module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 22
} | {
"line": 191,
"column": 6
} | [
{
"pp": "case inl\nφ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh✝ : ↑T0 ⊨ᵇ φ\nψ : Language.ring.Sentence\nh : ψ ∈ Theory.field ∪ genericMonicPolyHasRoot '' {n | 0 < n}\n⊢ Nonempty { s // ∀ q ∉ s, Theory.ACF ↑q ⊨ᵇ ψ }",
"ppTerm": "?inl",
"assigned": true,
... | [
"case inl\nφ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh✝ : ↑T0 ⊨ᵇ φ\nψ : Language.ring.Sentence\nh : ψ ∈ Theory.field ∪ genericMonicPolyHasRoot '' {n | 0 < n}\n⊢ ∀ q ∉ ∅, Theory.ACF ↑q ⊨ᵇ ψ"
] | refine ⟨⟨∅, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 209,
"column": 17
} | {
"line": 209,
"column": 20
} | {
"line": 210,
"column": 4
} | [
{
"pp": "φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\nf : (ψ : Language.ring.Sentence) → ψ ∈ Theory.ACF 0 → { s // ∀ q ∉ s, Theory.ACF ↑q ⊨ᵇ ψ }\ns : Finset Nat.Primes := T0.attach.biUnion fun φ ↦ ↑(f ↑φ ⋯)\np : Nat.Primes\nψ : Language.ring.Sentence\nh... | [
"φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\nf : (ψ : Language.ring.Sentence) → ψ ∈ Theory.ACF 0 → { s // ∀ q ∉ s, Theory.ACF ↑q ⊨ᵇ ψ }\ns : Finset Nat.Primes := T0.attach.biUnion fun φ ↦ ↑(f ↑φ ⋯)\np : Nat.Primes\nψ : Language.ring.Sentence\nhψ : ψ ∈ T0\n... | hpψ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 85
} | {
"line": 80,
"column": 2
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\n⊢ (∏ n, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees ≤ ∑ s, (Fintype.card K - 1) • {s}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [
"K : Type u_1\nσ : Type u_2\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees 1 ∪ ((X s - C (c s)) ^ (Fintype.card K - 1)).degrees ≤ (Fintype.card K - 1) • {s}"
] | refine degrees_prod_le.trans <| Finset.sum_le_sum fun s _ ↦ degrees_sub_le.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 203,
"column": 48
} | {
"line": 203,
"column": 79
} | {
"line": 203,
"column": 79
} | [
{
"pp": "case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ IsNoetherian K (R σ K)",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsNoetherian.iff_rank_lt_aleph0",
"Preorder.toLT",
"Cardinal",
"co... | [
"case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ Module.rank K (R σ K) < ℵ₀"
] | IsNoetherian.iff_rank_lt_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 160,
"column": 4
} | {
"line": 162,
"column": 44
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nh : x ∈ Ideal.comap i (pNilradical L p)\n⊢ ∃ n, x ^ p ^ n = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | obtain ⟨n, h⟩ := mem_pNilradical.1 <| Ideal.mem_comap.1 h
obtain ⟨m, h⟩ := mem_pNilradical.1 <| ker_le i p ((map_pow i x _).symm ▸ h)
exact ⟨n + m, by rwa [pow_add, pow_mul]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 160,
"column": 4
} | {
"line": 162,
"column": 44
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nh : x ∈ Ideal.comap i (pNilradical L p)\n⊢ ∃ n, x ^ p ^ n = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | obtain ⟨n, h⟩ := mem_pNilradical.1 <| Ideal.mem_comap.1 h
obtain ⟨m, h⟩ := mem_pNilradical.1 <| ker_le i p ((map_pow i x _).symm ▸ h)
exact ⟨n + m, by rwa [pow_add, pow_mul]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 63
} | {
"line": 200,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : CommSemiring K\ninst✝² : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝¹ : ExpChar L p\ninst✝ : PerfectRing L p\nx : K\nh✝ : x ∈ pNilradical K p\nn : ℕ\nh : x ^ p ^ n = 0\n⊢ x ∈ ker i",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"iterateFro... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : CommSemiring K\ninst✝² : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝¹ : ExpChar L p\ninst✝ : PerfectRing L p\nx : K\nh✝ : x ∈ pNilradical K p\nn : ℕ\nh : (iterateFrobeniusEquiv L p n).symm (i (x ^ p ^ n)) = (iterateFrobeniusEquiv L p n).symm (i 0)\n⊢ x ∈ ker i"
] | replace h := congr((iterateFrobeniusEquiv L p n).symm (i $h)) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.GroupTheory.CosetCover | {
"line": 268,
"column": 48
} | {
"line": 268,
"column": 61
} | {
"line": 268,
"column": 62
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\ninst✝ : DecidablePred FiniteIndex\nD : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k\nhD : D.FiniteIndex\nhD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex ... | [
"case pos\nG : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\ninst✝ : DecidablePred FiniteIndex\nD : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k\nhD : D.FiniteIndex\nhD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i\nt... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Nontrivial | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 66
} | {
"line": 43,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nha : Function.Injective ⇑(algebraMap R A)\nhb : Function.Injective ⇑(algebraMap R B)\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\nthis : IsDomain R\nFR : Typ... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nha : Function.Injective ⇑(algebraMap R A)\nhb : Function.Injective ⇑(algebraMap R B)\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\nthis✝ : IsDomain R\nFR : Type u_1 := Fr... | letI : CompatibleSMul FR R FA FB := CompatibleSMul.isScalarTower | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.FieldTheory.KummerExtension | {
"line": 522,
"column": 6
} | {
"line": 524,
"column": 87
} | {
"line": 525,
"column": 6
} | [
{
"pp": "case refine_2\nK : Type u\ninst✝³ : Field K\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\na : K\nα : L\nha : α ^ finrank K L = (algebraMap K L) a\nhα : K⟮α⟯ = ⊤\nq : K [X]\nhq : q.Monic\nhq' : (aeval α) q = 0\n⊢ (X ^ finrank K L - C a).degree ≤ (minpoly K α).degr... | [
"case refine_2\nK : Type u\ninst✝³ : Field K\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\na : K\nα : L\nha : α ^ finrank K L = (algebraMap K L) a\nhα : K⟮α⟯ = ⊤\nq : K [X]\nhq : q.Monic\nhq' : (aeval α) q = 0\n⊢ finrank K L ≤ finrank K ↥⊤"
] | rw [degree_X_pow_sub_C finrank_pos,
degree_eq_natDegree (minpoly.ne_zero (IsIntegral.of_finite K α)),
← IntermediateField.adjoin.finrank (IsIntegral.of_finite K α), hα, Nat.cast_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.NormalizedTrace | {
"line": 73,
"column": 57
} | {
"line": 73,
"column": 70
} | {
"line": 73,
"column": 71
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : CharZero F\ninst✝ : FiniteDimensional F K\na : K\nh : ↑(Module.finrank (↥F⟮a⟯) K) ≠ 0\n⊢ normalizedTraceAux F K a =\n ↑(Module.finrank (↥F⟮a⟯) K) * (trace F ↥F⟮a⟯) (AdjoinSimple.gen F a) /\n ↑(Module.... | [
"F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : CharZero F\ninst✝ : FiniteDimensional F K\na : K\nh : ↑(Module.finrank (↥F⟮a⟯) K) ≠ 0\n⊢ normalizedTraceAux F K a =\n ↑(Module.finrank (↥F⟮a⟯) K) * (trace F ↥F⟮a⟯) (AdjoinSimple.gen F a) /\n (↑(Module.finrank F ↥... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Subalgebra | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 66
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nA : Subalgebra R S\nx y : ↥⊥\na b : ↥A\nx' : R\nhx : (algebraMap R S) x' = ↑x\n⊢ (toSubmodule A).lTensorOne ((x * y) ⊗ₜ[R] (a * b)) =\n ... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nA : Subalgebra R S\nx y : ↥⊥\na b : ↥A\nx' : R\nhx : (algebraMap R ↥⊥) x' = x\n⊢ (toSubmodule A).lTensorOne ((x * y) ⊗ₜ[R] (a * b)) =\n (toSubmod... | replace hx : algebraMap R _ x' = x := Subtype.val_injective hx | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.RingTheory.LinearDisjoint | {
"line": 629,
"column": 2
} | {
"line": 629,
"column": 36
} | {
"line": 630,
"column": 2
} | [
{
"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 : 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✝ : Field (A ⊗[R] B) := H.toField... | haveI := hfa.isDomain fa.toRingHom | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.LinearAlgebra.TensorProduct.Subalgebra | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 66
} | {
"line": 96,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nA : Subalgebra R S\na b : ↥A\nx y : ↥⊥\nx' : R\nhx : (algebraMap R S) x' = ↑x\n⊢ (toSubmodule A).rTensorOne ((a * b) ⊗ₜ[R] (x * y)) =\n ... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nA : Subalgebra R S\na b : ↥A\nx y : ↥⊥\nx' : R\nhx : (algebraMap R ↥⊥) x' = x\n⊢ (toSubmodule A).rTensorOne ((a * b) ⊗ₜ[R] (x * y)) =\n (toSubmod... | replace hx : algebraMap R _ x' = x := Subtype.val_injective hx | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 21
} | {
"line": 224,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra E K\ninst✝² : IsScalarTower F E K\nS : Set K\ninst✝¹ : IsPurelyInseparable F E\nM : IntermediateField F K := adjoin F S\ninst✝ : Algebra.IsAlgebraic F ↥... | [
"F : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra E K\ninst✝² : IsScalarTower F E K\nS : Set K\ninst✝¹ : IsPurelyInseparable F E\nM : IntermediateField F K := adjoin F S\ninst✝ : Algebra.IsAlgebraic F ↥M\nL : Inter... | set L := adjoin E S | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 159,
"column": 57
} | {
"line": 159,
"column": 78
} | {
"line": 160,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ¬∃ c, f = C c\nthis✝ : UniqueFactorizationMonoid ↥K[f]\nH : (f.minpolyX ↥K[f]).natDegree = 0\nthis : (f.minpolyX ↥K⟮f⟯).natDegree = 0\n⊢ False",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWith... | [
"K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ¬∃ c, f = C c\nthis✝ : UniqueFactorizationMonoid ↥K[f]\nH : (f.minpolyX ↥K[f]).natDegree = 0\nthis : max f.num.natDegree f.denom.natDegree = 0\n⊢ False"
] | f.natDegree_minpolyX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 69
} | {
"line": 86,
"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\ni : Fin (n + 1)\n⊢ vectorSpan ℝ (s.points '' {i}ᶜ) ⟂ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)",
"pp... | [] | exact (Submodule.isOrtho_orthogonal_right _).mono_right inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 123,
"column": 7
} | {
"line": 123,
"column": 69
} | {
"line": 125,
"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)\nθ : Real.Angle\nx✝ : V\n⊢ (o.rotation θ).symm x✝ = (o.rotation (-θ)) x✝",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"LinearIsometryEq... | [] | simp [o.rotation_apply, o.rotation_symm_apply, sub_eq_add_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 43
} | {
"line": 276,
"column": 44
} | [
{
"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 y : V\nh : ‖x‖ = ‖y‖\n⊢ (o.rotation (o.oangle x y)) x = y",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv... | [
"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 y : V\nh : ‖x‖ = ‖y‖\n⊢ o.oangle ((o.rotation (o.oangle x y)) x) y = 0",
"case mpr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\nin... | rw [o.eq_iff_oangle_eq_zero_of_norm_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 222,
"column": 4
} | {
"line": 229,
"column": 19
} | {
"line": 231,
"column": 0
} | [
{
"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\n⊢ q ∈ s ∧ p... | [] | rintro ⟨hqs, hpq⟩
have hq : q ∈ mk' p s.directionᗮ := by
rwa [mem_mk', ← neg_mem_iff, neg_vsub_eq_vsub_rev]
suffices q ∈ ({(orthogonalProjection s p : P)} : Set P) by
simpa [eq_comm] using this
rw [← inter_eq_singleton_orthogonalProjection]
simp only [Set.mem_inter_iff, SetLike.mem_coe]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Projection | {
"line": 222,
"column": 4
} | {
"line": 229,
"column": 19
} | {
"line": 231,
"column": 0
} | [
{
"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\n⊢ q ∈ s ∧ p... | [] | rintro ⟨hqs, hpq⟩
have hq : q ∈ mk' p s.directionᗮ := by
rwa [mem_mk', ← neg_mem_iff, neg_vsub_eq_vsub_rev]
suffices q ∈ ({(orthogonalProjection s p : P)} : Set P) by
simpa [eq_comm] using this
rw [← inter_eq_singleton_orthogonalProjection]
simp only [Set.mem_inter_iff, SetLike.mem_coe]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 367,
"column": 2
} | {
"line": 376,
"column": 48
} | {
"line": 378,
"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\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\n⊢ -(s.height i * s.height j) < ⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoo... | [] | obtain rfl | hij := eq_or_ne i j
· rw [real_inner_self_eq_norm_sq]
refine lt_of_lt_of_le (b := 0) ?_ ?_
· rw [neg_lt_zero]
positivity
· positivity
rw [neg_lt]
refine lt_of_abs_lt ?_
rw [abs_neg]
exact abs_inner_vsub_altitudeFoot_lt_mul s hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 367,
"column": 2
} | {
"line": 376,
"column": 48
} | {
"line": 378,
"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\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\n⊢ -(s.height i * s.height j) < ⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoo... | [] | obtain rfl | hij := eq_or_ne i j
· rw [real_inner_self_eq_norm_sq]
refine lt_of_lt_of_le (b := 0) ?_ ?_
· rw [neg_lt_zero]
positivity
· positivity
rw [neg_lt]
refine lt_of_abs_lt ?_
rw [abs_neg]
exact abs_inner_vsub_altitudeFoot_lt_mul s hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 331,
"column": 42
} | {
"line": 345,
"column": 15
} | {
"line": 347,
"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)\nf : V ≃ₗᵢ[ℝ] V\nhd : 0 < LinearMap.det ↑f.toLinearEquiv\n⊢ ∃ θ, f = o.rotation θ",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"instInn... | [] | by
haveI : Nontrivial V := nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
use o.oangle x (f x)
apply LinearIsometryEquiv.toLinearEquiv_injective
apply LinearEquiv.toLinearMap_injective
apply (o.basisRightAngleRotation x hx).ext
intro i
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Projection | {
"line": 474,
"column": 69
} | {
"line": 483,
"column": 10
} | {
"line": 485,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace 𝕜 P\ninst✝³ : Nonempty ↥s₁\ninst✝² : Nonempty ↥s₂\ninst✝¹ : s₁.direction.HasOrthogonalProjection\n... | [] | by
rw [reflection_apply', reflection_apply']
constructor
· intro h
rw [← @vsub_eq_zero_iff_eq V, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_comm, add_sub_assoc,
vsub_sub_vsub_cancel_right, ←
two_smul 𝕜 ((orthogonalProjection s₁ p : P) -ᵥ orthogonalProjection s₂ p), smul_eq_zero] at h
simpa u... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 97
} | {
"line": 86,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : x ≠ 0 ∨ y ≠ 0\nhxy : ‖x + y‖ ^ 2 ≠ 0\n⊢ angle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGr... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : x ≠ 0 ∨ y ≠ 0\nhxy : ‖x + y‖ ^ 2 ≠ 0\n⊢ Real.arcsin √((‖x + y‖ ^ 2 - ‖x‖ ^ 2) / ‖x + y‖ ^ 2) = Real.arcsin (‖y‖ / ‖x + y‖)"
] | rw [angle_add_eq_arccos_of_inner_eq_zero h,
Real.arccos_eq_arcsin (div_nonneg (norm_nonneg _) (norm_nonneg _)), div_pow, one_sub_div hxy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 284,
"column": 22
} | {
"line": 284,
"column": 64
} | {
"line": 284,
"column": 65
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, -y⟫ = 0\n⊢ Real.sin (angle x (x + -y)) * ‖x + -y‖ = ‖y‖",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, -y⟫ = 0\n⊢ ‖-y‖ = ‖y‖"
] | sin_angle_add_mul_norm_of_inner_eq_zero h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 373,
"column": 2
} | {
"line": 376,
"column": 50
} | {
"line": 378,
"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\nh : ∠ p₁ p₂ p₃ = π / 2\n⊢ ∠ p₂ p₃ p₁ ≤ π / 2",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZero... | [] | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]
exact angle_add_le_pi_div_two_of_inner_eq_zero h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 373,
"column": 2
} | {
"line": 376,
"column": 50
} | {
"line": 378,
"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\nh : ∠ p₁ p₂ p₃ = π / 2\n⊢ ∠ p₂ p₃ p₁ ≤ π / 2",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZero... | [] | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]
exact angle_add_le_pi_div_two_of_inner_eq_zero h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 634,
"column": 35
} | {
"line": 636,
"column": 80
} | {
"line": 638,
"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)\nw x y z : V\nhw : w ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\nhs : (o.oangle w x).sign = (o.oangle y z).sign\n⊢ InnerProductGeometry.angle w x = InnerProductGeometry.... | [] | by
refine ⟨fun h => o.oangle_eq_of_angle_eq_of_sign_eq h hs, fun h => ?_⟩
rw [o.angle_eq_abs_oangle_toReal hw hx, o.angle_eq_abs_oangle_toReal hy hz, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 540,
"column": 2
} | {
"line": 542,
"column": 20
} | {
"line": 544,
"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 = ↑(Real.arccos ... | [] | rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_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.RightAngle | {
"line": 565,
"column": 2
} | {
"line": 567,
"column": 43
} | {
"line": 569,
"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 : (∡ p₂ p₃ p₁).sign = 1\n⊢ ∡ p₂ p₃ p₁ = ↑(Real.arctan ... | [] | rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(right_ne_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": 820,
"column": 6
} | {
"line": 820,
"column": 31
} | {
"line": 821,
"column": 4
} | [
{
"pp": "case 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\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 ×... | [] | simpa [hz] using (h' 0).1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 820,
"column": 6
} | {
"line": 820,
"column": 31
} | {
"line": 821,
"column": 4
} | [
{
"pp": "case 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\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 ×... | [] | simpa [hz] using (h' 0).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 820,
"column": 6
} | {
"line": 820,
"column": 31
} | {
"line": 821,
"column": 4
} | [
{
"pp": "case 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\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 ×... | [] | simpa [hz] using (h' 0).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 690,
"column": 2
} | {
"line": 692,
"column": 52
} | {
"line": 694,
"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 : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).tan * dist p₃ ... | [] | rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
tan_angle_mul_dist_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (right_ne_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.Affine | {
"line": 694,
"column": 2
} | {
"line": 751,
"column": 72
} | {
"line": 753,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\... | [] | · let s : Set (P × P × P) :=
(fun x : line[ℝ, p₁, p₂] × V => (x.1, p₅, x.2 +ᵥ (x.1 : P))) ''
Set.univ ×ˢ {v | SameRay ℝ (p₂ -ᵥ p₁) v ∧ v ≠ 0}
have hco : IsConnected s :=
haveI : ConnectedSpace line[ℝ, p₁, p₂] := AddTorsor.connectedSpace _ _
(isConnected_univ.prod (isConnected_setOf_sameRay... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 26
} | {
"line": 194,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝¹ : MetricSpace E\ninst✝ : MetricSpace F\nf : E → F\nhf : Isometry f\nps : Set E\nc : E\nr : ℝ\nhc : ∀ p ∈ ps, dist p c = r\np : E\nhp : p ∈ ps\n⊢ dist (f p) (f c) = r",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
... | [] | rw [hf.dist_eq, hc p hp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 556,
"column": 4
} | {
"line": 556,
"column": 29
} | {
"line": 557,
"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\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₂ s.center < s.radius\nh : p₁ = p₂\n⊢ 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫",
"ppTerm": "?pos✝",
... | [
"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\ns : Sphere P\np₁ p₂ : P\nhp₁ : dist p₂ s.center = s.radius\nhp₂ : dist p₂ s.center < s.radius\nh : p₁ = p₂\n⊢ 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫"
] | rw [h, mem_sphere] at hp₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 22
} | {
"line": 93,
"column": 22
} | [
{
"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\ns : Sphere P\nas : AffineSubspace ℝ P\nhr : s.center ∈ s\nhm : s.center ∈ as\n⊢ s.radius = 0 ∧ s.center ∈ as",
"ppTerm": "?refine_1",
"a... | [
"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\ns : Sphere P\nas : AffineSubspace ℝ P\nhr : s.radius = 0\nhm : s.center ∈ as\n⊢ s.radius = 0 ∧ s.center ∈ as"
] | center_mem_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 48
} | {
"line": 313,
"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\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 : ... | have hr : 0 ≤ s.radius := dist_nonneg.trans hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 163,
"column": 9
} | {
"line": 163,
"column": 14
} | {
"line": 163,
"column": 14
} | [
{
"pp": "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\ns : Sphere P\np : P\nhs : s.center ∈ s\nhsp : p = s.center\n⊢ p ∈ s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"congr... | [
"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\ns : Sphere P\np : P\nhs : p ∈ s\nhsp : p = s.center\n⊢ p ∈ s"
] | ← hsp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 167,
"column": 2
} | {
"line": 170,
"column": 19
} | {
"line": 172,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\nas : AffineSubspace ℝ P\nh : s.IsTangent as\np : P\nhp : p ∈ as\n⊢ s.radius ≤ dist p s.center",
"ppTerm": "?m.28",
"assigned": true,
... | [] | obtain ⟨x, h⟩ := h
refine le_of_sq_le_sq ?_ dist_nonneg
rw [h.dist_sq_eq_of_mem hp, le_add_iff_nonneg_right]
exact sq_nonneg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 167,
"column": 2
} | {
"line": 170,
"column": 19
} | {
"line": 172,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\nas : AffineSubspace ℝ P\nh : s.IsTangent as\np : P\nhp : p ∈ as\n⊢ s.radius ≤ dist p s.center",
"ppTerm": "?m.28",
"assigned": true,
... | [] | obtain ⟨x, h⟩ := h
refine le_of_sq_le_sq ?_ dist_nonneg
rw [h.dist_sq_eq_of_mem hp, le_add_iff_nonneg_right]
exact sq_nonneg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 327,
"column": 69
} | {
"line": 327,
"column": 91
} | {
"line": 328,
"column": 6
} | [
{
"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\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.cent... | [
"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\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 :... | vadd_right_cancel_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 60
} | {
"line": 261,
"column": 0
} | [
{
"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\ns : Sphere P\np : P\nas : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥as\ninst✝ : as.direction.HasOrthogonalProjection\nh : s.IsTangentAt p as\n⊢ p = ↑((orthogonal... | [] | refine h.eq_of_isTangentAt ?_
have h' := h.isTangent
rwa [isTangent_iff_isTangentAt_orthogonalProjection] at h' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 60
} | {
"line": 261,
"column": 0
} | [
{
"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\ns : Sphere P\np : P\nas : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥as\ninst✝ : as.direction.HasOrthogonalProjection\nh : s.IsTangentAt p as\n⊢ p = ↑((orthogonal... | [] | refine h.eq_of_isTangentAt ?_
have h' := h.isTangent
rwa [isTangent_iff_isTangentAt_orthogonalProjection] at h' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 359,
"column": 8
} | {
"line": 359,
"column": 22
} | {
"line": 359,
"column": 22
} | [
{
"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\ns₁ s₂ : Sphere P\nh₁ : s₁.center ∈ s₁\nh₂ : s₁.center ∈ s₂\n⊢ s₁.radius = 0 ∧ s₁.center ∈ s₂",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"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\ns₁ s₂ : Sphere P\nh₁ : s₁.radius = 0\nh₂ : s₁.center ∈ s₂\n⊢ s₁.radius = 0 ∧ s₁.center ∈ s₂"
] | center_mem_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 378,
"column": 8
} | {
"line": 378,
"column": 22
} | {
"line": 378,
"column": 22
} | [
{
"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\ns₁ s₂ : Sphere P\nh₁ : s₁.center ∈ s₁\nh₂ : s₁.center ∈ s₂\n⊢ s₁.radius = 0 ∧ s₁.center ∈ s₂",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"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\ns₁ s₂ : Sphere P\nh₁ : s₁.radius = 0\nh₂ : s₁.center ∈ s₂\n⊢ s₁.radius = 0 ∧ s₁.center ∈ s₂"
] | center_mem_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 371,
"column": 13
} | {
"line": 371,
"column": 35
} | {
"line": 371,
"column": 36
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center < s.radius\nhpc : p ≠ s.center\nv : ↥(s.orthRadius p).direction\nhv : ∀ (w : ↥(s... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center < s.radius\nhpc : p ≠ s.center\nv : ↥(s.orthRadius p).direction\nhv : ∀ (w : ↥(s.orthRadius ... | vadd_right_cancel_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 88
} | {
"line": 126,
"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\np₁ p₂ p₃ : P\ns : Sphere P\ninst✝ : Fact (finrank ℝ V = 2)\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhne₁₂ : p₁ ≠ p₂\nhne₂₃ : p₂ ≠ p₃\nhangle : ∠ p₁ p₂ p₃ = ... | [
"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\np₁ p₂ p₃ : P\ns : Sphere P\ninst✝ : Fact (finrank ℝ V = 2)\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhne₁₂ : p₁ ≠ p₂\nhne₂₃ : p₂ ≠ p₃\nhangle : ∠ p₁ p₂ p₃ = π / 2\nthis✝... | have := eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two
(Fact.out : finrank ℝ V = 2) hne hne₁₃ hp₁ hp₃ hp₂
hd.left_mem hd.right_mem (angle_eq_pi_div_two_iff_mem_sphere_ofDiameter.mp hangle) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 474,
"column": 9
} | {
"line": 474,
"column": 49
} | {
"line": 474,
"column": 49
} | [
{
"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\nh : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄\nhn : ¬Collinear ℝ {p₁, p₂, p₄}\n⊢ ¬Collinear... | [
"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\nh : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄\nhn : ¬Collinear ℝ {p₁, p₂, p₄}\n⊢ ¬Collinear ℝ {p₁, p₂, ... | ← collinear_iff_of_two_zsmul_oangle_eq h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 510,
"column": 11
} | {
"line": 510,
"column": 51
} | {
"line": 510,
"column": 51
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄\nhc : Collinear ℝ {p₁, p₂, p₄}\nhe : ¬p₁ = p₄... | [
"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\nh : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄\nhc : Collinear ℝ {p₁, p₂, p₄}\nhe : ¬p₁ = p₄\n⊢ Collinea... | ← collinear_iff_of_two_zsmul_oangle_eq h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 596,
"column": 2
} | {
"line": 596,
"column": 30
} | {
"line": 597,
"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\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\nfs : Finset (Fin (n + 1))\nm : ℕ\nhfs : #fs = m + 1\nhne... | [
"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\nfs : Finset (Fin (n + 1))\nm : ℕ\nhfs : #fs = m + 1\nhne : m ≠ n\nhm... | rw [range_face_points] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 736,
"column": 4
} | {
"line": 736,
"column": 83
} | {
"line": 736,
"column": 83
} | [
{
"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⊢ (affineSpan ℝ (Set.range (s.faceOppos... | [
"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 ↔ s.excenterWeights... | s.sOppSide_affineSpan_faceOpposite_point_right_iff h.sum_excenterWeights_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Congruence | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 80
} | {
"line": 87,
"column": 2
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | have h_bac' : ¬Collinear ℝ {b', a', c'} := by simpa [Set.insert_comm] using h' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Similarity | {
"line": 221,
"column": 38
} | {
"line": 223,
"column": 24
} | {
"line": 225,
"column": 0
} | [
{
"pp": "ι : Type u_1\nP₁ : Type u_3\nP₂ : Type u_4\nv₁ : ι → P₁\nv₂ : ι → P₂\ninst✝¹ : PseudoMetricSpace P₁\ninst✝ : PseudoMetricSpace P₂\n⊢ Similar v₁ v₂ ↔ ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": ... | [] | by
simp_rw [similar_iff_exists_pairwise_edist_eq, edist_nndist]
exact_mod_cast Iff.rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 51
} | {
"line": 65,
"column": 52
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | div_eq_div_iff (by positivity) (by positivity), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 28
} | {
"line": 50,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhr : ∀ (i j : Fin (n + 1)), i ≠ j → dist (s.points i) (s.points ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhr : dist (s.points i₁) (s.points i₂) = 0\n⊢ False"
] | replace hr := hr _ _ h₁₂ | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 44
} | {
"line": 199,
"column": 4
} | [
{
"pp": "n : ℕ\ni₁ i₂ : Fin (n + 3)\nh : i₁ ≠ i₂\ni : Fin (n + 2 + 1)\n⊢ #{i₁, i₂}ᶜ = n + 1",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"False",
"Fintype.card_fin",
"eq_false",
"congrArg",
"Compl.compl",
"Finset",
"Finset.card_compl",
"in... | [] | simp [card_compl, Fintype.card_fin, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 209,
"column": 42
} | {
"line": 209,
"column": 79
} | {
"line": 209,
"column": 79
} | [
{
"pp": "n : ℕ\ni₁ i₂ : Fin (n + 3)\nh : i₁ ≠ i₂\n⊢ ∑ x, mongePointWeightsWithCircumcenter n x - ∑ x, centroidWeightsWithCircumcenter {i₁, i₂}ᶜ x = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Finset.univ",
"Affine.Simplex.sum_mongePointWeigh... | [
"n : ℕ\ni₁ i₂ : Fin (n + 3)\nh : i₁ ≠ i₂\n⊢ 1 - ∑ x, centroidWeightsWithCircumcenter {i₁, i₂}ᶜ x = 0"
] | sum_mongePointWeightsWithCircumcenter | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 51
} | {
"line": 71,
"column": 52
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | div_eq_div_iff (by positivity) (by positivity), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 42
} | {
"line": 84,
"column": 42
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | ← div_eq_div_iff dist_a'b' dist_b'c' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 7
} | {
"line": 108,
"column": 2
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"case h\nV₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b ... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 284,
"column": 2
} | {
"line": 285,
"column": 8
} | {
"line": 286,
"column": 2
} | [
{
"pp": "case e'_2.e'_5.e'_9\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⊢ {i₁, i₂}ᶜ = Finset.map e.toEmbedding {e.symm i₁, e.s... | [
"case e'_3.e'_5\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⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"case e'_3.e'_6\... | · ext i
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 95
} | {
"line": 330,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ : Fin (n + 3)\np : P\nh : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂\ni₂ : Fin (n + 3)\nhne : i₁ ≠ i₂\n⊢ p -ᵥ 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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ : Fin (n + 3)\np : P\nh : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂\ni₂ : Fin (n + 3)\nhne : i₁ ≠ i₂\n⊢ p ∈ s.mongePlane i₁... | rw [← s.direction_mongePlane, vsub_right_mem_direction_iff_mem s.mongePoint_mem_mongePlane] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Group.Growth.LinearLowerBound | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 40
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nX : Finset G\nhX₁ : 1 ∈ X\nhX : X.Nontrivial\nn : ℕ\nhn : n ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}\nhmn : 0 ≤ n\nhm : ↑X ^ (0 - 1) = ↑(closure ↑X)\n⊢ ↑X ^ (n - 1) = ↑(closure ↑X)",
"ppTerm": "?inl",
"assigned": true,
"usedConsta... | [] | simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
hX.coe.not_subset_singleton] at hm | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Group.Growth.LinearLowerBound | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 40
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nX : Finset G\nhX₁ : 1 ∈ X\nhX : X.Nontrivial\nn : ℕ\nhn : n ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}\nhmn : 0 ≤ n\nhm : ↑X ^ (0 - 1) = ↑(closure ↑X)\n⊢ ↑X ^ (n - 1) = ↑(closure ↑X)",
"ppTerm": "?inl",
"assigned": true,
"usedConsta... | [] | simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
hX.coe.not_subset_singleton] at hm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Group.Growth.LinearLowerBound | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 40
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nX : Finset G\nhX₁ : 1 ∈ X\nhX : X.Nontrivial\nn : ℕ\nhn : n ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}\nhmn : 0 ≤ n\nhm : ↑X ^ (0 - 1) = ↑(closure ↑X)\n⊢ ↑X ^ (n - 1) = ↑(closure ↑X)",
"ppTerm": "?inl",
"assigned": true,
"usedConsta... | [] | simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
hX.coe.not_subset_singleton] at hm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Group.Growth.QuotientInter | {
"line": 64,
"column": 2
} | {
"line": 78,
"column": 52
} | {
"line": 80,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : DecidableEq G\nH : Subgroup G\ninst✝¹ : DecidablePred fun x ↦ x ∈ H\ninst✝ : H.Normal\nA : Finset G\nhAsymm : A⁻¹ = A\n⊢ #A ≤ #(image (⇑(QuotientGroup.mk' H)) A) * #({x ∈ A ^ 2 | x ∈ H})",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [] | classical
set π := QuotientGroup.mk' H
rw [card_eq_sum_card_image π]
refine sum_le_card_nsmul _ _ _ <| forall_mem_image.2 fun a ha ↦ ?_
calc
#{a' ∈ A | π a' = π a}
_ ≤ #({a' ∈ A | π a' = π a}⁻¹ * {a' ∈ A | π a' = π a}) :=
card_le_card_mul_left ⟨a⁻¹, by simpa⟩
_ ≤ #{x ∈ A⁻¹ * A | x ∈ H} := by
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Geometry.Group.Growth.QuotientInter | {
"line": 64,
"column": 2
} | {
"line": 78,
"column": 52
} | {
"line": 80,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : DecidableEq G\nH : Subgroup G\ninst✝¹ : DecidablePred fun x ↦ x ∈ H\ninst✝ : H.Normal\nA : Finset G\nhAsymm : A⁻¹ = A\n⊢ #A ≤ #(image (⇑(QuotientGroup.mk' H)) A) * #({x ∈ A ^ 2 | x ∈ H})",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [] | classical
set π := QuotientGroup.mk' H
rw [card_eq_sum_card_image π]
refine sum_le_card_nsmul _ _ _ <| forall_mem_image.2 fun a ha ↦ ?_
calc
#{a' ∈ A | π a' = π a}
_ ≤ #({a' ∈ A | π a' = π a}⁻¹ * {a' ∈ A | π a' = π a}) :=
card_le_card_mul_left ⟨a⁻¹, by simpa⟩
_ ≤ #{x ∈ A⁻¹ * A | x ∈ H} := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Group.Growth.QuotientInter | {
"line": 64,
"column": 2
} | {
"line": 78,
"column": 52
} | {
"line": 80,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : DecidableEq G\nH : Subgroup G\ninst✝¹ : DecidablePred fun x ↦ x ∈ H\ninst✝ : H.Normal\nA : Finset G\nhAsymm : A⁻¹ = A\n⊢ #A ≤ #(image (⇑(QuotientGroup.mk' H)) A) * #({x ∈ A ^ 2 | x ∈ H})",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [] | classical
set π := QuotientGroup.mk' H
rw [card_eq_sum_card_image π]
refine sum_le_card_nsmul _ _ _ <| forall_mem_image.2 fun a ha ↦ ?_
calc
#{a' ∈ A | π a' = π a}
_ ≤ #({a' ∈ A | π a' = π a}⁻¹ * {a' ∈ A | π a' = π a}) :=
card_le_card_mul_left ⟨a⁻¹, by simpa⟩
_ ≤ #{x ∈ A⁻¹ * A | x ∈ H} := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 317,
"column": 67
} | {
"line": 317,
"column": 96
} | {
"line": 317,
"column": 96
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_6\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh : IsManifold... | [] | (i.1.extend I).right_inv hx.1 | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 315,
"column": 4
} | {
"line": 318,
"column": 54
} | {
"line": 319,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_6\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh : IsManifold I (n + 1) M\nt... | [] | refine ((contDiffOn_fderiv_coord_change (I := I) i j).congr fun x hx => ?_).mono ?_
· rw [PartialEquiv.trans_source'] at hx
simp_rw [Function.comp_apply, tangentBundleCore_coordChange, (i.1.extend I).right_inv hx.1]
· exact (i.1.extend_image_source_inter j.1).subset | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 315,
"column": 4
} | {
"line": 318,
"column": 54
} | {
"line": 319,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_6\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh : IsManifold I (n + 1) M\nt... | [] | refine ((contDiffOn_fderiv_coord_change (I := I) i j).congr fun x hx => ?_).mono ?_
· rw [PartialEquiv.trans_source'] at hx
simp_rw [Function.comp_apply, tangentBundleCore_coordChange, (i.1.extend I).right_inv hx.1]
· exact (i.1.extend_image_source_inter j.1).subset | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential | {
"line": 126,
"column": 4
} | {
"line": 129,
"column": 52
} | {
"line": 130,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : ChartedSpace H M\nE' : Type u_5\ninst✝⁵ : NormedAddCo... | [] | intro z hz
apply (hs z hz.1).inter'
apply (hf z hz.1).preimage_mem_nhdsWithin
exact (isOpen_extChartAt_source y).mem_nhds hz.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential | {
"line": 126,
"column": 4
} | {
"line": 129,
"column": 52
} | {
"line": 130,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : ChartedSpace H M\nE' : Type u_5\ninst✝⁵ : NormedAddCo... | [] | intro z hz
apply (hs z hz.1).inter'
apply (hf z hz.1).preimage_mem_nhdsWithin
exact (isOpen_extChartAt_source y).mem_nhds hz.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 281,
"column": 2
} | {
"line": 284,
"column": 72
} | {
"line": 285,
"column": 2
} | [
{
"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... | have hφx : φ.source ∈ 𝓝 (e.extend I x) := by
simp_rw [φ, extendCoordChange, PartialEquiv.trans_source, PartialEquiv.symm_source,
Filter.inter_mem_iff, mem_interior_iff_mem_nhds.1 hx, true_and, e'.extend_source]
exact e.extend_preimage_mem_nhds hex <| e'.open_source.mem_nhds hex' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 359,
"column": 2
} | {
"line": 385,
"column": 42
} | {
"line": 387,
"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... | [] | wlog _ : IsRCLikeNormedField 𝕜
· simp [IsInteriorPoint, I'.range_eq_univ_of_not_isRCLikeNormedField ‹_›]
let _ := IsRCLikeNormedField.rclike 𝕜
let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
let _ : NormedSpace ℝ E' := NormedSpace.restrictScalars ℝ 𝕜 E'
-- Write everything in terms of extende... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 359,
"column": 2
} | {
"line": 385,
"column": 42
} | {
"line": 387,
"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... | [] | wlog _ : IsRCLikeNormedField 𝕜
· simp [IsInteriorPoint, I'.range_eq_univ_of_not_isRCLikeNormedField ‹_›]
let _ := IsRCLikeNormedField.rclike 𝕜
let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
let _ : NormedSpace ℝ E' := NormedSpace.restrictScalars ℝ 𝕜 E'
-- Write everything in terms of extende... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 281,
"column": 2
} | {
"line": 290,
"column": 43
} | {
"line": 291,
"column": 2
} | [
{
"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... | [
"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 : NormedAddCommGroup E'\ni... | have :
∀ᶠ y in 𝓝[range (I.prod I')] extChartAt (I.prod I') x x,
(extChartAt I x.1 ∘ Prod.fst ∘ (extChartAt (I.prod I') x).symm) y = y.1 := by
/- porting note: was
apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)
mfld_set_tac
-/
filter_upwards [extChartAt_targe... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 57
} | {
"line": 571,
"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... | [] | rw [mfderiv_id, mfderiv_const, ContinuousLinearMap.inr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 106,
"column": 2
} | {
"line": 112,
"column": 73
} | {
"line": 114,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace 𝕜 F\ninst✝¹⁰ : TopologicalSpace (TotalSpace F E)\ninst✝⁹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝⁸ : NormedAddCommGroup EB\ninst✝⁷ : ... | [] | intro x
unfold zeroSection
rw [mdifferentiableAt_section]
apply (mdifferentiableAt_const (c := 0)).congr_of_eventuallyEq
filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)] with y hy
using congr_arg Prod.snd <| (trivializationAt F E x).zeroSection 𝕜 hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 106,
"column": 2
} | {
"line": 112,
"column": 73
} | {
"line": 114,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace 𝕜 F\ninst✝¹⁰ : TopologicalSpace (TotalSpace F E)\ninst✝⁹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝⁸ : NormedAddCommGroup EB\ninst✝⁷ : ... | [] | intro x
unfold zeroSection
rw [mdifferentiableAt_section]
apply (mdifferentiableAt_const (c := 0)).congr_of_eventuallyEq
filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)] with y hy
using congr_arg Prod.snd <| (trivializationAt F E x).zeroSection 𝕜 hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 76
} | {
"line": 408,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : FiberBundle F E\ni... | [] | apply mdifferentiableAt_add_section hs <| mdifferentiableAt_neg_section ht | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 63
} | {
"line": 452,
"column": 2
} | [
{
"pp": "case hV\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W V₁ : (... | [
"case hV₁\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W V₁ : (x : M) → Ta... | · exact hV.differentiableWithinAt_mpullbackWithin_vectorField | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.GroupLieAlgebra | {
"line": 128,
"column": 11
} | {
"line": 128,
"column": 33
} | {
"line": 128,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Group G\ninst✝ : LieGroup I ... | [
"𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Group G\ninst✝ : LieGroup I (minSmoothne... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 63
} | {
"line": 485,
"column": 2
} | [
{
"pp": "case e_6.hV\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns t : Set M\nx : M\nV W ... | [
"case e_6.hW\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns t : Set M\nx : M\nV W : (x : M) → ... | · exact hV.differentiableWithinAt_mpullbackWithin_vectorField | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Homeomorph.TransferInstance | {
"line": 41,
"column": 10
} | {
"line": 41,
"column": 26
} | {
"line": 41,
"column": 26
} | [
{
"pp": "case e'_4\nR : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : TopologicalSpace β\ne : α ≃ β\nthis : TopologicalSpace α := e.topologicalSpace\n⊢ e.topologicalSpace = TopologicalSpace.coinduced (⇑e.symm) inst✝",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equ... | [
"case e'_4\nR : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : TopologicalSpace β\ne : α ≃ β\nthis : TopologicalSpace α := e.topologicalSpace\n⊢ e.topologicalSpace = TopologicalSpace.induced (⇑e) inst✝"
] | e.coinduced_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 548,
"column": 2
} | {
"line": 552,
"column": 83
} | {
"line": 553,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹³ : TopologicalSpace H\nE : Type u_3\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nH' : Type u_5\ninst✝⁸ : Topologi... | [
"𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹³ : TopologicalSpace H\nE : Type u_3\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nH' : Type u_5\ninst✝⁸ : TopologicalSpace H'\... | have : mfderiv[range I] (extChartAt I x₀).symm (extChartAt I x₀ x₀) =
mfderiv[(extChartAt I x₀).symm ⁻¹' s ∩ range I] (extChartAt I x₀).symm (extChartAt I x₀ x₀) :=
(MDifferentiableWithinAt.mfderivWithin_mono
(mdifferentiableWithinAt_extChartAt_symm (mem_extChartAt_target x₀))
(UniqueDiffWithinAt.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 220,
"column": 4
} | {
"line": 223,
"column": 34
} | {
"line": 224,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Convex.convex_isRCLikeNormedField",
... | [] | simp only [instIsRCLikeNormedField, ↓reduceDIte]
apply Convex.convex_isRCLikeNormedField
rw [range_euclideanQuadrant]
exact EuclideanQuadrant.convex | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 220,
"column": 4
} | {
"line": 223,
"column": 34
} | {
"line": 224,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Convex.convex_isRCLikeNormedField",
... | [] | simp only [instIsRCLikeNormedField, ↓reduceDIte]
apply Convex.convex_isRCLikeNormedField
rw [range_euclideanQuadrant]
exact EuclideanQuadrant.convex | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 40
} | {
"line": 277,
"column": 40
} | [
{
"pp": "x y : ℝ\nh : Fact (x < y)\n⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄, x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨min ((↑x_1).ofLp 0 + x) y, ⋯⟩ ∈ {z | ↑z < y}",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Lattice.to... | [
"x y : ℝ\nh : Fact (x < y)\n⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄, (↑x_1).ofLp 0 < y - x → (↑x_1).ofLp 0 + x < y ∨ y < y"
] | simp only [min_lt_iff, mem_setOf_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 30
} | {
"line": 326,
"column": 31
} | [
{
"pp": "x y : ℝ\nhxy : Fact (x < y)\n⊢ ↑((IccLeftChart x y).extend (𝓡∂ 1)) ⊥ ∈ frontier (range ↑(𝓡∂ 1))",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"frontier",
"Real",
... | [
"x y : ℝ\nhxy : Fact (x < y)\n⊢ 0 ∈ frontier (range ↑(𝓡∂ 1))"
] | IccLeftChart_extend_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 351,
"column": 4
} | {
"line": 357,
"column": 62
} | {
"line": 358,
"column": 2
} | [
{
"pp": "x✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nx y : ℝ\nh : Fact (x < y)\n⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,\n x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨toLp 2 fun x_2 ↦ y - ↑⟨max (y - (↑x_1).ofLp 0) x, ⋯⟩, ⋯⟩ = x_1",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.to... | [] | rintro ⟨z, hz⟩ h'z
rw [Subtype.mk_eq_mk]
ext i
dsimp at hz h'z
have A : x ≤ y - z 0 := by linarith
rw [Subsingleton.elim i 0]
simp only [Fin.isValue, A, sup_of_le_left, sub_sub_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 351,
"column": 4
} | {
"line": 357,
"column": 62
} | {
"line": 358,
"column": 2
} | [
{
"pp": "x✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nx y : ℝ\nh : Fact (x < y)\n⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,\n x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨toLp 2 fun x_2 ↦ y - ↑⟨max (y - (↑x_1).ofLp 0) x, ⋯⟩, ⋯⟩ = x_1",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.to... | [] | rintro ⟨z, hz⟩ h'z
rw [Subtype.mk_eq_mk]
ext i
dsimp at hz h'z
have A : x ≤ y - z 0 := by linarith
rw [Subsingleton.elim i 0]
simp only [Fin.isValue, A, sup_of_le_left, sub_sub_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 457,
"column": 33
} | {
"line": 457,
"column": 60
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case inl\nx✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM✝ : Type u_3\ninst✝² : TopologicalSpace M✝\ninst✝¹ : ChartedSpace H M✝\nx y : ℝ\ninst✝ : Fact (x < y)\nn : ℕ∞ω\nM :\n... | [
"case inl.inl\nx✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM✝ : Type u_3\ninst✝² : TopologicalSpace M✝\ninst✝¹ : ChartedSpace H M✝\nx y : ℝ\ninst✝ : Fact (x < y)\nn : ℕ∞ω\nM :\n ContDi... | rcases he' with (rfl | rfl) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 457,
"column": 33
} | {
"line": 457,
"column": 60
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case inr\nx✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM✝ : Type u_3\ninst✝² : TopologicalSpace M✝\ninst✝¹ : ChartedSpace H M✝\nx y : ℝ\ninst✝ : Fact (x < y)\nn : ℕ∞ω\nM :\n... | [
"case inr.inl\nx✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM✝ : Type u_3\ninst✝² : TopologicalSpace M✝\ninst✝¹ : ChartedSpace H M✝\nx y : ℝ\ninst✝ : Fact (x < y)\nn : ℕ∞ω\nM :\n ContDi... | rcases he' with (rfl | rfl) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.Immersion | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 96
} | {
"line": 496,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\nE''' : Type u_4\nE'' : Type u\nF : Type u_5\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\ninst✝¹² : NormedAddCommGroup E'''\ninst✝¹¹ : NormedSpace 𝕜 E'''\ni... | [
"𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\nE''' : Type u_4\nE'' : Type u\nF : Type u_5\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\ninst✝¹⁴ : NormedAddCommGroup E''\ninst✝¹³ : NormedSpace 𝕜 E''\ninst✝¹² : NormedAddCommGroup E'''\ninst✝¹¹ : NormedSpace 𝕜 E'''\ninst✝¹⁰ : Nor... | have : hφ.domChart.source ∈ 𝓝 (f x) := hφ.domChart.open_source.mem_nhds hφ.mem_domChart_source | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Instances.Sphere | {
"line": 130,
"column": 2
} | {
"line": 132,
"column": 67
} | {
"line": 133,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nw : E\nhw : w ∈ (ℝ ∙ v)ᗮ\nh₁ : 0 < ‖w‖ ^ 2 + 4\n⊢ stereoInvFunAux v w ∈ sphere 0 1",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.nor... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv : E\nhv : ‖v‖ = 1\nw : E\nhw : w ∈ (ℝ ∙ v)ᗮ\nh₁ : 0 < ‖w‖ ^ 2 + 4\n⊢ ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4"
] | suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4 by
simp only [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, this,
abs_of_pos h₁, stereoInvFunAux_apply, inv_mul_cancel₀ h₁.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
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