∀ (a_1 : Fin 2), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n⊢ Term.realize v (func ringFunc.mul a) = (FreeCo... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 13
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case func.neg\nα : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CompatibleRing R\np : FreeCommRing α\nv : α → R\na : Fin 1 → ring.Term α\nih : ∀ (a_1 : Fin 1), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n⊢ Term.realize v (func ringFunc.neg a) = (FreeCo... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 13
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case func.zero\nα : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CompatibleRing R\np : FreeCommRing α\nv : α → R\na : Fin 0 → ring.Term α\nih : ∀ (a_1 : Fin 0), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n⊢ Term.realize v (func ringFunc.zero a) = (Free... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 13
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case func.one\nα : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CompatibleRing R\np : FreeCommRing α\nv : α → R\na : Fin 0 → ring.Term α\nih : ∀ (a_1 : Fin 0), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n⊢ Term.realize v (func ringFunc.one a) = (FreeCo... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 92,
"column": 20
} | {
"line": 92,
"column": 29
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ng : α → β\nv : β → M\nl✝ : ℕ\nf : L.Functions l✝\nts : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v (relabel g (ts a)) = realize (v ∘ g) (ts a)\n⊢ realize v (relabel g (func f ts)) = realize (v ∘ g) (func f ts)",... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 92,
"column": 20
} | {
"line": 92,
"column": 29
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ng : α → β\nv : β → M\nl✝ : ℕ\nf : L.Functions l✝\nts : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v (relabel g (ts a)) = realize (v ∘ g) (ts a)\n⊢ realize v (relabel g (func f ts)) = realize (v ∘ g) (func f ts)",... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 92,
"column": 20
} | {
"line": 92,
"column": 29
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ng : α → β\nv : β → M\nl✝ : ℕ\nf : L.Functions l✝\nts : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v (relabel g (ts a)) = realize (v ∘ g) (ts a)\n⊢ realize v (relabel g (func f ts)) = realize (v ∘ g) (func f ts)",... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 28
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ntf : α → L.Term β\nv : β → M\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v ((_ts✝ a).subst tf) = realize (fun a ↦ realize v (tf a)) (_ts✝ a)\n⊢ realize v ((func _f✝ _ts✝).subs... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 28
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ntf : α → L.Term β\nv : β → M\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v ((_ts✝ a).subst tf) = realize (fun a ↦ realize v (tf a)) (_ts✝ a)\n⊢ realize v ((func _f✝ _ts✝).subs... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 28
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case func\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\ntf : α → L.Term β\nv : β → M\nl✝ : ℕ\n_f✝ : L.Functions l✝\n_ts✝ : Fin l✝ → L.Term α\nih : ∀ (a : Fin l✝), realize v ((_ts✝ a).subst tf) = realize (fun a ↦ realize v (tf a)) (_ts✝ a)\n⊢ realize v ((func _f✝ _ts✝).subs... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.AbelRuffini | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 32
} | {
"line": 272,
"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\nhx : x ∈ solvableByRad F E\nn : ℕ\nhn : n ≠ 0\nhα : IsSolvable (minpoly F (x ^ n)).Gal\np : F[X] := minpoly F (x ^ n)\nhp : p.comp (X ^ n) ≠ 0\n⊢ IsSolvable (minpoly F x).Gal",
"ppTerm": "?m.54",
"assign... | [
"case x\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\nhx : x ∈ solvableByRad F E\nn : ℕ\nhn : n ≠ 0\nhα : IsSolvable (minpoly F (x ^ n)).Gal\np : F[X] := ⋯\nhp : p.comp (X ^ n) ≠ 0\n⊢ Fact (map (algebraMap F (SplittingField ?q)) (minpoly F x)).Splits",
"case hq\nF : ... | apply gal_isSolvable_of_splits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.ModelTheory.Semantics | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 28
} | {
"line": 284,
"column": 0
} | [
{
"pp": "case cons\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nn : ℕ\nv : α → M\nxs : Fin n → M\nφ : L.BoundedFormula α n\nl : List (L.BoundedFormula α n)\nih : (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ l).Realize v xs ↔ ∀ φ ∈ l, φ.Realize v xs\n⊢ (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ (φ :: l)).Reali... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 28
} | {
"line": 284,
"column": 0
} | [
{
"pp": "case cons\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nn : ℕ\nv : α → M\nxs : Fin n → M\nφ : L.BoundedFormula α n\nl : List (L.BoundedFormula α n)\nih : (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ l).Realize v xs ↔ ∀ φ ∈ l, φ.Realize v xs\n⊢ (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ (φ :: l)).Reali... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 28
} | {
"line": 284,
"column": 0
} | [
{
"pp": "case cons\nL : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nn : ℕ\nv : α → M\nxs : Fin n → M\nφ : L.BoundedFormula α n\nl : List (L.BoundedFormula α n)\nih : (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ l).Realize v xs ↔ ∀ φ ∈ l, φ.Realize v xs\n⊢ (List.foldr (fun x1 x2 ↦ x1 ⊓ x2) ⊤ (φ :: l)).Reali... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Substructures | {
"line": 289,
"column": 2
} | {
"line": 291,
"column": 33
} | {
"line": 293,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\n⊢ lift.{max u w, w} #↥((closure L).toFun s) ≤ #(L.Term ↑s)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.coe_sort_coe",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Cardi... | [] | rw [← SetLike.coe_sort_coe, coe_closure_eq_range_term_realize]
rw [← Cardinal.lift_id'.{w, max u w} #(L.Term s)]
exact Cardinal.mk_range_le_lift | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Substructures | {
"line": 289,
"column": 2
} | {
"line": 291,
"column": 33
} | {
"line": 293,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\n⊢ lift.{max u w, w} #↥((closure L).toFun s) ≤ #(L.Term ↑s)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.coe_sort_coe",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Cardi... | [] | rw [← SetLike.coe_sort_coe, coe_closure_eq_range_term_realize]
rw [← Cardinal.lift_id'.{w, max u w} #(L.Term s)]
exact Cardinal.mk_range_le_lift | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Substructures | {
"line": 837,
"column": 6
} | {
"line": 837,
"column": 14
} | {
"line": 838,
"column": 4
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\nf g : M →[L] N\nn : ℕ\nfn : L.Functions n\nx : Fin n → M\nhx : ∀ (i : Fin n), x i ∈ {x | f x = g x}\nx✝ : Fin n\n⊢ f (x x✝) = g (x x✝)",
"ppTerm": "?m.5... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.ModelTheory.Definability | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 24
} | {
"line": 133,
"column": 0
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\nι : Type u_2\nf : ι → Set (α → M)\nhf : ∀ (i : ι), A.Definable L (f i)\ns✝ : Finset ι\ni : ι\ns : Finset ι\nx✝ : i ∉ s\nh : A.Definable L (s.inf f)\n⊢ A.Definable L (f i ⊓ s.inf f)",
"ppTerm": "?m.38",
"assigned": true,
... | [] | exact (hf i).inter h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 58
} | {
"line": 188,
"column": 4
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nA : Set M\nhA : MeetsDefinable A\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥((closure L).toFun A)\na : M\nhφ : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\nD : Set M := {y | φ.Realize default (Fin.snoc (Subtype.val ∘ x) y)}\nhD_ne : ... | [
"L : Language\nM : Type u_1\ninst✝ : L.Structure M\nA : Set M\nhA : MeetsDefinable A\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥((closure L).toFun A)\na : M\nhφ : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\nD : Set M := {y | φ.Realize default (Fin.snoc (Subtype.val ∘ x) y)}\nhD_ne : D.Nonempty\n... | simp only [Fin.isValue, mem_setOf_eq, Formula.relabel, Formula.Realize,
BoundedFormula.realize_subst, BoundedFormula.realize_relabel, Nat.add_zero, Fin.castAdd_zero,
Fin.cast_refl, Function.comp_id, Fin.natAdd_zero, D] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Skolem | {
"line": 56,
"column": 43
} | {
"line": 56,
"column": 57
} | {
"line": 56,
"column": 57
} | [
{
"pp": "case H\nL : Language\nn : ℕ\n⊢ lift.{max (max u v) ?u.64, u} #(L.Functions n) ≤ lift.{?u.64, max u v} #(L.BoundedFormula Empty (n + 1))",
"ppTerm": "?H",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.lift_mk_le",
"Cardinal",
"congrArg",
"Cardinal.lif... | [
"case H\nL : Language\nn : ℕ\n⊢ Nonempty (L.Functions n ↪ L.BoundedFormula Empty (n + 1))"
] | lift_mk_le.{v} | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 25
} | {
"line": 224,
"column": 2
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\nhψM : M ⊨ ψ\nhψS : ∃ i, φ.Realize i\n⊢ (D... | [
"L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\nhψM : M ⊨ ψ\nv' : Fin 1 → ↥S\nhv' : φ.Realize v'\n⊢ (... | obtain ⟨v', hv'⟩ := hψS | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.ModelTheory.Satisfiability | {
"line": 133,
"column": 2
} | {
"line": 145,
"column": 29
} | {
"line": 147,
"column": 0
} | [
{
"pp": "L : Language\nα : Type w\nT : L.Theory\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : L.Structure M\ninst✝ : M ⊨ T\nh : lift.{w', w} #↑s ≤ lift.{w, w'} #M\n⊢ ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable",
"ppTerm": "?m.11",
"assigned": true,
"used... | [] | haveI : Inhabited M := Classical.inhabited_of_nonempty inferInstance
rw [Cardinal.lift_mk_le'] at h
letI : (constantsOn α).Structure M := constantsOn.structure (Function.extend (↑) h.some default)
have : M ⊨ (L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s := by
refine ((LHom.onTheory_model _ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Satisfiability | {
"line": 133,
"column": 2
} | {
"line": 145,
"column": 29
} | {
"line": 147,
"column": 0
} | [
{
"pp": "L : Language\nα : Type w\nT : L.Theory\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : L.Structure M\ninst✝ : M ⊨ T\nh : lift.{w', w} #↑s ≤ lift.{w, w'} #M\n⊢ ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable",
"ppTerm": "?m.11",
"assigned": true,
"used... | [] | haveI : Inhabited M := Classical.inhabited_of_nonempty inferInstance
rw [Cardinal.lift_mk_le'] at h
letI : (constantsOn α).Structure M := constantsOn.structure (Function.extend (↑) h.some default)
have : M ⊨ (L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s := by
refine ((LHom.onTheory_model _ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 1061,
"column": 4
} | {
"line": 1065,
"column": 12
} | {
"line": 1066,
"column": 2
} | [
{
"pp": "case refine_1\nL : Language\nα : Type u'\nM : Type w\ninst✝ : L[[α]].Structure M\ns : Set α\nh :\n ∀ (φ : L[[α]].Sentence) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ (Set.diagonal α)ᶜ → ((L.con x).term.equal (L.con x_1).term).not = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i ↦ ↑(L.con i)) ... | [] | contrapose! ab
have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl
simp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal,
Term.realize_constants] at h'
exact h' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 1061,
"column": 4
} | {
"line": 1065,
"column": 12
} | {
"line": 1066,
"column": 2
} | [
{
"pp": "case refine_1\nL : Language\nα : Type u'\nM : Type w\ninst✝ : L[[α]].Structure M\ns : Set α\nh :\n ∀ (φ : L[[α]].Sentence) (x x_1 : α),\n (x, x_1) ∈ s ×ˢ s ∩ (Set.diagonal α)ᶜ → ((L.con x).term.equal (L.con x_1).term).not = φ → M ⊨ φ\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i ↦ ↑(L.con i)) ... | [] | contrapose! ab
have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl
simp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal,
Term.realize_constants] at h'
exact h' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Satisfiability | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 92
} | {
"line": 175,
"column": 2
} | [
{
"pp": "L : Language\nT : L.Theory\nκ : Cardinal.{w}\nM : Type w'\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Infinite M\nN : ((L.lhomWithConstants (Quotient.out κ)).onTheory T ∪ L.distinctConstantsTheory Set.univ).ModelType\nthis : ↑N ⊨ L.distinctConstantsTheory Set.univ\n⊢ lift.{max (max u v) w, w} #↑Se... | [
"L : Language\nT : L.Theory\nκ : Cardinal.{w}\nM : Type w'\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Infinite M\nN : ((L.lhomWithConstants (Quotient.out κ)).onTheory T ∪ L.distinctConstantsTheory Set.univ).ModelType\nthis : ↑N ⊨ L.distinctConstantsTheory Set.univ\n⊢ lift.{max u v w, max (max u v) w} (lift.{w... | refine
(card_le_of_model_distinctConstantsTheory L Set.univ N).trans (lift_le.{max u v w}.1 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Satisfiability | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 35
} | {
"line": 518,
"column": 4
} | [
{
"pp": "L : Language\nκ : Cardinal.{w}\nT : L.Theory\nh : κ.Categorical T\nh1 : ℵ₀ ≤ κ\nh2 : lift.{w, max u v} L.card ≤ lift.{max u v, w} κ\nhS : T.IsSatisfiable\nhT : ∀ (M : T.ModelType), Infinite ↑M\nφ : L.Sentence\nw✝ : T.ModelType\nh✝ : #↑w✝ = κ\n⊢ (∀ (M : T.ModelType), ↑M ⊨ φ) ∨ ∀ (M : T.ModelType), ↑M ⊨ ... | [
"L : Language\nκ : Cardinal.{w}\nT : L.Theory\nh : κ.Categorical T\nh1 : ℵ₀ ≤ κ\nh2 : lift.{w, max u v} L.card ≤ lift.{max u v, w} κ\nhS : T.IsSatisfiable\nhT : ∀ (M : T.ModelType), Infinite ↑M\nφ : L.Sentence\nw✝ : T.ModelType\nh✝ : #↑w✝ = κ\nMF : T.ModelType\nhMF : ¬↑MF ⊨ φ\nMT : T.ModelType\nhMT : ¬↑MT ⊨ Formula... | by_contra! ⟨⟨MF, hMF⟩, MT, hMT⟩ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.FieldTheory.Differential.Basic | {
"line": 39,
"column": 93
} | {
"line": 42,
"column": 6
} | {
"line": 44,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\na b : R\nha : a ≠ 0\nhb : b ≠ 0\n⊢ logDeriv (a * b) = logDeriv a + logDeriv b",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Derivation",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.FieldSimp... | [] | by
unfold logDeriv
simp [field]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Differential.Basic | {
"line": 55,
"column": 6
} | {
"line": 56,
"column": 15
} | {
"line": 58,
"column": 0
} | [
{
"pp": "case succ.inr\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\na : R\nn : ℕ\nh2 : logDeriv (a ^ n) = ↑n * logDeriv a\nhb : a ≠ 0\n⊢ logDeriv (a ^ (n + 1)) = ↑(n + 1) * logDeriv a",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"Mu... | [] | rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>
simp [hb] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.Differential.Basic | {
"line": 55,
"column": 6
} | {
"line": 56,
"column": 15
} | {
"line": 58,
"column": 0
} | [
{
"pp": "case succ.inr\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\na : R\nn : ℕ\nh2 : logDeriv (a ^ n) = ↑n * logDeriv a\nhb : a ≠ 0\n⊢ logDeriv (a ^ (n + 1)) = ↑(n + 1) * logDeriv a",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"Mu... | [] | rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>
simp [hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Basic | {
"line": 55,
"column": 6
} | {
"line": 56,
"column": 15
} | {
"line": 58,
"column": 0
} | [
{
"pp": "case succ.inr\nR : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\na : R\nn : ℕ\nh2 : logDeriv (a ^ n) = ↑n * logDeriv a\nhb : a ≠ 0\n⊢ logDeriv (a ^ (n + 1)) = ↑(n + 1) * logDeriv a",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"Mu... | [] | rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>
simp [hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 98,
"column": 15
} | {
"line": 98,
"column": 50
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case refine_2\nK : Type u_1\nσ : Type u_2\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\nn : σ\nh : n ∉ Finset.univ\n⊢ (Fintype.card K - 1) * Multiset.count n {n} = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"HMul.... | [] | exact (h <| Finset.mem_univ _).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 181,
"column": 10
} | {
"line": 182,
"column": 33
} | {
"line": 183,
"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 ... | [] | by_cases h : u i = 0 <;>
simp [logDeriv_prod, h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 181,
"column": 10
} | {
"line": 182,
"column": 33
} | {
"line": 183,
"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 ... | [] | by_cases h : u i = 0 <;>
simp [logDeriv_prod, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 181,
"column": 10
} | {
"line": 182,
"column": 33
} | {
"line": 183,
"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 ... | [] | by_cases h : u i = 0 <;>
simp [logDeriv_prod, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Invariant.Basic | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 28
} | {
"line": 199,
"column": 6
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nG : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Group G\ninst✝³ : MulSemiringAction G B\ninst✝² : IsInvariant A B G\ninst✝¹ : Finite G\ninst✝ : SMulCommClass G A B\nP✝ Q✝ : Ideal B\nhP✝ : P✝.IsPrime\nhQ✝ : Q✝.IsPrime\nhPQ✝ : Ideal.under... | [
"A : Type u_1\nB : Type u_2\nG : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Group G\ninst✝³ : MulSemiringAction G B\ninst✝² : IsInvariant A B G\ninst✝¹ : Finite G\ninst✝ : SMulCommClass G A B\nP✝ Q✝ : Ideal B\nhP✝ : P✝.IsPrime\nhQ✝ : Q✝.IsPrime\nhPQ✝ : Ideal.under A P✝ = Idea... | obtain ⟨g, -, hg⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.Galois.IsGaloisGroup | {
"line": 144,
"column": 40
} | {
"line": 144,
"column": 63
} | {
"line": 144,
"column": 63
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nA' : Type u_3\nB : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra A B\ninst✝³ : MulSemiringAction G B\nC : Type u_5\ninst✝² : CommSemiring C\ninst✝¹ : Algebra C B\nhA : IsGaloisGroup G A B\ninst✝ : FaithfulSMul A B\nx : A\nx✝ : G\n... | [] | by rw [smul_algebraMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 105,
"column": 22
} | {
"line": 105,
"column": 41
} | {
"line": 105,
"column": 41
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\np : ℕ\nhchar : ExpChar D p\na : D\nha : a ∉ k\nhinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\nm : ℕ\nhm : 1 ≤ m ∧ a ^ p ^ m ∈ k\nn : ℕ\nhn : p ^ m ≤ n\ninter : (⇑((ad (↥k) D) a))^[p ^ m] = 0\n⊢ Function.const D 0 = 0",
"ppT... | [
"D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\np : ℕ\nhchar : ExpChar D p\na : D\nha : a ∉ k\nhinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\nm : ℕ\nhm : 1 ≤ m ∧ a ^ p ^ m ∈ k\nn : ℕ\nhn : p ^ m ≤ n\ninter : (⇑((ad (↥k) D) a))^[p ^ m] = 0\n⊢ 0 = 0"
] | Function.const_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 526,
"column": 29
} | {
"line": 565,
"column": 81
} | {
"line": 567,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_4\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.Is... | [] | by
have := Algebra.IsInvariant.isIntegral A B G
have := isAlgebraic_of_isFractionRing (A ⧸ P) (B ⧸ Q) K L
constructor
intro x
obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (B ⧸ Q) x
obtain ⟨b, a, ha, h⟩ := (Algebra.IsAlgebraic.isAlgebraic (R := A ⧸ P) y).exists_smul_eq_mul x hy
obtain ⟨a, rfl⟩ :... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.MvRatFunc.Rank | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 46
} | {
"line": 40,
"column": 2
} | [
{
"pp": "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\nhinj : Function.Injective ⇑(algebraMap R K)\nh1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K\ni : σ\nhx : Transcendental F ((algebraMap... | [
"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\nhinj : Function.Injective ⇑(algebraMap R K)\nh1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K\ni : σ\nhx : Transcendental F ((algebraMap R K) (MvPol... | rw [lift_id'.{v, u}, lift_umax.{v, u}] at h2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 182,
"column": 49
} | {
"line": 185,
"column": 33
} | {
"line": 187,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsSeparable K L\nc : ConjRootClass K L\ninst✝ : Fintype ↑c.carrier\n⊢ c.minpoly.aroots L = c.carrier.toFinset.val",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Multiset.... | [] | by
classical
simp_rw [← rootSet_minpoly_eq_carrier, rootSet_def, Finset.toFinset_coe, Multiset.toFinset_val,
c.nodup_aroots_minpoly.dedup] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LinearDisjoint | {
"line": 648,
"column": 49
} | {
"line": 648,
"column": 90
} | {
"line": 648,
"column": 90
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nA : Type v\ninst✝⁷ : CommRing A\nB : Type w\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Flat R A\ninst✝² : Flat R B\ninst✝¹ : Algebra.Transcendental R A\ninst✝ : Algebra.Transcendental R B\nH : IsField (A ⊗[R] B)\nthis✝⁴ : Field (A ⊗[R] B)... | [
"R : Type u\ninst✝⁸ : CommRing R\nA : Type v\ninst✝⁷ : CommRing A\nB : Type w\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Flat R A\ninst✝² : Flat R B\ninst✝¹ : Algebra.Transcendental R A\ninst✝ : Algebra.Transcendental R B\nH : IsField (A ⊗[R] B)\nthis✝⁴ : Field (A ⊗[R] B) := H.toFiel... | ← Algebra.adjoin_singleton_eq_range_aeval | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 670,
"column": 2
} | {
"line": 673,
"column": 67
} | {
"line": 675,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type v\ninst✝⁵ : CommRing A\nB : Type w\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Flat R A\ninst✝ : Flat R B\nH : IsField (A ⊗[R] B)\n⊢ Algebra.IsAlgebraic R A ∨ Algebra.IsAlgebraic R B",
"ppTerm": "?m.31",
"assigned": true,
... | [] | by_contra! h
simp_rw [← Algebra.transcendental_iff_not_isAlgebraic] at h
obtain ⟨_, _⟩ := h
exact Algebra.TensorProduct.not_isField_of_transcendental R A B H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LinearDisjoint | {
"line": 670,
"column": 2
} | {
"line": 673,
"column": 67
} | {
"line": 675,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type v\ninst✝⁵ : CommRing A\nB : Type w\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Flat R A\ninst✝ : Flat R B\nH : IsField (A ⊗[R] B)\n⊢ Algebra.IsAlgebraic R A ∨ Algebra.IsAlgebraic R B",
"ppTerm": "?m.31",
"assigned": true,
... | [] | by_contra! h
simp_rw [← Algebra.transcendental_iff_not_isAlgebraic] at h
obtain ⟨_, _⟩ := h
exact Algebra.TensorProduct.not_isField_of_transcendental R A B H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 34
} | {
"line": 367,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\nhsep : ∀ (i : ι), IsSeparable F (v i)\nh : LinearIndependent F v\n⊢ LinearIndependent F fun x ↦ v x ^ q ^ n",
"ppTerm": "?m.29",
"assigned": true,
"usedConsta... | [
"F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\nhsep : ∀ (i : ι), IsSeparable F (v i)\nh : LinearIndependent F v\nE' : IntermediateField F E := adjoin F (Set.range v)\n⊢ LinearIndependent F fun x ↦ v x ^ q ^ n"
] | let E' := adjoin F (Set.range v) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 18
} | {
"line": 65,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\n⊢ (aeval X) (f.minpolyX ↥K⟮f⟯) = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RatFunc.minpolyX._proof_1",
"NonAs... | [] | simp only [aeval_sub, aeval_map_algebraMap, aeval_X_left_eq_algebraMap, map_mul, aeval_C,
IntermediateField.algebraMap_apply, coe_algebraMap]
nth_rw 2 [← num_div_denom f]
rw [div_mul_cancel₀ _ (algebraMap_ne_zero f.denom_ne_zero)]
exact sub_self _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 18
} | {
"line": 65,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\n⊢ (aeval X) (f.minpolyX ↥K⟮f⟯) = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RatFunc.minpolyX._proof_1",
"NonAs... | [] | simp only [aeval_sub, aeval_map_algebraMap, aeval_X_left_eq_algebraMap, map_mul, aeval_C,
IntermediateField.algebraMap_apply, coe_algebraMap]
nth_rw 2 [← num_div_denom f]
rw [div_mul_cancel₀ _ (algebraMap_ne_zero f.denom_ne_zero)]
exact sub_self _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Relrank | {
"line": 230,
"column": 47
} | {
"line": 237,
"column": 27
} | {
"line": 239,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\nh1 : A ≤ B\nh2 : B ≤ C\n⊢ A.relrank B * B.relrank C = A.relrank C",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subfield.toDivisionRing",
"SubsemiringClass.nontrivial",
"Subfield.relrank",
... | [] | by
have h3 := h1.trans h2
rw [relrank_eq_rank_of_le h1, relrank_eq_rank_of_le h2, relrank_eq_rank_of_le h3]
letI : Algebra A B := (inclusion h1).toAlgebra
letI : Algebra B C := (inclusion h2).toAlgebra
letI : Algebra A C := (inclusion h3).toAlgebra
haveI : IsScalarTower A B C := IsScalarTower.of_algebraMap_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Relrank | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 69
} | {
"line": 262,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\nh : B ≤ C\n⊢ A.relfinrank B * 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_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.FieldTheory.Relrank | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 69
} | {
"line": 262,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\nh : B ≤ C\n⊢ A.relfinrank B * 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_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 69
} | {
"line": 262,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\nh : B ≤ C\n⊢ A.relfinrank B * 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_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Relrank | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 69
} | {
"line": 487,
"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\nh : B ≤ C\n⊢ A.relfinrank B * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.FieldTheory.Relrank | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 69
} | {
"line": 487,
"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\nh : B ≤ C\n⊢ A.relfinrank B * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 69
} | {
"line": 487,
"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\nh : B ≤ C\n⊢ A.relfinrank B * B.relfinrank C = (A ⊓ B).relfinrank C",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"HMul.hMul",
... | [] | simpa using! congr(toNat $(relrank_mul_relrank_eq_inf_relrank A h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Diffeology.Basic | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 40
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case e'_3\nX : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : DiffeologicalSpace X\ninst✝³ : IsDTopologyCompatible X\ninst✝² : TopologicalSpace Y\ninst✝¹ : DiffeologicalSpace Y\ninst✝ : IsDTopologyCompatible Y\nf : X → Y\nhf : DSmooth f\n⊢ inst✝⁵ = dTopology",
"ppTerm": "?e'_3",
... | [
"case e'_4\nX : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : DiffeologicalSpace X\ninst✝³ : IsDTopologyCompatible X\ninst✝² : TopologicalSpace Y\ninst✝¹ : DiffeologicalSpace Y\ninst✝ : IsDTopologyCompatible Y\nf : X → Y\nhf : DSmooth f\n⊢ inst✝² = dTopology"
] | · rw [IsDTopologyCompatible.dTop_eq X] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 61,
"column": 52
} | {
"line": 64,
"column": 17
} | {
"line": 66,
"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\nm n : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\n⊢ (s.reindex e).altitude = s.altitude ∘ ⇑e.symm",
"ppTerm": "?m.56",
"assigned": true,
"... | [] | by
ext i
simp_rw [altitude, reindex_points, Set.image_comp, Equiv.image_compl]
simp [altitude] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Projection | {
"line": 145,
"column": 95
} | {
"line": 152,
"column": 12
} | {
"line": 154,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ ↑((orthogonalProjecti... | [] | by
constructor
· exact fun h => h ▸ orthogonalProjection_mem p
· intro h
have hp : p ∈ (s : Set P) ∩ mk' p s.directionᗮ := ⟨h, self_mem_mk' p _⟩
rw [inter_eq_singleton_orthogonalProjection p] at hp
symm
exact hp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Projection | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 50
} | {
"line": 228,
"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... | rw [← inter_eq_singleton_orthogonalProjection] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 50
} | {
"line": 383,
"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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\n⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| < s.heigh... | [] | exact abs_inner_vsub_altitudeFoot_lt_mul _ hij | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 50
} | {
"line": 383,
"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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\n⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| < s.heigh... | [] | exact abs_inner_vsub_altitudeFoot_lt_mul _ hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 50
} | {
"line": 383,
"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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\n⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| < s.heigh... | [] | exact abs_inner_vsub_altitudeFoot_lt_mul _ hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 33
} | {
"line": 234,
"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 : P\nhp₁p₂ : p₁ ≠ p₂\n⊢ InnerProductGeometry.angle (p₁ -ᵥ p₂) (-(p₁ -ᵥ p₂)) = π",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"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 : P\nhp₁p₂ : p₁ ≠ p₂\n⊢ p₁ -ᵥ p₂ ≠ 0"
] | apply angle_self_neg_of_nonzero | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 37
} | {
"line": 271,
"column": 2
} | [
{
"pp": "case inl\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ l... | [] | simp [-neg_vsub_eq_vsub_rev, hlt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 37
} | {
"line": 271,
"column": 2
} | [
{
"pp": "case inl\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ l... | [] | simp [-neg_vsub_eq_vsub_rev, hlt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 37
} | {
"line": 271,
"column": 2
} | [
{
"pp": "case inl\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ l... | [] | simp [-neg_vsub_eq_vsub_rev, hlt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 31
} | {
"line": 420,
"column": 4
} | [
{
"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\np₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃ ∨ Wbtw ℝ p₂ p₃ p₁ ∨ Wbtw ℝ p₃ p₁ p₂\nh₁₂ : ¬p₁ = p₂\nh₃₂ : p₃ = p₂\n⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ ... | [
"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\np₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃ ∨ Wbtw ℝ p₂ p₃ p₁ ∨ Wbtw ℝ p₃ p₁ p₂\nh₁₂ : ¬p₁ = p₂\nh₃₂ : ¬p₃ = p₂\n⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π"
] | · exact Or.inr (Or.inl h₃₂) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 19
} | {
"line": 425,
"column": 4
} | [
{
"pp": "case mpr.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\nhy : y ≠ 0\n⊢ ‖x‖ = ‖y‖ ∧ SameRay ℝ x y → x = y",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"case mpr.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\nhy : y ≠ 0\nh₁ : ‖x‖ = ‖y‖\nh₂ : SameRay ℝ x y\n⊢ x = y"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 449,
"column": 4
} | {
"line": 450,
"column": 71
} | {
"line": 451,
"column": 2
} | [
{
"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\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ ↑(↑‖y‖ ^ 2 * (o.kahler x) z).arg = ↑((o.kahler x) z).arg",
"ppTerm": "?m.61",
"assigned": true,
"usedCon... | [] | congr 1
exact mod_cast Complex.arg_real_mul _ (by positivity : 0 < ‖y‖ ^ 2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 449,
"column": 4
} | {
"line": 450,
"column": 71
} | {
"line": 451,
"column": 2
} | [
{
"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\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ ↑(↑‖y‖ ^ 2 * (o.kahler x) z).arg = ↑((o.kahler x) z).arg",
"ppTerm": "?m.61",
"assigned": true,
"usedCon... | [] | congr 1
exact mod_cast Complex.arg_real_mul _ (by positivity : 0 < ‖y‖ ^ 2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 730,
"column": 21
} | {
"line": 730,
"column": 84
} | {
"line": 732,
"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 : V\nh : o.oangle x y = ↑(π / 2)\n⊢ ⟪y, x⟫ = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Inner.i... | [] | rw [real_inner_comm, o.inner_eq_zero_of_oangle_eq_pi_div_two h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 730,
"column": 21
} | {
"line": 730,
"column": 84
} | {
"line": 732,
"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 : V\nh : o.oangle x y = ↑(π / 2)\n⊢ ⟪y, x⟫ = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Inner.i... | [] | rw [real_inner_comm, o.inner_eq_zero_of_oangle_eq_pi_div_two h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 730,
"column": 21
} | {
"line": 730,
"column": 84
} | {
"line": 732,
"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 : V\nh : o.oangle x y = ↑(π / 2)\n⊢ ⟪y, x⟫ = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Inner.i... | [] | rw [real_inner_comm, o.inner_eq_zero_of_oangle_eq_pi_div_two h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 949,
"column": 20
} | {
"line": 949,
"column": 45
} | {
"line": 949,
"column": 45
} | [
{
"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\nh : ‖x‖ = ‖y‖\nhn : ¬x = y\n| (o.oangle (y - x) y).sign",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Orientation.oangle_sign... | [
"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\nh : ‖x‖ = ‖y‖\nhn : ¬x = y\n| (o.oangle y x).sign"
] | oangle_sign_sub_left_swap | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 664,
"column": 91
} | {
"line": 667,
"column": 98
} | {
"line": 669,
"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)\n⊢ (∡ p₃ p₁ p₂).cos * dist p₁ p₃ = dist p₁ p₂",
"ppTer... | [] | by
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.cos_coe, dist_comm p₁ p₃,
cos_angle_mul_dist_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Basic | {
"line": 134,
"column": 4
} | {
"line": 135,
"column": 48
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case refine_1\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\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 : ... | · intro i
fin_cases i <;> simp [b, hc.symm, hp.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Basic | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 15
} | [
{
"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\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 : 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\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 : p₁ ∈ s\nhp₂s : ... | ← hbs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 78,
"column": 2
} | {
"line": 80,
"column": 6
} | {
"line": 82,
"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\np : P\n⊢ ↑((orthogonalProjection (s.orthRadius p)) s.center) = p",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Nor... | [] | simp_rw [orthRadius, coe_orthogonalProjection_eq_iff_mem]
rw [← Submodule.neg_mem_iff]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 78,
"column": 2
} | {
"line": 80,
"column": 6
} | {
"line": 82,
"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\np : P\n⊢ ↑((orthogonalProjection (s.orthRadius p)) s.center) = p",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Nor... | [] | simp_rw [orthRadius, coe_orthogonalProjection_eq_iff_mem]
rw [← Submodule.neg_mem_iff]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Bisector | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 11
} | {
"line": 168,
"column": 4
} | [
{
"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\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np p' : P\ns₁ s₂ : AffineSubspace ℝ P\nhp'₁ : p' ∈ s₁\nhp'₂ : p' ∈ s₂\nhne : ↑((orthogonalProjec... | [
"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\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np p' : P\ns₁ s₂ : AffineSubspace ℝ P\nhp'₁ : p' ∈ s₁\nhp'₂ : p' ∈ s₂\nhne : ↑((orthogonalProjection s₁) p) ... | have h₁'' : (orthogonalProjection s₁ p : P) = (orthogonalProjection (s₁ ⊓ s₂) p : P) := by
rw [← orthogonalProjection_orthogonalProjection_of_le inf_le_left, eq_comm,
orthogonalProjection_eq_self_iff]
grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 74
} | {
"line": 100,
"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\ns : AffineSubspace ℝ P\ninst✝ : s.direction.HasOrthogonalProjection\nps : Set P\nhnps : ps.Nonempty\np : P\nhps : ps ⊆ ↑s\nhp : p ∉ s\nthis : Nonempty ↥s\ncc ... | [] | rwa [mem_affineSpan_insert_iff (orthogonalProjection_mem p)] at hcc₃ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 74
} | {
"line": 100,
"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\ns : AffineSubspace ℝ P\ninst✝ : s.direction.HasOrthogonalProjection\nps : Set P\nhnps : ps.Nonempty\np : P\nhps : ps ⊆ ↑s\nhp : p ∉ s\nthis : Nonempty ↥s\ncc ... | [] | rwa [mem_affineSpan_insert_iff (orthogonalProjection_mem p)] at hcc₃ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 74
} | {
"line": 100,
"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\ns : AffineSubspace ℝ P\ninst✝ : s.direction.HasOrthogonalProjection\nps : Set P\nhnps : ps.Nonempty\np : P\nhps : ps ⊆ ↑s\nhp : p ∉ s\nthis : Nonempty ↥s\ncc ... | [] | rwa [mem_affineSpan_insert_iff (orthogonalProjection_mem p)] at hcc₃ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 285,
"column": 4
} | {
"line": 293,
"column": 8
} | {
"line": 294,
"column": 2
} | [
{
"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 : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nh : p₁ ≠ p₂\n⊢ ‖midpoint ℝ p₁ p₂ -ᵥ p₁‖ / (1 ... | [] | norm_cast
rw [one_div, div_inv_eq_mul, ← mul_self_inj (by positivity) (by positivity),
norm_add_sq_eq_norm_sq_add_norm_sq_real (o.inner_smul_rotation_pi_div_two_right _ _),
← mul_assoc, mul_comm, mul_comm _ (√_), ← mul_assoc, ← mul_assoc,
Real.mul_self_sqrt (by positivity), norm_smul, LinearIsomet... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 285,
"column": 4
} | {
"line": 293,
"column": 8
} | {
"line": 294,
"column": 2
} | [
{
"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 : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nh : p₁ ≠ p₂\n⊢ ‖midpoint ℝ p₁ p₂ -ᵥ p₁‖ / (1 ... | [] | norm_cast
rw [one_div, div_inv_eq_mul, ← mul_self_inj (by positivity) (by positivity),
norm_add_sq_eq_norm_sq_add_norm_sq_real (o.inner_smul_rotation_pi_div_two_right _ _),
← mul_assoc, mul_comm, mul_comm _ (√_), ← mul_assoc, ← mul_assoc,
Real.mul_self_sqrt (by positivity), norm_smul, LinearIsomet... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 98
} | {
"line": 329,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ (s.reindex e).circumcenter = s.circumcenter",
"ppTerm": "?m.46",
"assigned": true,
"us... | [] | simp_rw [circumcenter, circumsphere_reindex] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 98
} | {
"line": 329,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ (s.reindex e).circumcenter = s.circumcenter",
"ppTerm": "?m.46",
"assigned": true,
"us... | [] | simp_rw [circumcenter, circumsphere_reindex] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 98
} | {
"line": 329,
"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\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ (s.reindex e).circumcenter = s.circumcenter",
"ppTerm": "?m.46",
"assigned": true,
"us... | [] | simp_rw [circumcenter, circumsphere_reindex] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 383,
"column": 2
} | {
"line": 393,
"column": 98
} | {
"line": 395,
"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... | [] | 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_orthogonalProjectio... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 383,
"column": 2
} | {
"line": 393,
"column": 98
} | {
"line": 395,
"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... | [] | 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_orthogonalProjectio... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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