module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.JacobsonNoether | {
"line": 79,
"column": 2
} | {
"line": 84,
"column": 51
} | [
{
"pp": "D : Type u_1\ninst✝² : DivisionRing D\ninst✝¹ : Algebra.IsAlgebraic (↥k) D\np : ℕ\ninst✝ : ExpChar D p\na : D\nha : a ∉ k\nhinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\n⊢ ∃ n, 1 ≤ n ∧ a ^ p ^ n ∈ k",
"usedConstants": [
"Iff.mpr",
"MulOne.toOne",
"False",
"Subring.instSetLi... | obtain ⟨n, hn⟩ := exists_pow_mem_center_of_inseparable p a hinsep
have nzero : n ≠ 0 := by
rintro rfl
rw [pow_zero, pow_one] at hn
exact ha hn
exact ⟨n, ⟨Nat.one_le_iff_ne_zero.mpr nzero, hn⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 79,
"column": 2
} | {
"line": 84,
"column": 51
} | [
{
"pp": "D : Type u_1\ninst✝² : DivisionRing D\ninst✝¹ : Algebra.IsAlgebraic (↥k) D\np : ℕ\ninst✝ : ExpChar D p\na : D\nha : a ∉ k\nhinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\n⊢ ∃ n, 1 ≤ n ∧ a ^ p ^ n ∈ k",
"usedConstants": [
"Iff.mpr",
"MulOne.toOne",
"False",
"Subring.instSetLi... | obtain ⟨n, hn⟩ := exists_pow_mem_center_of_inseparable p a hinsep
have nzero : n ≠ 0 := by
rintro rfl
rw [pow_zero, pow_one] at hn
exact ha hn
exact ⟨n, ⟨Nat.one_le_iff_ne_zero.mpr nzero, hn⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing K\ninst✝⁷ : CommRing L\ninst✝⁶ : CommRing M\ninst✝⁵ : CommRing N\ni : K →+* L\nj : K →+* M\nk : K →+* N\nf : L →+* M\ng : L →+* N\np : ℕ\ninst✝⁴ : ExpChar M p\ninst✝³ : ExpChar K p\ninst✝² : PerfectRing M p\ninst✝¹ : IsPRadical i... | nth_rw 1 [iterateFrobeniusEquiv_symm_add_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing K\ninst✝⁷ : CommRing L\ninst✝⁶ : CommRing M\ninst✝⁵ : CommRing N\ni : K →+* L\nj : K →+* M\nk : K →+* N\nf : L →+* M\ng : L →+* N\np : ℕ\ninst✝⁴ : ExpChar M p\ninst✝³ : ExpChar K p\ninst✝² : PerfectRing M p\ninst✝¹ : IsPRadical i... | nth_rw 1 [iterateFrobeniusEquiv_symm_add_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 167,
"column": 28
} | {
"line": 167,
"column": 48
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : a ∉ k\nha₀ : a ≠ 0\nb : D\nhb1 : ((ad (↥k) D) a) b ≠ 0\nn : ℕ\nhn : 0 < n\nc : D := (⇑((ad (↥k) D) a))^[n] b\nhb : c ≠ 0 ∧ (⇑((ad (↥k... | inv_mul_cancel₀ ha₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 171,
"column": 55
} | {
"line": 171,
"column": 75
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : a ∉ k\nha₀ : a ≠ 0\nb : D\nhb1 : ((ad (↥k) D) a) b ≠ 0\nn : ℕ\nhn : 0 < n\nc : D := (⇑((ad (↥k) D) a))^[n] b\nhb : c ≠ 0 ∧ (⇑((ad (↥k... | inv_mul_cancel₀ ha₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.LinearIndependent | {
"line": 41,
"column": 11
} | {
"line": 41,
"column": 20
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\nH : ∀ (p : F[X]), (aeval x) p = 0 → p = 0\ns : Finset F\nm : F → F\ni : F\nhi : i ∈ s\nhnz : ∀ (a : F), x - (algebraMap F E) a ≠ 0\nb : E := ∏ j ∈ s, (x - (algebraMap F E) j)\nh1 : ∀ i ∈ s, m i • (b * (x - (alge... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.CosetCover | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 62
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nD H : Subgroup G\ninst✝ : D.FiniteIndex\nhD_le_H : D ≤ H\nt : Set ↥H\nht : IsComplement t ↑(D.subgroupOf H) ∧ 1 ∈ t\n⊢ ∃ t, IsComplement ↑t ↑(D.subgroupOf H) ∧ ⋃ g ∈ t, ↑g • ↑D = ↑H",
"usedConstants": [
"Iff.mpr",
"Subgroup.subgroupOf",
"Subgroup.Fi... | have hf : t.Finite := ht.1.finite_left_iff.mpr inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.CosetCover | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (... | cases (Set.mem_union _ _ _).mp (hcovers.superset (Set.mem_univ y)) with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 270,
"column": 8
} | {
"line": 270,
"column": 35
} | [
{
"pp": "case pos\nK : Type u\ninst✝¹ : Field K\na : K\nh : a = 0\nhζ : (primitiveRoots 1 K).Nonempty\nH : Irreducible (X ^ 1 - C a)\ninst✝ : NeZero 1\n__spread✝⁻⁰ : ↥(rootsOfUnity 1 K) →* Gal(K[1√a]/K) := autAdjoinRootXPowSubC 1 a\ne : Gal(K[1√a]/K)\nthis✝ : Fact (Irreducible (X ^ 1 - C a))\nthis : Algebra K K... | ← AdjoinRoot.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CosetCover | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 23
} | [
{
"pp": "G : 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\... | Finset.sum_sigma, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 330,
"column": 47
} | {
"line": 330,
"column": 78
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nn : ℕ\nhζ✝ : (primitiveRoots n K).Nonempty\na : K\nH✝ : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα✝ : α ^ n = (algebraMap K L) a\nhζ : (primitiveRoots n K).Nonempty\nH : Irreducible... | IsSplittingField.adjoin_rootSet | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 490,
"column": 4
} | {
"line": 490,
"column": 59
} | [
{
"pp": "case refine_2\nK : Type u\ninst✝⁵ : Field K\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\nhK : (primitiveRoots (finrank K L) K).Nonempty\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\nζ : K\nhζ : IsPrimitiveRoot ζ (finrank K L)\nσ : Gal(L/K)\nhσ : Function.S... | apply IsGalois.intermediateFieldEquivSubgroup.injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.TensorProduct.Nontrivial | {
"line": 43,
"column": 2
} | {
"line": 45,
"column": 34
} | [
{
"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 : Ty... | exact Algebra.TensorProduct.mapOfCompatibleSMul FR R R FA FB |>.comp
(Algebra.TensorProduct.map (IsScalarTower.toAlgHom R A FA) (IsScalarTower.toAlgHom R B FB))
|>.toRingHom.domain_nontrivial | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 437,
"column": 36
} | {
"line": 437,
"column": 44
} | [
{
"pp": "case a.h.h\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nM' : Submodule R S\nh : M' ≤ M\ninst✝ : Flat R ↥N\ni : ↥M' ⊗[R] ↥N →ₗ[R] S := M.mulMap N ∘ₗ LinearMap.rTensor (↥N) (inclusion h)\nhi : Function.Injective ⇑i\nx✝¹ ... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 448,
"column": 36
} | {
"line": 448,
"column": 44
} | [
{
"pp": "case a.h.h\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nN' : Submodule R S\nh : N' ≤ N\ninst✝ : Flat R ↥M\ni : ↥M ⊗[R] ↥N' →ₗ[R] S := M.mulMap N ∘ₗ LinearMap.lTensor (↥M) (inclusion h)\nhi : Function.Injective ⇑i\nx✝¹ ... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CosetCover | {
"line": 348,
"column": 73
} | {
"line": 353,
"column": 87
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\nh : ∀ i ∈ s, (H i).FiniteIndex → s.card < (H i).index\nhs : s.Nonempty\nhs' : 0 < s.card\ni : ι\nhi : i ∈ s\n⊢ (↑(H i).index)⁻¹ < (↑s.card)⁻¹",
"usedConstants": [
... | by
cases eq_or_ne (H i).index 0 with
| inl hindex =>
rwa [hindex, Nat.cast_zero, inv_zero, inv_pos, Nat.cast_pos]
| inr hindex =>
exact inv_strictAnti₀ (by exact_mod_cast hs') (by exact_mod_cast h i hi ⟨hindex⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.Exponent | {
"line": 303,
"column": 64
} | {
"line": 305,
"column": 50
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : HasExponent K L\np : ℕ\ninst✝ : ExpChar K p\nn : ℕ\nhn : exponent K L ≤ n\na : K\n⊢ (iterateFrobenius K L p hn) ((algebraMap K L) a) = a ^ p ^ n",
"usedConstants": [
"Eq.mpr",
"RingHom.instRin... | by
apply (algebraMap K L).injective
rw [map_pow, algebraMap_iterateFrobenius K p hn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 203,
"column": 31
} | {
"line": 203,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\n⊢ F⟮x⟯ ≤ perfectClosure F E ↔ (minpoly F x).natSepDegree = 1",
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOrder",
"congrArg",
"IntermediateField",
"perfectClosure",
... | adjoin_simple_le_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 209,
"column": 31
} | {
"line": 209,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nq : ℕ\nhF : ExpChar F q\nx : E\n⊢ F⟮x⟯ ≤ perfectClosure F E ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range",
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOrder",
"Subring.instSetLike",
... | adjoin_simple_le_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Degree | {
"line": 73,
"column": 66
} | {
"line": 75,
"column": 52
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nx y : K⟮X⟯\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (x * y).num.natDegree + (x.denom * y.denom).natDegree = x.num.natDegree + y.num.natDegree + (x * y).denom.natDegree",
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"RatFunc.denom",
"HMul.hMul",
... | ←
Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))
(mul_ne_zero x.denom_ne_zero y.denom_ne_zero), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Relrank | {
"line": 448,
"column": 95
} | {
"line": 449,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B : IntermediateField F E\nh : A ≤ B\n⊢ finrank F ↥A * A.relfinrank B = finrank F ↥B",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | by
simpa using congr(toNat $(rank_bot_mul_relrank h)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Relrank | {
"line": 518,
"column": 6
} | {
"line": 518,
"column": 54
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA : IntermediateField F E\n⊢ ⊥.relrank A = Module.rank F ↥A",
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOrder",
"Lattice.toSemilatticeSup",
"NonUnitalCommRing.toNonUnitalNonAsso... | ← rank_bot_mul_relrank (show ⊥ ≤ A from bot_le), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 69,
"column": 85
} | {
"line": 71,
"column": 37
} | [
{
"pp": "K : Type u_1\ninst✝⁹ : Field K\nf : K⟮X⟯\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra K A\ninst✝⁶ : Algebra (↥K [f]) A\nB : Type u_3\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra K B\ninst✝³ : Algebra (↥K [f]) B\ninst✝² : Algebra A B\ninst✝¹ : IsScalarTower K A B\ninst✝ : IsScalarTower (↥K [f]) A B\n⊢... | by
simp [minpolyX, Polynomial.map_map, ← IsScalarTower.algebraMap_eq,
← IsScalarTower.algebraMap_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 382,
"column": 2
} | {
"line": 384,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (Φ E).coeff (generatorIndex h) ≠ 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"RingHom.instRingHomClass",
"IsDomain.to_noZeroDivisors",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul... | apply_fun algebraMap K[X] K⟮X⟯
rw [map_zero, Φ_coeff_generatorIndex h]
exact mul_ne_zero_iff.mpr ⟨algebraMap_ne_zero (num_ne_zero (c_ne_zero h)), generator_ne_zero h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 382,
"column": 2
} | {
"line": 384,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (Φ E).coeff (generatorIndex h) ≠ 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"RingHom.instRingHomClass",
"IsDomain.to_noZeroDivisors",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul... | apply_fun algebraMap K[X] K⟮X⟯
rw [map_zero, Φ_coeff_generatorIndex h]
exact mul_ne_zero_iff.mpr ⟨algebraMap_ne_zero (num_ne_zero (c_ne_zero h)), generator_ne_zero h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 166,
"column": 6
} | {
"line": 167,
"column": 19
} | [
{
"pp": "case mpr.refine_1\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\np : P\nhne : p ≠ s.points i\nh : p -ᵥ s.points i ∈ (s.altitude i).direction\n⊢... | rw [Submodule.span_le]
simpa using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 166,
"column": 6
} | {
"line": 167,
"column": 19
} | [
{
"pp": "case mpr.refine_1\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\np : P\nhne : p ≠ s.points i\nh : p -ᵥ s.points i ∈ (s.altitude i).direction\n⊢... | rw [Submodule.span_le]
simpa using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 14
} | [
{
"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... | mem_mk', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 224,
"column": 11
} | {
"line": 224,
"column": 19
} | [
{
"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 q : P\nhqs : q ∈ s\nhpq : p ... | mem_mk', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 339,
"column": 10
} | {
"line": 339,
"column": 40
} | [
{
"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\nr : ℝ\nhr : r ≠ 0\nh : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.a... | rw [range_faceOpposite_points] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 342,
"column": 10
} | {
"line": 342,
"column": 40
} | [
{
"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\nr : ℝ\nhr : r ≠ 0\nh : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.a... | rw [range_faceOpposite_points] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 370,
"column": 83
} | {
"line": 375,
"column": 6
} | [
{
"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\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp... | by
rw [← mul_self_inj_of_nonneg dist_nonneg dist_nonneg, ←
mul_self_inj_of_nonneg dist_nonneg dist_nonneg,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p₃ hp₁,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p₃ hp₂]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Projection | {
"line": 422,
"column": 6
} | {
"line": 422,
"column": 23
} | [
{
"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 x : P\nhx : x ∈ s\n⊢ (reflec... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 429,
"column": 6
} | {
"line": 429,
"column": 23
} | [
{
"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⊢ (reflection s) p = (↑... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 16
} | [
{
"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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni : Fin (n + 1)\n⊢ -(s.height i * s.height i) < ⟪s.points i -ᵥ s.altitudeFoot i, s.points i -ᵥ s.alt... | rw [real_inner_self_eq_norm_sq]
refine lt_of_lt_of_le (b := 0) ?_ ?_
· rw [neg_lt_zero]
positivity
· positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 16
} | [
{
"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\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni : Fin (n + 1)\n⊢ -(s.height i * s.height i) < ⟪s.points i -ᵥ s.altitudeFoot i, s.points i -ᵥ s.alt... | rw [real_inner_self_eq_norm_sq]
refine lt_of_lt_of_le (b := 0) ?_ ?_
· rw [neg_lt_zero]
positivity
· positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 463,
"column": 6
} | {
"line": 463,
"column": 23
} | [
{
"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⊢ (reflection s) p = p ... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 243,
"column": 28
} | {
"line": 243,
"column": 50
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₃ p₁ = dist p₃ p₂\nm : P := midpoint ℝ p₁ p₂\nh1 : p₃ -ᵥ p₁ = p₃ -ᵥ m - (p₁ -ᵥ m)\n⊢ p₃ -ᵥ p₂ = p₃ -ᵥ m + ⅟2 • (p₁ -ᵥ p₂)",
"usedCo... | ← midpoint_vsub_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 26
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₃ p₁ = dist p₃ p₂\n⊢ ∠ p₃ (midpoint ℝ p₁ p₂) p₂ = π / 2",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace... | midpoint_comm p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 25
} | [
{
"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 : Sbtw ℝ p₁ p₂ p₃\n⊢ ∠ p₃ p₂ p₁ = π",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"congrArg",
"id",
... | ← h.angle₁₂₃_eq_pi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 171,
"column": 63
} | {
"line": 172,
"column": 67
} | [
{
"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\n⊢ o.oangle y x = -o.oangle x y",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NegZeroCla... | by
simp only [oangle, o.kahler_swap y x, Complex.arg_conj_coe_angle] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 136,
"column": 2
} | {
"line": 141,
"column": 49
} | [
{
"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\n⊢ Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖",
"usedConstants": [
"mul_self_nonneg",
"div_le_one_of_le₀",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrde... | rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,
Real.sin_arcsin (le_trans (by simp) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
(div_le_one_of_le₀ _ (norm_nonneg _))]
rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
norm_add_sq_eq_norm_sq_add_norm_sq_real h]
exact le_add_of_nonneg_left (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 136,
"column": 2
} | {
"line": 141,
"column": 49
} | [
{
"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\n⊢ Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖",
"usedConstants": [
"mul_self_nonneg",
"div_le_one_of_le₀",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrde... | rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,
Real.sin_arcsin (le_trans (by simp) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
(div_le_one_of_le₀ _ (norm_nonneg _))]
rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
norm_add_sq_eq_norm_sq_add_norm_sq_real h]
exact le_add_of_nonneg_left (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nhxy : ‖x + y‖ = 0\nh' : ‖x‖ = 0 ∧ ‖y‖ * ‖y‖ = 0\n⊢ ‖x‖ / ‖x + y‖ * ‖x + y‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real",
"instHD... | simp [h'.1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 418,
"column": 94
} | {
"line": 431,
"column": 8
} | [
{
"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\n⊢ x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"InnerProduct... | by
rw [oangle_eq_zero_iff_sameRay]
constructor
· rintro rfl
simp; rfl
· rcases eq_or_ne y 0 with (rfl | hy)
· simp
rintro ⟨h₁, h₂⟩
obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
have : ‖y‖ ≠ 0 := by simpa using hy
obtain rfl : r = 1 := by
apply mul_right_cancel₀ this
simpa ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 578,
"column": 20
} | {
"line": 578,
"column": 57
} | [
{
"pp": "case inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh0 : 0 ≤ InnerProductGeometry.angle x y\nhpi : InnerProductGeometry.angle x y ≤ π\nh : o.oangle x y = ↑(InnerProductGeomet... | Real.Angle.abs_toReal_coe_eq_self_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 812,
"column": 2
} | {
"line": 812,
"column": 68
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\n⊢ (o.oangle x (r ... | let s : Set (V × V) := (fun r' : ℝ => (x, r' • x + y)) '' Set.univ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 925,
"column": 2
} | {
"line": 926,
"column": 12
} | [
{
"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\nr₁ r₂ : ℝ\n⊢ (o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.... | simp_rw [o.oangle_rev y, Real.Angle.sign_neg, add_comm (r₁ • x), oangle_sign_smul_add_smul_right,
mul_neg] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 925,
"column": 2
} | {
"line": 926,
"column": 12
} | [
{
"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\nr₁ r₂ : ℝ\n⊢ (o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.... | simp_rw [o.oangle_rev y, Real.Angle.sign_neg, add_comm (r₁ • x), oangle_sign_smul_add_smul_right,
mul_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 925,
"column": 2
} | {
"line": 926,
"column": 12
} | [
{
"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\nr₁ r₂ : ℝ\n⊢ (o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.... | simp_rw [o.oangle_rev y, Real.Angle.sign_neg, add_comm (r₁ • x), oangle_sign_smul_add_smul_right,
mul_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 68
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : (-o).oangle y x = ↑(π / 2)\n⊢ ((-o).oangle x (x - y)).cos * ‖x - y‖ = ‖x‖",
"usedConstants": [
"AlternatingMap.instAddCommGroup",
"Alternat... | exact (-o).cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 648,
"column": 61
} | {
"line": 651,
"column": 89
} | [
{
"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).tan = dist p₃ p₂ / dist p₁ p₂",
"usedC... | 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.tan_coe,
tan_angle_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.Angle.Oriented.RightAngle | {
"line": 700,
"column": 4
} | {
"line": 701,
"column": 50
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₃ p₁ p₂).sign = 1\n⊢ Real.tan (∠ p₂ p₁ p₃) * dis... | tan_angle_mul_dist_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (left_ne_of_oangle_eq_pi_div_two h)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 878,
"column": 8
} | {
"line": 878,
"column": 43
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄\nhs : (affineSpan ℝ {p₁, p₂}).SSameSi... | angle_eq_abs_oangle_toReal h₁₂ h₃₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 98
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, p -ᵥ q⟫ = 0\n⊢ (signedDist v) p = (signedDist v) q",
"usedConstants": [
"Norm.norm",
"InnerProductSpace.toNormedSpace",... | ext r
simpa [NormedSpace.normalize, real_inner_smul_left, h] using signedDist_vadd_left v (p -ᵥ q) q r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 98
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, p -ᵥ q⟫ = 0\n⊢ (signedDist v) p = (signedDist v) q",
"usedConstants": [
"Norm.norm",
"InnerProductSpace.toNormedSpace",... | ext r
simpa [NormedSpace.normalize, real_inner_smul_left, h] using signedDist_vadd_left v (p -ᵥ q) q r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 172,
"column": 54
} | {
"line": 180,
"column": 19
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\n⊢ |((signedDist v) p) q| = dist p q ↔ q -ᵥ p ∈ ℝ ∙ v",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Rea... | by
rw [Submodule.mem_span_singleton]
rw [signedDist_apply_apply, dist_eq_norm_vsub', NormedSpace.normalize, real_inner_smul_left,
abs_mul, abs_inv, abs_norm]
by_cases h : v = 0
· simp [h, eq_comm (a := (0 : ℝ)), eq_comm (a := (0 : V))]
rw [inv_mul_eq_iff_eq_mul₀ (by positivity)]
rw [← Real.norm_eq_abs, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 933,
"column": 41
} | {
"line": 947,
"column": 9
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄ + ↑π\nhs : (affineSpan ℝ {p₁, p₂}).SO... | by
have h₃₂ : p₃ ≠ p₂ := by
rintro rfl
exact hs.left_notMem (right_mem_affineSpan_pair _ _ _)
have h₄₂ : p₄ ≠ p₂ := by
rintro rfl
exact hs.right_notMem (right_mem_affineSpan_pair _ _ _)
have ha' : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ (AffineEquiv.pointReflection ℝ p₂ p₄) := by
rw [oangle_pointReflection_right... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 180,
"column": 50
} | {
"line": 180,
"column": 59
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nhab : a ≠ b\nhp : p ∈ perpBisector a b\ns : ℝ\nh_wbtw : Wbtw ℝ a b ((AffineMap.lineMap a b) s)\n⊢ s * 0 - ⅟2 * 0 = 0",
"usedConstants": [
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 180,
"column": 60
} | {
"line": 180,
"column": 69
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nhab : a ≠ b\nhp : p ∈ perpBisector a b\ns : ℝ\nh_wbtw : Wbtw ℝ a b ((AffineMap.lineMap a b) s)\n⊢ 0 - ⅟2 * 0 = 0",
"usedConstants": [
"Eq.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 154,
"column": 6
} | {
"line": 155,
"column": 14
} | [
{
"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 ... | rw [← Fintype.coe_image_univ, hu]
simp [b] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Basic | {
"line": 154,
"column": 6
} | {
"line": 155,
"column": 14
} | [
{
"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 ... | rw [← Fintype.coe_image_univ, hu]
simp [b] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Basic | {
"line": 167,
"column": 61
} | {
"line": 167,
"column": 73
} | [
{
"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\nhc₁s : c₁ ∈ s\nhc₂s : c₂ ∈ s\nhp₁s : p₁ ∈ ... | simp [hp₂c₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 53,
"column": 18
} | {
"line": 53,
"column": 26
} | [
{
"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 x : P\n⊢ x ∈ mk' p (ℝ ∙ (p -ᵥ s.center))ᗮ ↔ ⟪x -ᵥ p, p -ᵥ s.center⟫ = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNor... | mem_mk', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 156,
"column": 44
} | {
"line": 163,
"column": 21
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\n⊢ s.IsTangent (s.orthRadius p) ↔ p ∈ s",
"usedConstants": [
"Iff.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
... | by
refine ⟨?_, fun h ↦ (isTangentAt_orthRadius_iff_mem.2 h).isTangent⟩
rintro ⟨q, hs, hsp, hle⟩
rw [orthRadius_le_orthRadius_iff] at hle
rcases hle with rfl | rfl
· exact hs
· rw [center_mem_orthRadius_iff] at hsp
rwa [← hsp] at hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 461,
"column": 6
} | {
"line": 464,
"column": 19
} | [
{
"pp": "case pos\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\ninst✝ : Nontrivial V\ns₁ s₂ : Sphere P\nh : dist s₁.center s₂.center = s₂.radius - s₁.radius\nh₁ : 0 ≤ s₁.radius\nh₂ : 0 ≤ s₂.radius\nh0 : s₁.center... | rw [h0, dist_self, eq_comm, sub_eq_zero, eq_comm] at h
have hs : s₁ = s₂ := by
ext <;> assumption
simp [hs, h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 461,
"column": 6
} | {
"line": 464,
"column": 19
} | [
{
"pp": "case pos\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\ninst✝ : Nontrivial V\ns₁ s₂ : Sphere P\nh : dist s₁.center s₂.center = s₂.radius - s₁.radius\nh₁ : 0 ≤ s₁.radius\nh₂ : 0 ≤ s₂.radius\nh0 : s₁.center... | rw [h0, dist_self, eq_comm, sub_eq_zero, eq_comm] at h
have hs : s₁ = s₂ := by
ext <;> assumption
simp [hs, h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Bisector | {
"line": 185,
"column": 2
} | {
"line": 188,
"column": 15
} | [
{
"pp": "case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\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 : ↑((or... | · exfalso
have h' := h.collinear
rw [Set.pair_comm] at h'
exact hc h' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 371,
"column": 40
} | {
"line": 371,
"column": 66
} | [
{
"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... | smul_eq_zero_iff_left hv0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 60
} | [
{
"pp": "case inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhpc : p ≠ s.center\nh : dist p s.center = s.radius\n⊢ (Metric.sphere s.center s.radius ... | simp [inter_orthRadius_eq_singleton_of_dist_eq_radius h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 60
} | [
{
"pp": "case inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhpc : p ≠ s.center\nh : dist p s.center = s.radius\n⊢ (Metric.sphere s.center s.radius ... | simp [inter_orthRadius_eq_singleton_of_dist_eq_radius h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 60
} | [
{
"pp": "case inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhpc : p ≠ s.center\nh : dist p s.center = s.radius\n⊢ (Metric.sphere s.center s.radius ... | simp [inter_orthRadius_eq_singleton_of_dist_eq_radius h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 265,
"column": 18
} | {
"line": 269,
"column": 90
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ... | by
convert tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center hp₁ hp₃ hp₁p₃
convert (Real.Angle.tan_eq_inv_of_two_zsmul_add_two_zsmul_eq_pi _).symm
rw [add_comm,
two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi hp₁ hp₂ hp₃ hp₁p₂.symm hp₂p₃ hp₁p₃] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 549,
"column": 56
} | {
"line": 553,
"column": 8
} | [
{
"pp": "n : ℕ\n⊢ ∑ i, circumcenterWeightsWithCircumcenter n i = 1",
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Finset.univ",
"Finset.sum_ite_eq'",
"Real.instZero",
"congrArg",
"HEq.refl",
"Finset",
"False.elim",
"AddMonoid.toAddZeroClass... | by
classical
convert sum_ite_eq' univ circumcenterIndex (Function.const _ (1 : ℝ)) with j
· cases j <;> simp [circumcenterWeightsWithCircumcenter]
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 760,
"column": 2
} | {
"line": 762,
"column": 25
} | [
{
"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\nps : Set P\nn : ℕ\ninst✝ : FiniteDimensional ℝ V\nhd : finrank ℝ V = n\nhc : Cospherical ps\n⊢ ∃ c, ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circ... | rw [← finrank_top, ← direction_top ℝ V P] at hd
refine exists_circumsphere_eq_of_cospherical_subset ?_ hd hc
exact Set.subset_univ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 760,
"column": 2
} | {
"line": 762,
"column": 25
} | [
{
"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\nps : Set P\nn : ℕ\ninst✝ : FiniteDimensional ℝ V\nhd : finrank ℝ V = n\nhc : Cospherical ps\n⊢ ∃ c, ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circ... | rw [← finrank_top, ← direction_top ℝ V P] at hd
refine exists_circumsphere_eq_of_cospherical_subset ?_ hd hc
exact Set.subset_univ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 60
} | [
{
"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))\n⊢ s.excenterWeightsUnnorm signsᶜ = -s.excenterWeightsUnnorm signs",
"usedConstan... | ext i
by_cases h : i ∈ signs <;> simp [excenterWeightsUnnorm, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 60
} | [
{
"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))\n⊢ s.excenterWeightsUnnorm signsᶜ = -s.excenterWeightsUnnorm signs",
"usedConstan... | ext i
by_cases h : i ∈ signs <;> simp [excenterWeightsUnnorm, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 34
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhy : y ≠ 0\nhx : x ≠ 0\nh✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))\nn : ℤ\nh : n ≤ -1\n⊢ π + -(2 * π) < 0 + 0",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWi... | linear_combination Real.pi_pos | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 39
} | [
{
"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 : 2 • o.oangle x (x - y) = 2 • o.oangle (y - x) y\nh0 : o.oangle x y ≠ 0\nhpi : o.oangle x y ≠ ↑π\nhs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).s... | rw [Real.Angle.two_zsmul_eq_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 97
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni : Fin (n + 1)\n⊢ 0 < (if i ∈ {i} then -1 else 1) * (s.height i)⁻¹ + ∑ i_1 ∈ {i}ᶜ, s.excen... | simp only [Finset.mem_singleton, ↓reduceIte, neg_mul, one_mul, lt_neg_add_iff_add_lt, add_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 393,
"column": 30
} | {
"line": 393,
"column": 70
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c a' b' c' : P\nr : ℝ\nh : ∠ a' b' c' = ∠ a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n⊢ r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 -... | by simp [pow_two, ← law_cos a b c]; ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Congruence | {
"line": 139,
"column": 48
} | {
"line": 139,
"column": 60
} | [
{
"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\nx✝¹ x✝ : ι\n⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝)",
"usedConstants": [
"Eq.mpr",
"NNDist... | edist_nndist | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Congruence | {
"line": 145,
"column": 44
} | {
"line": 145,
"column": 56
} | [
{
"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⊢ (Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂)) ↔\n Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)",
"usedConstants": [
... | edist_nndist | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 628,
"column": 2
} | {
"line": 628,
"column": 16
} | [
{
"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)\nhf : s.excenter signs ∉ affineSpan ℝ (S... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 668,
"column": 28
} | {
"line": 668,
"column": 38
} | [
{
"pp": "case refine_1.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs... | ← mul_neg, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 961,
"column": 64
} | {
"line": 964,
"column": 31
} | [
{
"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\np : P\nhp : p ∈ affineSpan ℝ (Set.range s.points)\n⊢ (∃ r, ∀ (i : Fin (n + 1)), (s.signedInfDist i) p = r) ↔ p = s... | by
convert s.exists_forall_signedInfDist_eq_iff_excenterExists_and_eq_excenter hp (signs := ∅)
· simp
· simp [excenterExists_empty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 77
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhz : ‖z‖ = 1\nH : angle x z ≠ π\nH0 : angle x z = angle x y + angle y z\nH1 : ¬angle x z = 0\nHxz : Real.sin (angle x z) ≠ 0\nH2 : ¬angle x y = 0\nH3 : ¬angle y z = 0\nH4 : ¬angl... | have H12 : ‖normalize (ortho y z)‖ = 1 := norm_normalize_eq_one_iff.mpr H10 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1196,
"column": 4
} | {
"line": 1196,
"column": 44
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhne : i ≠ j\n⊢ SignType.sign (s.excenterWeights {i} j) = 1",
"usedCo... | s.sign_excenterWeights_singleton_pos hne | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1212,
"column": 56
} | {
"line": 1214,
"column": 41
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhne : i ≠ j\n⊢ SignType.sign (s.touchpointWeights {i} j i) = -1",
"u... | by
rw [(s.excenterExists_singleton i).sign_touchpointWeights hne.symm,
s.sign_excenterWeights_singleton_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Similarity | {
"line": 215,
"column": 36
} | {
"line": 215,
"column": 48
} | [
{
"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\nx✝² : ℝ≥0\nx✝¹ x✝ : ι\n⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝)",
"usedConstants": [
... | edist_nndist | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Similarity | {
"line": 222,
"column": 49
} | {
"line": 222,
"column": 61
} | [
{
"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⊢ (∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = ↑r * edist (v₂ i₁) (v₂ i₂)) ↔\n ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁... | edist_nndist | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 103,
"column": 12
} | {
"line": 103,
"column": 44
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\n⊢ ↑n * ‖s.faceOppositeCentroid i -ᵥ (((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circum... | show (n : ℝ) = ‖(n : ℝ)‖ by simp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 223,
"column": 2
} | {
"line": 225,
"column": 89
} | [
{
"pp": "case h.e'_5.e_self\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 : Triangle ℝ P\ni : Fin 3\n⊢ (orthogonalProjection (affineSpan ℝ (Set.range (Simplex.faceOpposite s i).points))) (s.points i) =\n (ort... | rw [orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem,
Simplex.points_vsub_eulerPoint, Submodule.smul_mem_iff _ (by norm_num),
← orthocenter_eq_mongePoint, direction_affineSpan, Simplex.range_faceOpposite_points] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 86,
"column": 2
} | {
"line": 90,
"column": 55
} | [
{
"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... | have k_pos : 0 < k := by
rw [hk]
apply div_pos
· simp [dist_pos, ne₁₂_of_not_collinear h_not_col]
· simp [dist_pos, ne₁₂_of_not_collinear h_not_col'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 41
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nv : V\nhp : ⟪v, p -ᵥ s.center⟫ = 0\n⊢ (-2 * 0 / ⟪v, v⟫) • v +ᵥ p = p",
"usedConstants": [
"Eq.mpr",
"InnerP... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 225,
"column": 2
} | {
"line": 244,
"column": 55
} | [
{
"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 : Sphere P\nn : ℕ\ninst✝ : n.AtLeastTwo\nsx : Affine.Simplex ℝ P n\ni : Fin (n + 1)\nhi : sx.points i ∈ s\nhsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.c... | obtain ⟨w, hw, hw01, rfl⟩ := hp
let r : ℝ := (1 - w i)⁻¹
have hrpos : 0 < r := by simp [inv_pos, sub_pos, r, (hw01 i).2]
let p' : P := AffineMap.lineMap (sx.points i) (Finset.univ.affineCombination ℝ sx.points w) r
have hp' : (p' -ᵥ (sx.points i)) =
r • (Finset.univ.affineCombination ℝ sx.points w -ᵥ (sx.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 225,
"column": 2
} | {
"line": 244,
"column": 55
} | [
{
"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 : Sphere P\nn : ℕ\ninst✝ : n.AtLeastTwo\nsx : Affine.Simplex ℝ P n\ni : Fin (n + 1)\nhi : sx.points i ∈ s\nhsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.c... | obtain ⟨w, hw, hw01, rfl⟩ := hp
let r : ℝ := (1 - w i)⁻¹
have hrpos : 0 < r := by simp [inv_pos, sub_pos, r, (hw01 i).2]
let p' : P := AffineMap.lineMap (sx.points i) (Finset.univ.affineCombination ℝ sx.points w) r
have hp' : (p' -ᵥ (sx.points i)) =
r • (Finset.univ.affineCombination ℝ sx.points w -ᵥ (sx.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 526,
"column": 6
} | {
"line": 526,
"column": 10
} | [
{
"pp": "case right\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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\nh₁... | hui, | Lean.Elab.Tactic.evalRewriteSeq | null |
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