`Compfiles/Imo2023P2.lean`

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273 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2023, Problem 2

Let ABC be an acute-angled triangle with AB < AC. Let Ω be the circumcircle of ABC. Let S be the midpoint of the arc CB of Ω containing A. The perpendicular from A to BC meets BS at D and meets Ω again at E ≠ A. The line through D parallel to BC meets line BE at L. Denote the circumcircle of triangle BDL by ω. Let ω meet Ω again at P ≠ B. Prove that the line tangent to ω at P meets line BS on the internal angle bisector of ∠BAC.

Imports (2)

Imported by 1 file

Compfiles.lean

lemma Imo2023P2.aux₁ alternate version of Real.Angle.abs_toReal_add_abs_toReal_eq_pi_of_two_nsmul_add_eq_zero_of_sign_eq line 48 source
Source comment
alternate version of Real.Angle.abs_toReal_add_abs_toReal_eq_pi_of_two_nsmul_add_eq_zero_of_sign_eq

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, SignType, Real.Angle.sign, Ne, OfNat.ofNat, HSMul.hSMul, Nat, HAdd.hAdd, Real.pi
Proof uses: Or, Eq.mp, congrArg, propext, Real.Angle.two_nsmul_eq_zero_iff, Mathlib.Tactic.Contrapose.contrapose₂, id, Not, Neg.neg, SignType.self_eq_neg_iff, Real.Angle.sign_neg, add_eq_zero_iff_eq_neg, add_comm
51 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma aux₁ {a b : Real.Angle} (h₁ : a.sign = b.sign) (h₂ : a.sign ≠ 0)
  (h : 2 • (a + b) = 0) : a + b = Real.pi := by
…
```
Mentions: h
lemma Imo2023P2.aux₂ alternate version of Real.Angle.sign_pi_sub line 59 source
Source comment
alternate version of Real.Angle.sign_pi_sub

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, HAdd.hAdd, Real.pi, SignType, Real.Angle.sign
Proof uses: Eq.mpr, HSub.hSub, id, congrArg, Eq.mp, Eq.symm, propext, eq_sub_iff_add_eq, Real.Angle.sign_pi_sub
24 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma aux₂ {a b : Real.Angle} (h : a + b = Real.pi) : a.sign = b.sign := by
…
```
Mentions: h
lemma Imo2023P2.aux₃ line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, HAdd.hAdd, Real.pi, Ne, SignType, Real.Angle.sign, OfNat.ofNat, Real, abs, Real.Angle.toReal
Proof uses: HSMul.hSMul, Nat, Eq.mpr, id, congrArg, Real.Angle.two_nsmul_coe_pi, Imo2023P2.aux₂, Real.Angle.abs_toReal_add_abs_toReal_eq_pi_of_two_nsmul_add_eq_zero_of_sign_eq
35 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₄ line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Ne, SignType, Real.Angle.sign, OfNat.ofNat, Eq, HAdd.hAdd, Real, abs, Real.Angle.toReal
Proof uses: Neg.neg, Not, Or, Classical.em, Eq.mpr, id, congrArg, LT.lt, Eq.symm, propext, Real.Angle.toReal_neg_iff_sign_neg, rfl, Int, HSMul.hSMul, HMul.hMul, Nat, Real.pi, HSub.hSub, Exists, Eq.mp and 95 more
179 instances, coercions and auxiliary constants left out.

Tactic trace: 73 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₅ line 150 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, HAdd.hAdd, SignType, Real.Angle.sign, LT.lt, Real, abs, Real.Angle.toReal
Proof uses: dite, OfNat.ofNat, Eq.mpr, id, congrArg, Eq.symm, Or, Real.pi, propext, Real.Angle.sign_eq_zero_iff, Eq.mp, HSub.hSub, sub_eq_iff_eq_add', of_eq_true, Eq.trans, True, congr, congrFun', sub_self, eq_self and 99 more
170 instances, coercions and auxiliary constants left out.

Tactic trace: 72 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₆ line 218 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, SignType, Real.Angle.sign, Ne, OfNat.ofNat, HSMul.hSMul, Nat
Proof uses: HAdd.hAdd, Real.pi, Or, Eq.mp, congrArg, propext, Real.Angle.two_nsmul_eq_iff, Mathlib.Tactic.Contrapose.contrapose₂, id, Not, Neg.neg, SignType.self_eq_neg_iff, HSub.hSub, Real.Angle.sign_sub_pi, Eq.symm, sub_eq_iff_eq_add
35 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₇ line 228 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Exists, Int, Eq, HMul.hMul, LT.lt, abs, OfNat.ofNat
Proof uses: Eq.mpr, id, congrArg, mul_eq_zero_of_right, Mathlib.Tactic.Contrapose.contrapose₁, Not, Ne, LE.le, implies_congr, Eq.refl, abs_mul, Eq.symm, Int.cast_one, Int.cast_abs, propext, Int.cast_le, Int.one_le_abs, Trans.trans, le_abs_self, mul_one and 2 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₈ line 245 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, LT.lt, abs, HAdd.hAdd
Proof uses: Eq.mpr, And, Neg.neg, id, Eq, congrArg, propext, abs_lt, And.intro, Trans.trans, rfl, And.left, Iff.mp, Eq.symm, zero_add, add_le_add_left, add_le_add_right, add_zero, And.right
42 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₉ line 260 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, abs, HAdd.hAdd, LT.lt
Proof uses: Eq.symm, Eq.mpr, id, congrArg, Or, And, LE.le, OfNat.ofNat, propext, abs_add_eq_add_abs_iff, Mathlib.Tactic.Contrapose.contrapose₁, Not, Ne, implies_congr, Eq.trans, congr, Mathlib.Tactic.Push.not_and_eq, Eq.refl, ne_of_lt, dite and 6 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.aux₁₀ line 275 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, Eq, HAdd.hAdd, HSMul.hSMul, Nat, OfNat.ofNat, Real.pi, LT.lt, Real, abs, Real.Angle.toReal, HDiv.hDiv
Proof uses: Eq.mpr, id, congrArg, rfl, Trans.trans, LE.le, abs_add_le, add_lt_add, Eq.mp, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne', Mathlib.Tactic.Ring.Common.add_congr, NNRat.rawCast, Mathlib.Tactic.Ring.Common.div_congr, Mathlib.Tactic.Ring.Common.atom_pf, Eq.symm, Eq.refl and 60 more
168 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.angle_eq_of_oangle_eq line 317 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Real.Angle, EuclideanGeometry.oangle, Ne, EuclideanGeometry.angle
Proof uses: Eq.mpr, abs, Real.Angle.toReal, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Eq.refl
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.angle_eq_of_cos_angle_eq line 324 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Eq, Real.cos, EuclideanGeometry.angle
Proof uses: id, InnerProductGeometry.angle, VSub.vsub, Eq.mpr, Real.arccos, HDiv.hDiv, Inner.inner, HMul.hMul, Norm.norm, congrArg, Eq.mp, InnerProductGeometry.cos_angle, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.eq_of_mem_sphere_of_collinear line 332 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, EuclideanGeometry.Sphere, Membership.mem, Collinear, Insert.insert, Set, Singleton.singleton, Or, Eq
Proof uses: dite, Classical.propDecidable, Or.inl, Not, Or.inr, of_eq_true, Eq.trans, True, congr, congrArg, eq_self, true_or, or_true, AffineSubspace, affineSpan, Collinear.mem_affineSpan_of_mem_of_ne, EuclideanGeometry.Sphere.secondInter, VSub.vsub, Eq.mp, Eq.symm and 10 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.eq_max_of_max_ne_top line 357 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Submodule, Subtype, Membership.mem, Ne, Max.max, Top.top
Proof uses: Submodule.eq_of_le_of_finrank_eq, le_sup_left, Eq.mpr, id, congrArg, LE.le, Submodule.finrank_le, Eq.symm, Submodule.finrank_mono, Mathlib.Tactic.Contrapose.contrapose₄, Submodule.eq_top_of_finrank_eq, Fact.out, Eq.refl, Eq.mp
49 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.affineSpan_pair_finrank line 375 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Ne, Eq, Nat, Module.finrank, Subtype, Membership.mem, Submodule, AffineSubspace.direction, affineSpan, Insert.insert, Set, Singleton.singleton, OfNat.ofNat
Proof uses: Eq.mpr, vectorSpan, id, congrArg, direction_affineSpan, AffineIndependent, Fin, Nat.succ, Matrix.vecCons, Matrix.vecEmpty, affineIndependent_of_ne, Set.range, Eq.symm, Matrix.range_cons_cons_empty, AffineIndependent.finrank_vectorSpan, of_eq_true, Fintype.card, Eq.trans, True, congrFun' and 4 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.inter_nonempty_of_not_parallel line 384 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Ne, Not, AffineSubspace.Parallel, affineSpan, Insert.insert, Set, Singleton.singleton, Set.Nonempty, Inter.inter, AffineSubspace
Proof uses: Exists.intro, Membership.mem, mem_affineSpan, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self, true_or, AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top, Mathlib.Tactic.Contrapose.contrapose₂, Submodule, Max.max, AffineSubspace.direction, Top.top, id, Eq.mpr, And and 17 more
55 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.oangle_eq_zero_of_collinear line 411 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Collinear, Insert.insert, Set, Singleton.singleton, Or, Real.Angle, EuclideanGeometry.oangle
Proof uses: Wbtw, Collinear.wbtw_or_wbtw_or_wbtw, Or.inl, Wbtw.oangle₂₁₃_eq_zero, Wbtw.oangle₁₃₂_eq_zero, Or.inr, Wbtw.oangle₃₁₂_eq_zero
57 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Sphere.IsTangentAt_of_two_zsmul_oangle_eq line 422 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, EuclideanGeometry.Sphere, Membership.mem, Ne, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, EuclideanGeometry.Sphere.IsTangentAt and 4 more
Proof uses: EuclideanGeometry.Sphere.IsTangentAt_of_angle_eq_pi_div_two, HAdd.hAdd, EuclideanGeometry.Sphere.center, Real.pi, EuclideanGeometry.Sphere.two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi, Ne.symm, EuclideanGeometry.Sphere.ne_center_of_mem_of_mem_of_ne, Eq.mpr, EuclideanGeometry.angle, HDiv.hDiv, abs, Real.Angle.toReal, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Or, Neg.neg, propext, Real.Angle.abs_toReal_eq_pi_div_two_iff, Iff.mp and 6 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Sphere.dist_le_radius_of_sbtw line 439 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, EuclideanGeometry.Sphere, Sbtw, Membership.mem, LT.lt, Dist.dist, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius
Proof uses: Max.max, Sbtw.dist_lt_max_dist, Eq.mp, congrArg, max_self, Eq, propext, EuclideanGeometry.mem_sphere
32 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Collinear.sbtw_or_wbtw_or_wbtw line 450 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Collinear, Insert.insert, Set, Singleton.singleton, Or, Sbtw, Wbtw
Proof uses: Collinear.wbtw_or_wbtw_or_wbtw, dite, Classical.propDecidable, Or.inl, Not, Or.inr, Eq, Eq.mp, Ne, congrArg, propext, not_ne_iff, And, not_and_or, not_and, Sbtw.eq_1, Eq.mpr, id, wbtw_self_right, wbtw_self_left
37 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Collinear.sbtw_or_sbtw_or_wbtw_of_ne line 469 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Collinear, Insert.insert, Set, Singleton.singleton, Ne, Or, Sbtw, Wbtw
Proof uses: Collinear.wbtw_or_wbtw_or_wbtw, Classical.byContradiction, Not, Eq, Eq.mpr, id, congrArg, wbtw_self_right, wbtw_self_left, Ne.symm, False, And, Eq.mp, propext, not_ne_iff, not_and_or, Sbtw.eq_1, Eq.trans, Decidable.not_or_of_imp, Classical.propDecidable and 1 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.parallel_of_two_zsmul_oangle_eq reverse of EuclideanGeometry.two_zsmul_oangle_of_parallel line 488 source
Source comment
reverse of EuclideanGeometry.two_zsmul_oangle_of_parallel

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Ne, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, AffineSubspace.Parallel, affineSpan, Insert.insert and 2 more
Proof uses: Eq.mpr, And, Submodule, vectorSpan, Iff, EmptyCollection.emptyCollection, id, congrArg, propext, AffineSubspace.affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty, And.intro, Submodule.span, VSub.vsub, vectorSpan_pair, Eq.symm, Exists, Units, Submodule.span_singleton_eq_span_singleton, Or, Eq.mp and 32 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 37 steps
Open to load the checked before-and-after goal states.
```
lemma parallel_of_two_zsmul_oangle_eq {P₁ P₂ P₃ P₄ : Pt}
  (h₁₂ : P₁ ≠ P₂) (h₂₃ : P₂ ≠ P₃) (h₃₄ : P₃ ≠ P₄)
  (h : (2 : ℤ) • ∡ P₁ P₂ P₃ = (2 : ℤ) • ∡ P₄ P₃ P₂)
  : line[ℝ, P₁, P₂] ∥ line[ℝ, P₃, P₄] := by
…
```
Mentions: h
structure Imo2023P2.SphereOrder line 521 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin
13 instances, coercions and auxiliary constants left out.
def Imo2023P2.diag_inter_set line 526 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Set
Proof uses: Inter.inter, AffineSubspace, affineSpan, Insert.insert, Singleton.singleton
17 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.mem_diag_inter_set_comm line 529 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Membership.mem, Set, Imo2023P2.diag_inter_set
Proof uses: And.intro, AffineSubspace, affineSpan, Insert.insert, Singleton.singleton, And.left, Eq.mp, Inter.inter, congrArg, Imo2023P2.diag_inter_set.eq_1, Eq.mpr, id, Eq, Set.pair_comm, And.right
19 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.diag_inter_set_comm line 538 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Eq, Set, Imo2023P2.diag_inter_set
Proof uses: Set.ext, Iff.intro, Membership.mem, Imo2023P2.mem_diag_inter_set_comm
7 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.mem_diag_inter_set_rotate line 544 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Membership.mem, Set, Imo2023P2.diag_inter_set
Proof uses: And.intro, AffineSubspace, affineSpan, Insert.insert, Singleton.singleton, And.right, Eq.mp, Inter.inter, congrArg, Imo2023P2.diag_inter_set.eq_1, Eq.mpr, id, Eq, Set.pair_comm, And.left
19 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.diag_inter_set_rotate line 553 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Eq, Set, Imo2023P2.diag_inter_set
Proof uses: Set.ext, Iff.intro, Membership.mem, Imo2023P2.mem_diag_inter_set_rotate
7 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.comm line 566 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, id, congrArg, Set.pair_comm, Set.insert_comm, Imo2023P2.SphereOrder.cospherical, Ne, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Neg.neg, Real.Angle, EuclideanGeometry.oangle_rev, Real.Angle.sign_neg, propext and 6 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₁_ne_P₂ line 579 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: EuclideanGeometry.left_ne_of_oangle_sign_ne_zero, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₁_ne_P₃ line 582 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₂_ne_P₃ line 585 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: Ne.symm, EuclideanGeometry.right_ne_of_oangle_sign_ne_zero, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero line 588 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne, SignType, Real.Angle.sign, EuclideanGeometry.oangle
Proof uses: Eq.mpr, id, congrArg, Eq.symm, Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₃₄₁, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero
30 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₁_ne_P₄ line 592 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: EuclideanGeometry.right_ne_of_oangle_sign_ne_zero, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₃_ne_P₄ line 595 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: EuclideanGeometry.left_ne_of_oangle_sign_ne_zero, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.P₂_ne_P₄ line 598 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Mathlib.Tactic.Contrapose.contrapose₄, Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₃₄₁, Eq.mp, Neg.neg, congrArg, propext, SignType.self_eq_neg_iff, Real.Angle, Real.Angle.sign_neg, EuclideanGeometry.oangle_rev, Eq.symm
35 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.oangle₁₂₃_add_oangle₃₄₁_eq_pi line 605 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle, Real.pi
Proof uses: EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Imo2023P2.SphereOrder.cospherical, HSMul.hSMul, Int, EuclideanGeometry.Cospherical.two_zsmul_oangle_eq, Eq.mp, congrArg, Set.pair_comm, Ne.symm, Imo2023P2.SphereOrder.P₁_ne_P₂, Imo2023P2.SphereOrder.P₂_ne_P₃, Imo2023P2.SphereOrder.P₁_ne_P₄, Imo2023P2.SphereOrder.P₃_ne_P₄, Imo2023P2.aux₁, Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₃₄₁, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Neg.neg and 7 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.oangle₂₃₄_add_oangle₄₁₂_eq_pi line 613 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle, Real.pi
Proof uses: Eq.symm, EuclideanGeometry.oangle_add_oangle_add_oangle_eq_pi, Ne.symm, Imo2023P2.SphereOrder.P₂_ne_P₃, Imo2023P2.SphereOrder.P₁_ne_P₃, Imo2023P2.SphereOrder.P₁_ne_P₂, Imo2023P2.SphereOrder.P₁_ne_P₄, Imo2023P2.SphereOrder.P₃_ne_P₄, Imo2023P2.SphereOrder.oangle₁₂₃_add_oangle₃₄₁_eq_pi, Eq.mpr, HSub.hSub, id, congrArg, propext, sub_eq_zero, Neg.neg, neg_eq_zero, Eq.mp, sub_eq_iff_eq_add', EuclideanGeometry.oangle_add
53 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.rotate line 627 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, id, congrArg, Set.pair_comm, Set.insert_comm, Imo2023P2.SphereOrder.cospherical, SignType, Real.Angle.sign, EuclideanGeometry.oangle, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Exists and 27 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.symm line 661 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.rotate
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.rotate' line 664 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.rotate
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.sign_oangle₂₃₄_ne_zero line 667 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Ne, SignType, Real.Angle.sign, EuclideanGeometry.oangle
Proof uses: Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.SphereOrder.rotate
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.angle₁₂₄_add_angle₄₂₃_eq_angle₁₂₃ line 670 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, HAdd.hAdd, EuclideanGeometry.angle
Proof uses: abs, Real.Angle.toReal, EuclideanGeometry.oangle, Real.pi, Imo2023P2.aux₃, Imo2023P2.SphereOrder.oangle₁₂₃_add_oangle₃₄₁_eq_pi, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.SphereOrder.oangle₂₃₄_add_oangle₄₁₂_eq_pi, Imo2023P2.SphereOrder.sign_oangle₂₃₄_ne_zero, Eq.symm, EuclideanGeometry.angle_add_angle_add_angle_eq_pi, Ne.symm, Imo2023P2.SphereOrder.P₃_ne_P₄, Imo2023P2.SphereOrder.P₁_ne_P₂, LE.le, EuclideanGeometry.angle_le_angle_add_angle, Eq.mpr, id, congrArg, add_comm and 61 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₂₃₄ line 698 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, SignType, Real.Angle.sign, EuclideanGeometry.oangle
Proof uses: Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle_add, Imo2023P2.SphereOrder.P₁_ne_P₂, Ne.symm, Imo2023P2.SphereOrder.P₂_ne_P₄, Imo2023P2.SphereOrder.P₂_ne_P₃, EuclideanGeometry.angle, Imo2023P2.SphereOrder.angle₁₂₄_add_angle₄₂₃_eq_angle₁₂₃, Ne, Eq.mpr, id, congrArg, Eq.symm, EuclideanGeometry.oangle_rotate_sign, Imo2023P2.SphereOrder.sign_oangle₂₃₄_ne_zero, Imo2023P2.aux₄, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Eq.mp, add_comm and 4 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.trans line 711 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.rotate, Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, id, congrArg, Set.pair_comm, EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Eq.symm, EuclideanGeometry.Cospherical.two_zsmul_oangle_eq, Imo2023P2.SphereOrder.cospherical, Ne.symm, Imo2023P2.SphereOrder.P₁_ne_P₂ and 42 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 44 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.trans' line 752 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.trans
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.trans'' line 757 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.trans, Imo2023P2.SphereOrder.trans'
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.exists_diag_inter line 762 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Exists, Membership.mem, Set, Imo2023P2.diag_inter_set
Proof uses: Imo2023P2.inter_nonempty_of_not_parallel, Imo2023P2.SphereOrder.P₁_ne_P₃, Imo2023P2.SphereOrder.P₂_ne_P₄, AffineSubspace.Parallel, affineSpan, Insert.insert, Singleton.singleton, AffineSubspace.Parallel.refl, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, EuclideanGeometry.two_zsmul_oangle_of_parallel, Eq.mp, congrArg, Set.pair_comm, SignType, Real.Angle.sign, Eq.mpr, id and 45 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 37 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.diag_inter_ne_P₁ line 803 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Membership.mem, Set, Imo2023P2.diag_inter_set, Ne
Proof uses: AffineSubspace, affineSpan, Insert.insert, Singleton.singleton, Set.mem_of_mem_inter_right, Collinear, collinear_insert_of_mem_affineSpan_pair, Eq.mp, congrArg, EuclideanGeometry.Cospherical, Imo2023P2.SphereOrder.cospherical, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Exists, False, propext, EuclideanGeometry.cospherical_iff_exists_sphere and 19 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.diag_inter_sbtw line 833 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Membership.mem, Set, Imo2023P2.diag_inter_set, Sbtw
Proof uses: Classical.byContradiction, Not, Collinear, Insert.insert, Singleton.singleton, Eq.mpr, id, congrArg, Set.insert_comm, collinear_insert_of_mem_affineSpan_pair, And.left, AffineSubspace, affineSpan, Eq.mp, Inter.inter, And, propext, Set.mem_inter_iff, Imo2023P2.diag_inter_set.eq_1, Or and 44 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 55 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.SphereOrder.diag_inter_sbtw' line 889 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.SphereOrder, Membership.mem, Set, Imo2023P2.diag_inter_set, Sbtw
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.SphereOrder.rotate, Eq.mp, congrArg, Imo2023P2.diag_inter_set_rotate
30 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.sphereOrder_of_sbtw_diag_inter line 897 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Ne, Sbtw, Imo2023P2.SphereOrder
Proof uses: Imo2023P2.SphereOrder.mk, SignType, Real.Angle.sign, EuclideanGeometry.oangle, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Exists, False, Eq.mp, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Membership.mem, Set.mem_insert, Set.mem_insert_of_mem, Or, Imo2023P2.eq_of_mem_sphere_of_collinear and 24 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Sphere.antipode line 930 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, EuclideanGeometry.Sphere
Proof uses: EuclideanGeometry.Sphere.secondInter, VSub.vsub, EuclideanGeometry.Sphere.center
10 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Sphere.antipode_mem_sphere line 935 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, EuclideanGeometry.Sphere, Membership.mem, Imo2023P2.Sphere.antipode
Proof uses: Eq.mpr, EuclideanGeometry.Sphere.secondInter, VSub.vsub, EuclideanGeometry.Sphere.center, id, Eq, congrArg, Imo2023P2.Sphere.antipode.eq_1, propext, EuclideanGeometry.Sphere.secondInter_mem
12 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Sphere.antipode_isDiameter line 941 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, EuclideanGeometry.Sphere, Membership.mem, EuclideanGeometry.Sphere.IsDiameter, Imo2023P2.Sphere.antipode
Proof uses: Eq.mpr, EuclideanGeometry.Sphere.secondInter, VSub.vsub, EuclideanGeometry.Sphere.center, id, Eq, congrArg, Imo2023P2.Sphere.antipode.eq_1, And, Wbtw, propext, EuclideanGeometry.Sphere.isDiameter_iff_mem_and_mem_and_wbtw, And.intro, Iff.mpr, EuclideanGeometry.Sphere.secondInter_mem, EuclideanGeometry.Sphere.wbtw_secondInter, LE.le, Dist.dist, EuclideanGeometry.Sphere.radius, OfNat.ofNat and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
structure Imo2023P2.Imo2023q2Cfg line 951 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin
13 instances, coercions and auxiliary constants left out.
def Imo2023P2.Imo2023q2Cfg.ABC line 977 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Affine.Triangle
Proof uses: Affine.Simplex.mk, Matrix.vecCons, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.C, Matrix.vecEmpty, Imo2023P2.Imo2023q2Cfg.h_ABC
30 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.B_ne_A line 980 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Function.Injective.ne, Nat.succ, Matrix.vecCons, Imo2023P2.Imo2023q2Cfg.C, Matrix.vecEmpty, AffineIndependent.injective, Imo2023P2.Imo2023q2Cfg.h_ABC, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.C_ne_A line 983 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Function.Injective.ne, Nat.succ, Matrix.vecCons, Imo2023P2.Imo2023q2Cfg.B, Matrix.vecEmpty, AffineIndependent.injective, Imo2023P2.Imo2023q2Cfg.h_ABC, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.B_ne_C line 986 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Function.Injective.ne, Nat.succ, Matrix.vecCons, Imo2023P2.Imo2023q2Cfg.A, Matrix.vecEmpty, AffineIndependent.injective, Imo2023P2.Imo2023q2Cfg.h_ABC, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.S_ne_B line 989 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.B
Proof uses: Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B_ne_C, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mpr, id, congrArg, Eq.symm, Dist.dist, propext, dist_eq_zero, And.left, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, Eq.mp
30 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.S_ne_C line 997 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.B_ne_C, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mpr, id, congrArg, Eq.symm, Dist.dist, propext, dist_eq_zero, Eq.mp, dist_comm, And.left, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω
30 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.S_ne_A line 1005 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Dist.dist, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, And.left, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, LT.lt, Imo2023P2.Imo2023q2Cfg.h_AB_lt_BC, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl and 3 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.A_in_Ω line 1013 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.A
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_Ω, And.left, Eq.mp, And, congrArg, propext, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.B_in_Ω line 1018 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.B
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.A, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_Ω, And.left, And.right, And, Eq.mp, congrArg, propext, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.C_in_Ω line 1023 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.C
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_Ω, And.right, And, Eq.mp, congrArg, propext, Set.singleton_subset_iff, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.A_opposite_BC line 1028 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, Set.range, HAdd.hAdd and 4 more
Proof uses: Set.ext, Eq.mpr, Iff, Membership.mem, Or, id, congrArg, propext, Set.mem_insert_iff, Set.mem_singleton_iff, Exists, Set.mem_range, Iff.intro, Exists.intro, Fin.succAbove, Fin.cast, Nat.sub_one_add_one, NeZero.ne, Affine.Simplex.faceOpposite_point_eq_point_succAbove, of_eq_true and 25 more
59 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAC_acute line 1049 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, HDiv.hDiv, Real.pi
Proof uses: Affine.Simplex.AcuteAngled, Affine.Simplex.mk, Matrix.vecCons, Matrix.vecEmpty, Imo2023P2.Imo2023q2Cfg.h_ABC, Imo2023P2.Imo2023q2Cfg.h_acute_ABC, Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, And.right, And, Eq.mp, Affine.Simplex.points, HAdd.hAdd, propext, Affine.Triangle.acuteAngled_iff_angle_lt
48 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_ACB_acute line 1056 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, HDiv.hDiv, Real.pi
Proof uses: Affine.Simplex.AcuteAngled, Affine.Simplex.mk, Matrix.vecCons, Matrix.vecEmpty, Imo2023P2.Imo2023q2Cfg.h_ABC, Imo2023P2.Imo2023q2Cfg.h_acute_ABC, Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, And.left, And.right, And, Eq.mp, Affine.Simplex.points, HAdd.hAdd, propext, Affine.Triangle.acuteAngled_iff_angle_lt
48 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CBA_acute line 1063 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, HDiv.hDiv, Real.pi
Proof uses: Affine.Simplex.AcuteAngled, Affine.Simplex.mk, Matrix.vecCons, Matrix.vecEmpty, Imo2023P2.Imo2023q2Cfg.h_ABC, Imo2023P2.Imo2023q2Cfg.h_acute_ABC, Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, And.left, And, Eq.mp, Affine.Simplex.points, HAdd.hAdd, propext, Affine.Triangle.acuteAngled_iff_angle_lt
48 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_ACB_lt_angle_CBA line 1070 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B
Proof uses: AffineIndependent, Nat.succ, Matrix.vecCons, Matrix.vecEmpty, Imo2023P2.Imo2023q2Cfg.h_ABC, Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, Dist.dist, propext, EuclideanGeometry.angle_lt_iff_dist_lt, Eq.mp, Not, Collinear, Insert.insert, Set, Singleton.singleton, affineIndependent_iff_not_collinear_set, Imo2023P2.Imo2023q2Cfg.h_AB_lt_BC
35 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CBS_eq_angle_SBC line 1077 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S
Proof uses: EuclideanGeometry.angle_eq_angle_of_dist_eq, Eq.symm, Dist.dist, And.left, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_BSC_eq_oangle_BAC line 1082 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Eq.symm, HSMul.hSMul, Int, EuclideanGeometry.Sphere.two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.A_in_Ω, And.right, Dist.dist, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.S_ne_B and 25 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BSC_eq_angle_BAC line 1095 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Imo2023P2.angle_eq_of_oangle_eq, Imo2023P2.Imo2023q2Cfg.oangle_BSC_eq_oangle_BAC, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_B, Imo2023P2.Imo2023q2Cfg.S_ne_C, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CBS_eq line 1099 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, HDiv.hDiv, HAdd.hAdd, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Eq.mpr, HMul.hMul, id, congrArg, propext, eq_div_iff_mul_eq, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Nat.cast_zero, Bool, Bool.false, mul_two, Imo2023P2.Imo2023q2Cfg.angle_CBS_eq_angle_SBC, Real.pi, EuclideanGeometry.angle_add_angle_add_angle_eq_pi, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_C, Imo2023P2.Imo2023q2Cfg.C_ne_A and 7 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CBS_lt_angle_CBA line 1109 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Eq.mpr, HDiv.hDiv, HAdd.hAdd, id, congrArg, Imo2023P2.Imo2023q2Cfg.angle_CBS_eq, HMul.hMul, propext, div_lt_iff₀, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Bool.false, mul_two, add_lt_add_iff_left, Imo2023P2.Imo2023q2Cfg.angle_ACB_lt_angle_CBA
82 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_SBA_eq_sign_oangle_CBS line 1113 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle_add, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.Imo2023q2Cfg.S_ne_B, Imo2023P2.Imo2023q2Cfg.B_ne_A, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.angle_CBS_lt_angle_CBA, Imo2023P2.Imo2023q2Cfg.h_S_A, Eq.symm, Imo2023P2.aux₅, Eq.mp, abs, Real.Angle.toReal, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal
34 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC line 1123 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω, And, Membership.mem and 31 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.M line 1146 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg
Proof uses: Imo2023P2.Sphere.antipode, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.S
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.M_in_Ω line 1148 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.Sphere.antipode_mem_sphere, Imo2023P2.Imo2023q2Cfg.S, And.right, Dist.dist, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω
29 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.isDiameter_SM line 1152 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.Sphere.IsDiameter, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.Sphere.antipode_isDiameter, And.right, Dist.dist, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Ω_radius_ne_zero line 1156 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.B_ne_A, Mathlib.Tactic.Contrapose.contrapose₄, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Eq.mpr, EuclideanGeometry.Sphere.center, id, congrArg, Eq.mp, Dist.dist, propext, dist_eq_zero, EuclideanGeometry.mem_sphere, Eq.refl
32 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_SOM line 1164 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.S, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.M
Proof uses: EuclideanGeometry.Sphere.IsDiameter.sbtw, Imo2023P2.Imo2023q2Cfg.isDiameter_SM, Imo2023P2.Imo2023q2Cfg.Ω_radius_ne_zero
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.O_ne_A line 1167 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.A
Proof uses: EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω_radius_ne_zero, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mp, Dist.dist, congrArg, Iff.mp, Membership.mem, EuclideanGeometry.Sphere, EuclideanGeometry.mem_sphere, Imo2023P2.Imo2023q2Cfg.A_in_Ω, dist_comm, Eq.symm, propext, dist_eq_zero
30 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.S_ne_M line 1173 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Eq.mpr, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, propext, EuclideanGeometry.Sphere.IsDiameter.left_ne_right_iff_radius_ne_zero, Imo2023P2.Imo2023q2Cfg.isDiameter_SM, Imo2023P2.Imo2023q2Cfg.Ω_radius_ne_zero
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_SBM_eq line 1177 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.M, HDiv.hDiv, Real.pi
Proof uses: Eq.mpr, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, propext, EuclideanGeometry.Sphere.thales_theorem, Imo2023P2.Imo2023q2Cfg.isDiameter_SM, Imo2023P2.Imo2023q2Cfg.B_in_Ω
41 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_SCM_eq line 1181 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.M, HDiv.hDiv, Real.pi
Proof uses: Eq.mpr, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, propext, EuclideanGeometry.Sphere.thales_theorem, Imo2023P2.Imo2023q2Cfg.isDiameter_SM, Imo2023P2.Imo2023q2Cfg.C_in_Ω
41 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.BM_eq_CM line 1185 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Dist.dist, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Eq.mpr, id, congrArg, dist_comm, HMul.hMul, Eq.symm, propext, mul_self_inj, dist_nonneg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.S, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_SBM_eq, Imo2023P2.Imo2023q2Cfg.angle_SCM_eq, HSub.hSub, Eq.mp, HAdd.hAdd, sub_eq_iff_eq_add', EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two and 9 more
55 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.M_ne_B line 1195 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B
Proof uses: Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B_ne_C, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mpr, id, congrArg, Eq.symm, Dist.dist, propext, dist_eq_zero, Eq.mp, dist_comm, Imo2023P2.Imo2023q2Cfg.BM_eq_CM
29 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.M_ne_C line 1203 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.B_ne_C, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mpr, id, congrArg, Eq.symm, Dist.dist, propext, dist_eq_zero, Eq.mp, Imo2023P2.Imo2023q2Cfg.BM_eq_CM, dist_comm
29 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CMS_lt line 1211 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.S, HDiv.hDiv, Real.pi
Proof uses: EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two, Imo2023P2.Imo2023q2Cfg.angle_SCM_eq, Imo2023P2.Imo2023q2Cfg.M_ne_C
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_SMB_lt line 1215 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B, HDiv.hDiv, Real.pi
Proof uses: Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two, Imo2023P2.Imo2023q2Cfg.angle_SBM_eq, Imo2023P2.Imo2023q2Cfg.M_ne_B
42 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_CMS_eq_oangle_SMB line 1220 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.B
Proof uses: HSMul.hSMul, Int, EuclideanGeometry.Sphere.two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Imo2023P2.Imo2023q2Cfg.M_in_Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, And.right, Dist.dist, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, Imo2023P2.Imo2023q2Cfg.M_ne_C, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_M, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.Imo2023q2Cfg.S_ne_B and 20 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BSCM line 1235 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.rotate', Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω, And and 35 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM line 1259 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.rotate, Imo2023P2.SphereOrder.trans, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC, Imo2023P2.Imo2023q2Cfg.sphereOrder_BSCM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BACM line 1262 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.symm, Imo2023P2.SphereOrder.trans'', Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC, Imo2023P2.SphereOrder.rotate, Imo2023P2.Imo2023q2Cfg.sphereOrder_BSCM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.M_ne_A line 1265 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Ne.symm, Imo2023P2.SphereOrder.P₂_ne_P₄, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.X_set line 1268 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set
Proof uses: Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.exists_X line 1270 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Exists, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set
Proof uses: Imo2023P2.SphereOrder.exists_diag_inter, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_ne_A line 1273 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.SphereOrder.rotate, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM, Eq.mp, Imo2023P2.diag_inter_set, congrArg, Imo2023P2.diag_inter_set_rotate, Imo2023P2.Imo2023q2Cfg.X_set.eq_1
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_ne_S line 1279 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM, Eq.mp, Imo2023P2.diag_inter_set, congrArg, Imo2023P2.diag_inter_set_rotate, Imo2023P2.Imo2023q2Cfg.X_set.eq_1
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_ne_M line 1285 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.SphereOrder.rotate', Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM, Eq.mp, Imo2023P2.diag_inter_set, congrArg, Eq.symm, Imo2023P2.diag_inter_set_rotate, Imo2023P2.Imo2023q2Cfg.X_set.eq_1
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AXM line 1291 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw', Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_BXS line 1295 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_ne_O line 1299 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.SphereOrder.sign_oangle₂₃₄_ne_zero, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM, Ne.symm, EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero, Eq.mp, congrArg, Real.Angle, Eq.symm, Sbtw.oangle_eq_left, Sbtw.symm, Imo2023P2.Imo2023q2Cfg.sbtw_SOM, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AXM and 1 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_BAM_eq_oangle_MAC line 1308 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.C
Proof uses: HSMul.hSMul, Int, EuclideanGeometry.Sphere.two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Imo2023P2.Imo2023q2Cfg.M_in_Ω, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.M_ne_A, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.Imo2023q2Cfg.M_ne_C, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.M_ne_B, Dist.dist, Imo2023P2.Imo2023q2Cfg.BM_eq_CM, EuclideanGeometry.oangle_eq_oangle_of_dist_eq, Eq.mp, congrArg and 14 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAM_eq_angle_MAC line 1328 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.angle_eq_of_oangle_eq, Imo2023P2.Imo2023q2Cfg.oangle_BAM_eq_oangle_MAC, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.M_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_BAX_eq_oangle_XAC line 1333 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A and 1 more
Proof uses: Eq.mpr, HAdd.hAdd, Imo2023P2.Imo2023q2Cfg.M, id, congrArg, Eq.symm, EuclideanGeometry.oangle_add, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.M_ne_A, Imo2023P2.Imo2023q2Cfg.X_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.oangle_BAM_eq_oangle_MAC, add_comm, propext, add_right_cancel_iff, Neg.neg, EuclideanGeometry.oangle_rev, Wbtw.oangle₂₁₃_eq_zero, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AXM and 8 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAX_eq_angle_XAC line 1343 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.angle_eq_of_oangle_eq, Imo2023P2.Imo2023q2Cfg.oangle_BAX_eq_oangle_XAC, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.X_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAM_acute line 1349 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, HDiv.hDiv, Real.pi
Proof uses: Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.angle_BAC_acute, lt_of_lt_of_le', HAdd.hAdd, Imo2023P2.SphereOrder.angle₁₂₄_add_angle₄₂₃_eq_angle₁₂₃, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACM, Eq.mpr, LE.le, id, congrArg, Eq.symm, propext, le_add_iff_nonneg_right, EuclideanGeometry.angle_nonneg
62 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAM_eq_angle_BAX line 1356 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, Or, Real.Angle, EuclideanGeometry.oangle, Wbtw, propext, EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtw, Imo2023P2.Imo2023q2Cfg.M_ne_A, Imo2023P2.Imo2023q2Cfg.X_ne_A, Or.inr, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AXM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAM_eq line 1364 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, HDiv.hDiv, HSub.hSub, Real.pi and 1 more
Proof uses: HAdd.hAdd, Imo2023P2.SphereOrder.angle₁₂₄_add_angle₄₂₃_eq_angle₁₂₃, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACM, Eq.mpr, id, congrArg, Eq.mp, HMul.hMul, Eq.symm, propext, eq_div_iff_mul_eq, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Nat.cast_zero, Bool, Bool.false, mul_two, Imo2023P2.Imo2023q2Cfg.angle_BAM_eq_angle_MAC and 54 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.XO_lt_Ω_radius line 1373 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, LT.lt, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω and 1 more
Proof uses: Imo2023P2.Sphere.dist_le_radius_of_sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sbtw_AXM, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.M_in_Ω
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_in_Ω line 1377 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.P
Proof uses: Set, Inter.inter, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_P, And.right, Eq.mp, And, congrArg, propext, Set.mem_inter_iff
29 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.B_in_ω line 1382 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.B
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.D, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.L, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_ω, And.left, Eq.mp, And, congrArg, propext, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.D_in_ω line 1387 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.D
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.L, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_ω, And.left, And.right, And, Eq.mp, congrArg, propext, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.L_in_ω line 1392 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.L
Proof uses: LE.le, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.D, Singleton.singleton, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_ω, And.right, And, Eq.mp, congrArg, propext, Set.singleton_subset_iff, Set.insert_subset_iff
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_in_ω line 1397 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.P
Proof uses: Set, Inter.inter, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.h_P, And.left, Eq.mp, And, congrArg, propext, Set.mem_inter_iff
29 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.ω_radius_ne_zero line 1402 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.ω
Proof uses: Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.h_P_ne_B, Mathlib.Tactic.Contrapose.contrapose₄, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.B_in_ω, Imo2023P2.Imo2023q2Cfg.P_in_ω, Eq.mpr, EuclideanGeometry.Sphere.center, id, congrArg, Eq.mp, Dist.dist, propext, dist_eq_zero, EuclideanGeometry.mem_sphere, Eq.refl
32 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_X line 1410 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, Imo2023P2.Imo2023q2Cfg.P
Proof uses: LT.lt, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.XO_lt_Ω_radius, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.P_in_Ω, congrArg, Eq.symm, Eq.mp, propext and 1 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.perp_A_BC_eq_AE line 1419 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, AffineSubspace, Imo2023P2.Imo2023q2Cfg.perp_A_BC, affineSpan, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.A, Singleton.singleton and 1 more
Proof uses: Eq.symm, Membership.mem, Inter.inter, Metric.sphere, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_E, LE.le, affineSpan_le_of_subset_coe, Eq.mpr, And, id, congrArg, propext, Set.pair_subset_iff, And.intro, And.right, Submodule.IsOrtho, AffineSubspace.direction and 26 more
64 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_in_Ω line 1444 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Set, Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.perp_A_BC, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_E, And.right, Eq.mp, And, congrArg, propext, Set.mem_inter_iff
33 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.H line 1449 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg
Proof uses: Affine.Simplex.altitudeFoot, Imo2023P2.Imo2023q2Cfg.ABC, HAdd.hAdd
31 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.collinear_HBC line 1451 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C
Proof uses: collinear_insert_of_mem_affineSpan_pair, Eq.mpr, Membership.mem, AffineSubspace, affineSpan, Set.range, HAdd.hAdd, HSub.hSub, Affine.Simplex.points, Affine.Simplex.faceOpposite, Imo2023P2.Imo2023q2Cfg.ABC, id, congrArg, Imo2023P2.Imo2023q2Cfg.A_opposite_BC, Affine.Simplex.altitudeFoot_mem_affineSpan_faceOpposite
45 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_AHx_eq line 1456 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, EuclideanGeometry.angle and 4 more
Proof uses: Eq.mpr, InnerProductGeometry.angle, VSub.vsub, id, congrArg, EuclideanGeometry.angle.eq_1, Inner.inner, Eq.symm, propext, InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, Submodule.IsOrtho.inner_eq, vectorSpan, Set.image, HAdd.hAdd, Affine.Simplex.points, Imo2023P2.Imo2023q2Cfg.ABC, Compl.compl, AffineSubspace.direction, Affine.Simplex.altitude and 26 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_AHC_eq line 1470 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.C, HDiv.hDiv, Real.pi
Proof uses: Imo2023P2.Imo2023q2Cfg.angle_AHx_eq, of_eq_true, Membership.mem, Set, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Eq.trans, Or, True, congrArg, eq_self, or_true
38 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_AHB_eq line 1474 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.B, HDiv.hDiv, Real.pi
Proof uses: Imo2023P2.Imo2023q2Cfg.angle_AHx_eq, of_eq_true, Membership.mem, Set, Insert.insert, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, Eq.trans, Or, True, congr, congrArg, eq_self, true_or
38 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.not_sbtw_BCH line 1478 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Not, Sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.H
Proof uses: EuclideanGeometry.angle, Real.pi, Sbtw.angle₁₂₃_eq_pi, Mathlib.Tactic.Contrapose.contrapose₁, False, Eq.mpr, True, Ne, id, implies_congr, not_false_eq_true, Eq.refl, ne_of_lt, Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.oangle_add, Imo2023P2.Imo2023q2Cfg.B_ne_C, Ne.symm and 50 more
95 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.not_sbtw_CBH line 1494 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Not, Sbtw, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.H
Proof uses: EuclideanGeometry.angle, Real.pi, Sbtw.angle₁₂₃_eq_pi, Mathlib.Tactic.Contrapose.contrapose₁, False, Eq.mpr, True, Ne, id, implies_congr, not_false_eq_true, Eq.refl, ne_of_lt, Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.oangle_add, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_C and 50 more
95 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.wbtw_BHC line 1509 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Wbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Not, Sbtw, Imo2023P2.Imo2023q2Cfg.not_sbtw_BCH, Imo2023P2.Imo2023q2Cfg.not_sbtw_CBH, Or, Imo2023P2.Collinear.sbtw_or_sbtw_or_wbtw_of_ne, Imo2023P2.Imo2023q2Cfg.collinear_HBC, Imo2023P2.Imo2023q2Cfg.B_ne_C, Eq.mpr, id, congrArg, propext, wbtw_comm, False.elim, Eq.mp, sbtw_comm
32 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.H_ne_B line 1517 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.B
Proof uses: LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_CBA_acute, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, congrArg, Eq.symm, EuclideanGeometry.angle_comm, Imo2023P2.Imo2023q2Cfg.angle_AHC_eq
41 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.H_ne_C line 1523 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.C
Proof uses: LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.B, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_ACB_acute, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, congrArg, Eq.symm, Imo2023P2.Imo2023q2Cfg.angle_AHB_eq
41 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_BHC line 1529 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Eq.mpr, And, Wbtw, Ne, id, congrArg, Sbtw.eq_1, And.intro, Imo2023P2.Imo2023q2Cfg.wbtw_BHC, Imo2023P2.Imo2023q2Cfg.H_ne_B, Imo2023P2.Imo2023q2Cfg.H_ne_C
30 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.AE_eq_altitude line 1536 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, AffineSubspace, affineSpan, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.A, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.E and 3 more
Proof uses: AffineSubspace.ext_of_direction_eq, Eq.mpr, Submodule, AffineSubspace.direction, Min.min, Submodule.orthogonal, vectorSpan, Set.image, Affine.Simplex.points, Compl.compl, Set.range, id, congrArg, Affine.Simplex.direction_altitude, HSub.hSub, Affine.Simplex.faceOpposite, Eq.symm, Affine.Simplex.range_faceOpposite_points, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.C and 45 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_AHE line 1562 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.H, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Eq.mpr, id, congrArg, Set.insert_comm, collinear_insert_of_mem_affineSpan_pair, Membership.mem, AffineSubspace, affineSpan, Affine.Simplex.altitude, Imo2023P2.Imo2023q2Cfg.ABC, HAdd.hAdd, Imo2023P2.Imo2023q2Cfg.AE_eq_altitude, Affine.Simplex.altitudeFoot_mem_altitude
44 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.HO_lt_Ω_radius line 1568 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, Dist.dist, Imo2023P2.Imo2023q2Cfg.H, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius
Proof uses: Imo2023P2.Sphere.dist_le_radius_of_sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.sbtw_BHC, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AHE line 1572 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.E
Proof uses: EuclideanGeometry.sbtw_of_collinear_of_dist_center_lt_radius, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.collinear_AHE, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.HO_lt_Ω_radius, Imo2023P2.Imo2023q2Cfg.E_in_Ω, Ne.symm, Imo2023P2.Imo2023q2Cfg.h_E_ne_A
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.H_ne_A line 1576 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.sbtw_AHE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BACE line 1579 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.sphereOrder_of_sbtw_diag_inter, Imo2023P2.Imo2023q2Cfg.H, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω, And and 13 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_BAE_eq_sign_oangle_BAM line 1592 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Eq.mpr, id, congrArg, EuclideanGeometry.oangle_rotate_sign, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₂₃₄, Imo2023P2.SphereOrder.rotate', Imo2023P2.Imo2023q2Cfg.sphereOrder_BACE, Eq.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAE_eq_BAH line 1599 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.H
Proof uses: Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, Or, Real.Angle, EuclideanGeometry.oangle, Wbtw, propext, EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtw, Imo2023P2.Imo2023q2Cfg.h_E_ne_A, Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.sbtw_AHE, Or.inr, Sbtw.wbtw
28 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_HBA_eq_CBA line 1606 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Eq.mpr, id, congrArg, EuclideanGeometry.angle_comm, Eq.symm, Or, Real.Angle, EuclideanGeometry.oangle, Wbtw, propext, EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtw, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_C, Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.sbtw_BHC, Or.inr, Sbtw.wbtw
28 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAE_eq line 1615 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, HSub.hSub, HDiv.hDiv, Real.pi and 1 more
Proof uses: Eq.mpr, Imo2023P2.Imo2023q2Cfg.H, id, congrArg, Imo2023P2.Imo2023q2Cfg.angle_BAE_eq_BAH, Eq.symm, Imo2023P2.Imo2023q2Cfg.angle_HBA_eq_CBA, HAdd.hAdd, propext, eq_sub_iff_add_eq, add_comm, EuclideanGeometry.angle_add_angle_add_angle_eq_pi, Ne.symm, Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.sbtw_BHC, Eq.mp, Imo2023P2.Imo2023q2Cfg.angle_AHB_eq, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow and 48 more
102 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAE_lt_angle_BAM line 1622 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Eq.mpr, HDiv.hDiv, HSub.hSub, Real.pi, Imo2023P2.Imo2023q2Cfg.C, id, congrArg, Imo2023P2.Imo2023q2Cfg.angle_BAM_eq, Imo2023P2.Imo2023q2Cfg.angle_BAE_eq, Eq.symm, propext, sub_pos, HAdd.hAdd, HMul.hMul, Neg.neg, Eq.trans, HPow.hPow, Nat.rawCast, NNRat.rawCast, Rat.rawCast and 79 more
139 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_EAM_eq_sign_oangle_BAE line 1628 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B
Proof uses: Real.Angle, HAdd.hAdd, EuclideanGeometry.oangle_add, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.h_E_ne_A, Imo2023P2.Imo2023q2Cfg.M_ne_A, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.angle_BAE_lt_angle_BAM, Imo2023P2.Imo2023q2Cfg.sign_oangle_BAE_eq_sign_oangle_BAM, Eq.symm, Imo2023P2.aux₅, Eq.mp, abs, Real.Angle.toReal, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal
34 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BAME line 1638 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.rotate, Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω, And and 18 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE line 1655 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.rotate, Imo2023P2.SphereOrder.trans', Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACE, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_ASME line 1658 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.trans'', Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BAME, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_ne_S line 1661 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Ne.symm, Imo2023P2.SphereOrder.P₃_ne_P₄, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_ne_B line 1664 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.B
Proof uses: Ne.symm, Imo2023P2.SphereOrder.P₁_ne_P₄, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_ne_C line 1667 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Ne.symm, Imo2023P2.SphereOrder.P₃_ne_P₄, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACE
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_ne_H line 1670 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.H
Proof uses: Ne.symm, Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sbtw_AHE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.D_in_diag_inter_BASE line 1673 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.E and 1 more
Proof uses: Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.perp_A_BC, affineSpan, Insert.insert, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.h_D, And.intro, And.right, Eq.mpr, id, congrArg, Eq.symm, Imo2023P2.Imo2023q2Cfg.perp_A_BC_eq_AE, And.left
35 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.D_ne_B line 1680 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.B
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE, Imo2023P2.Imo2023q2Cfg.D_in_diag_inter_BASE
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_BDS line 1683 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE, Imo2023P2.Imo2023q2Cfg.D_in_diag_inter_BASE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_ADE line 1686 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw', Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASE, Imo2023P2.Imo2023q2Cfg.D_in_diag_inter_BASE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.D_ne_A line 1689 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.sbtw_ADE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.D_ne_S line 1692 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.sbtw_BDS
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.E_ne_D line 1695 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Ne.symm, Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sbtw_ADE
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.DO_lt_Ω_radius line 1698 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, Dist.dist, Imo2023P2.Imo2023q2Cfg.D, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius
Proof uses: Imo2023P2.Sphere.dist_le_radius_of_sbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sbtw_BDS, Imo2023P2.Imo2023q2Cfg.B_in_Ω, And.right, Imo2023P2.Imo2023q2Cfg.C, Membership.mem, Set, Metric.sphere, Imo2023P2.Imo2023q2Cfg.h_S_Ω
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_D line 1702 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.D
Proof uses: LT.lt, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.DO_lt_Ω_radius, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.P_in_Ω, congrArg, Eq.symm, Eq.mp and 2 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.L_ne_D line 1710 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Membership.mem, Set, Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Insert.insert, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.h_L, Imo2023P2.Imo2023q2Cfg.perp_A_BC, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.h_D, Collinear.mem_affineSpan_of_mem_of_ne, collinear_insert_of_mem_affineSpan_pair, Eq.mp, congrArg, And.right, And and 40 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.prll_D_BC_eq_LD line 1737 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.L, Singleton.singleton and 1 more
Proof uses: Membership.mem, Inter.inter, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.h_L, Eq.symm, LE.le, affineSpan_le_of_subset_coe, Eq.mpr, And, id, congrArg, propext, Set.pair_subset_iff, And.intro, Set.mem_of_mem_inter_left, And.left, AffineSubspace.Parallel, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.h_prll_D_BC and 17 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_S line 1757 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.D, Singleton.singleton, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_BDS, Or, Imo2023P2.eq_of_mem_sphere_of_collinear, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.B_in_ω, Imo2023P2.Imo2023q2Cfg.D_in_ω, Imo2023P2.Imo2023q2Cfg.P_in_ω, Eq.mp, congrArg, Eq.symm, Mathlib.Tactic.Contrapose.contrapose₁, False and 13 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_BLE line 1768 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.L, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Membership.mem, Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Imo2023P2.Imo2023q2Cfg.h_L, Eq.mp, congrArg, Set.insert_comm, collinear_insert_of_mem_affineSpan_pair, And.right, And, propext, Set.mem_inter_iff
39 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.L_ne_E line 1776 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Submodule.IsOrtho, AffineSubspace.direction, Imo2023P2.Imo2023q2Cfg.perp_A_BC, affineSpan, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.C, And.left, Membership.mem, AffineSubspace, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.h_perp_A_BC, AffineSubspace.Parallel, Imo2023P2.Imo2023q2Cfg.prll_D_BC, And.right, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.h_prll_D_BC, Collinear.mem_affineSpan_of_mem_of_ne and 42 more
75 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.H_ne_D line 1792 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.D
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC, Mathlib.Tactic.Contrapose.contrapose₄, Eq.mpr, id, congrArg, Eq.symm, EuclideanGeometry.oangle_rotate_sign, Real.Angle, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_BHC, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_BDS and 3 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_EHD line 1799 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.H, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Membership.mem, AffineSubspace, affineSpan, Imo2023P2.Imo2023q2Cfg.A, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AHE, of_eq_true, Eq.trans, Or, True, congrArg, eq_self, or_true, congr, true_or, Imo2023P2.Imo2023q2Cfg.h_E_ne_A, Imo2023P2.Imo2023q2Cfg.sbtw_ADE, mem_affineSpan and 1 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_EHD line 1807 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Or, Wbtw, Imo2023P2.Collinear.sbtw_or_wbtw_or_wbtw, Imo2023P2.Imo2023q2Cfg.collinear_EHD, False.elim, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Wbtw.oangle_sign_eq_of_ne_left, Imo2023P2.Imo2023q2Cfg.H_ne_D, Mathlib.Tactic.Contrapose.contrapose₁, False, Eq.mpr, Not, True, Ne, id, implies_congr, not_false_eq_true and 31 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.not_collinear_EBH line 1838 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Not, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.B, Singleton.singleton and 1 more
Proof uses: Imo2023P2.Imo2023q2Cfg.C, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_BHC, Eq.mpr, SignType, Real.Angle.sign, EuclideanGeometry.oangle, id, congrArg, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Or, Real.Angle, Real.pi, Real.Angle.sign_eq_zero_iff, HSMul.hSMul, Real.Angle.two_nsmul_eq_zero_iff, HAdd.hAdd and 14 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_LDE_eq_oangle_BHE line 1851 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.H
Proof uses: Eq.symm, EuclideanGeometry.oangle_eq_of_parallel, Membership.mem, AffineSubspace, affineSpan, Insert.insert, Set, Singleton.singleton, Mathlib.Tactic.Contrapose.contrapose₁, Collinear, False, Eq.mpr, Not, True, id, implies_congr, not_false_eq_true, Eq.refl, Imo2023P2.Imo2023q2Cfg.not_collinear_EBH, Eq.mp and 34 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_ELD_eq_oangle_EBH line 1872 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.H
Proof uses: Eq.symm, EuclideanGeometry.oangle_eq_of_parallel, Membership.mem, AffineSubspace, affineSpan, Insert.insert, Set, Singleton.singleton, Mathlib.Tactic.Contrapose.contrapose₁, Collinear, False, Eq.mpr, Not, True, id, implies_congr, not_false_eq_true, Eq.refl, Imo2023P2.Imo2023q2Cfg.not_collinear_EBH, Eq.mp and 34 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.BE_lt_LE line 1894 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, Dist.dist, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.L
Proof uses: Not, Collinear, Insert.insert, Set, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.H, Imo2023P2.Imo2023q2Cfg.not_collinear_EBH, Imo2023P2.Imo2023q2Cfg.D, Eq.mpr, SignType, Real.Angle.sign, EuclideanGeometry.oangle, id, congrArg, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Real.Angle, Imo2023P2.Imo2023q2Cfg.oangle_ELD_eq_oangle_EBH, HDiv.hDiv and 30 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.not_wbtw_BLE line 1920 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Not, Wbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Eq.mpr, LT.lt, Dist.dist, HAdd.hAdd, id, congrArg, Eq.symm, propext, dist_lt_dist_add_dist_iff, lt_add_of_nonneg_of_lt, dist_nonneg, Imo2023P2.Imo2023q2Cfg.BE_lt_LE
43 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.LO_gt_Ω_radius line 1924 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, Dist.dist, Imo2023P2.Imo2023q2Cfg.L, EuclideanGeometry.Sphere.center
Proof uses: Not, Wbtw, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.not_wbtw_BLE, Mathlib.Tactic.Contrapose.contrapose₂, Eq.mpr, LE.le, id, implies_congr, Eq.refl, EuclideanGeometry.wbtw_of_collinear_of_dist_center_le_radius, Imo2023P2.Imo2023q2Cfg.collinear_BLE, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.E_in_Ω, Ne.symm, Imo2023P2.Imo2023q2Cfg.E_ne_B
38 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.L_ne_P line 1931 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.P
Proof uses: LT.lt, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.LO_gt_Ω_radius, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.P_in_Ω, congrArg, Eq.mp, propext and 1 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.L_ne_B line 1939 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.B
Proof uses: LT.lt, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.LO_gt_Ω_radius, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_eq, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.B_in_Ω, congrArg, Eq.mp, propext and 1 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_LPS line 1947 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.P, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Eq.mpr, SignType, Real.Angle.sign, EuclideanGeometry.oangle, id, congrArg, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Or, Real.Angle, Real.pi, Real.Angle.sign_eq_zero_iff, HSMul.hSMul, Int, Real.Angle.two_zsmul_eq_zero_iff, HAdd.hAdd, Imo2023P2.Imo2023q2Cfg.B, EuclideanGeometry.oangle_add, Imo2023P2.Imo2023q2Cfg.L_ne_P and 57 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.F line 1968 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg
Proof uses: Imo2023P2.Sphere.antipode, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.A
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.F_in_Ω line 1970 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.Sphere.antipode_mem_sphere, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.A_in_Ω
27 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.isDiameter_AF line 1974 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.Sphere.IsDiameter, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.Sphere.antipode_isDiameter, Imo2023P2.Imo2023q2Cfg.A_in_Ω
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AOF line 1978 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.F
Proof uses: EuclideanGeometry.Sphere.IsDiameter.sbtw, Imo2023P2.Imo2023q2Cfg.isDiameter_AF, Imo2023P2.Imo2023q2Cfg.Ω_radius_ne_zero
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_A line 1981 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Ne.symm, Sbtw.left_ne_right, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.sbtw_AOF
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_O line 1984 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: Ne.symm, Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sbtw_AOF
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_B line 1987 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.B
Proof uses: LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_ACB_acute, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, ge_of_eq, EuclideanGeometry.Sphere.IsDiameter, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.isDiameter_AF, Membership.mem, EuclideanGeometry.Sphere and 5 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_C line 1996 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.C
Proof uses: LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_CBA_acute, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, ge_of_eq, EuclideanGeometry.Sphere.IsDiameter, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.isDiameter_AF, Membership.mem, EuclideanGeometry.Sphere and 6 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_ASF_eq line 2005 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, HDiv.hDiv, Real.pi
Proof uses: Eq.mpr, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, propext, EuclideanGeometry.Sphere.thales_theorem, Imo2023P2.Imo2023q2Cfg.isDiameter_AF, And.right, Dist.dist, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.B, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω
43 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_CAO_add_oangle_ABC line 2009 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, HAdd.hAdd, HSMul.hSMul, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center and 3 more
Proof uses: EuclideanGeometry.Sphere.two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.B_ne_C, Ne.symm, Imo2023P2.Imo2023q2Cfg.C_ne_A
38 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.acute_angle_CAO line 2018 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, HDiv.hDiv and 1 more
Proof uses: Eq.mpr, abs, Real.Angle.toReal, EuclideanGeometry.oangle, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.O_ne_A, EuclideanGeometry.Sphere.abs_oangle_center_right_toReal_lt_pi_div_two, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω
44 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_CAO_add_angle_ABC line 2024 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, HAdd.hAdd, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.B and 2 more
Proof uses: Real.Angle, HSMul.hSMul, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.oangle_CAO_add_oangle_ABC, LT.lt, Imo2023P2.Imo2023q2Cfg.acute_angle_CAO, Imo2023P2.Imo2023q2Cfg.angle_CBA_acute, Eq.mpr, abs, Real.Angle.toReal, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.O_ne_A, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.aux₁₀, Eq.mp and 1 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_BAO_add_oangle_ACB line 2033 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, HAdd.hAdd, HSMul.hSMul, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center and 3 more
Proof uses: EuclideanGeometry.Sphere.two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.C_ne_A, Ne.symm, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.Imo2023q2Cfg.B_ne_A
38 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.acute_angle_BAO line 2042 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, LT.lt, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, HDiv.hDiv and 1 more
Proof uses: Eq.mpr, abs, Real.Angle.toReal, EuclideanGeometry.oangle, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.O_ne_A, EuclideanGeometry.Sphere.abs_oangle_center_right_toReal_lt_pi_div_two, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω
44 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAO_add_angle_ACB line 2048 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, HAdd.hAdd, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.C and 2 more
Proof uses: Real.Angle, HSMul.hSMul, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.oangle_BAO_add_oangle_ACB, LT.lt, Imo2023P2.Imo2023q2Cfg.acute_angle_BAO, Imo2023P2.Imo2023q2Cfg.angle_ACB_acute, Eq.mpr, abs, Real.Angle.toReal, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.O_ne_A, Ne.symm, Imo2023P2.Imo2023q2Cfg.C_ne_A, Imo2023P2.Imo2023q2Cfg.B_ne_C, Imo2023P2.aux₁₀, Eq.mp
58 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_BAO_add_angle_OAC_eq_angle_BAC line 2056 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, HAdd.hAdd, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.C
Proof uses: HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_CAO_add_angle_ABC, Imo2023P2.Imo2023q2Cfg.angle_BAO_add_angle_ACB, Eq.mpr, HSub.hSub, id, congrArg, Eq.mp, EuclideanGeometry.angle_comm, Eq.symm, propext, eq_sub_iff_add_eq, Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt, Not.intro, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, HMul.hMul, inferInstance, AddMonoidWithOne and 77 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_BAO_ne_zero line 2065 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, EuclideanGeometry.Sphere.center and 1 more
Proof uses: Eq.mpr, Imo2023P2.Imo2023q2Cfg.F, id, congrArg, Real.Angle, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, Or, Imo2023P2.eq_of_mem_sphere_of_collinear, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.F_in_Ω, Eq.mp, Collinear, Insert.insert, Set, Singleton.singleton, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Mathlib.Tactic.Contrapose.contrapose₁ and 12 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_OAC_ne_zero line 2076 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, SignType, Real.Angle.sign, EuclideanGeometry.oangle, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.A and 1 more
Proof uses: Eq.mpr, Imo2023P2.Imo2023q2Cfg.F, id, congrArg, Real.Angle, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, Or, Imo2023P2.eq_of_mem_sphere_of_collinear, Imo2023P2.Imo2023q2Cfg.F_in_Ω, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω, Eq.mp, Collinear, Insert.insert, Set, Singleton.singleton, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Mathlib.Tactic.Contrapose.contrapose₁ and 12 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sign_oangle_BAF_eq_sign_oangle_FAC line 2087 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Eq.mpr, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, Real.Angle, Eq.symm, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, Sbtw.oangle_eq_left, HAdd.hAdd, EuclideanGeometry.oangle_add, Imo2023P2.Imo2023q2Cfg.B_ne_A, Imo2023P2.Imo2023q2Cfg.O_ne_A, Imo2023P2.Imo2023q2Cfg.C_ne_A, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.angle_BAO_add_angle_OAC_eq_angle_BAC, Ne, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.Imo2023q2Cfg.E and 10 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_BACF line 2100 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.SphereOrder.rotate, Imo2023P2.SphereOrder.mk, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω, And and 20 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_ASCF line 2117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.SphereOrder.trans'', Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BACF, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASC
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_S line 2120 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Ne.symm, Imo2023P2.SphereOrder.P₂_ne_P₄, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASCF
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_ASFM line 2123 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.sphereOrder_of_sbtw_diag_inter, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, And, Membership.mem and 16 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_AFME line 2135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.rotate', Imo2023P2.SphereOrder.trans, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASME, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASFM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_E line 2138 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.P₂_ne_P₄, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_AFME
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_BAE_eq_oangle_FAC line 2141 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Eq.mpr, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, id, congrArg, Eq.symm, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, EuclideanGeometry.oangle_eq_of_angle_eq_of_sign_eq, EuclideanGeometry.angle, HSub.hSub, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_BAE_eq, HAdd.hAdd, propext, sub_eq_iff_eq_add, EuclideanGeometry.angle_comm, Imo2023P2.Imo2023q2Cfg.angle_CAO_add_angle_ABC, Eq.refl and 7 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_SEF_eq_oangle_EFS line 2153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Eq.mpr, Imo2023P2.Imo2023q2Cfg.B, id, congrArg, EuclideanGeometry.Sphere.two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.Ω, And.right, Dist.dist, Imo2023P2.Imo2023q2Cfg.C, Membership.mem, Set, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.h_S_Ω, Imo2023P2.Imo2023q2Cfg.E_in_Ω, Imo2023P2.Imo2023q2Cfg.B_in_Ω, Imo2023P2.Imo2023q2Cfg.F_in_Ω, Imo2023P2.Imo2023q2Cfg.E_ne_S, Ne.symm and 23 more
64 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.K'_set line 2170 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set
Proof uses: Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Insert.insert, Imo2023P2.Imo2023q2Cfg.S, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.F
33 instances, coercions and auxiliary constants left out.
def Imo2023P2.Imo2023q2Cfg.K_set line 2171 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set
Proof uses: Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.F
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.K'_ne_D line 2173 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Ne, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Collinear, Insert.insert, Imo2023P2.Imo2023q2Cfg.S, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.F, collinear_insert_of_mem_affineSpan_pair, And.right, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Eq.mp, Inter.inter, And, congrArg, propext, Set.mem_inter_iff, Imo2023P2.Imo2023q2Cfg.K'_set.eq_1, Imo2023P2.Imo2023q2Cfg.B, Wbtw.collinear, Sbtw.wbtw and 35 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.two_zsmul_oangle_ADK' line 2195 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle and 3 more
Proof uses: AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, And.left, affineSpan, Insert.insert, Imo2023P2.Imo2023q2Cfg.S, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.E, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_ADE, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self and 38 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.K'_ne_A line 2220 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Ne, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.D, Real.pi, Imo2023P2.Imo2023q2Cfg.two_zsmul_oangle_ADK', Mathlib.Tactic.Contrapose.contrapose₃, id, Not, Eq.mpr, congrArg, EuclideanGeometry.oangle_self_left_right, smul_zero, Ne.symm, Real.Angle.pi_ne_zero
51 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.K'_ne_S line 2227 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Ne, Imo2023P2.Imo2023q2Cfg.S
Proof uses: AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, And.left, affineSpan, Insert.insert, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.F, Collinear, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.D, collinear_insert_of_mem_affineSpan_pair, Eq.mp, congrArg, Imo2023P2.Imo2023q2Cfg.prll_D_BC_eq_LD, Collinear.mem_affineSpan_of_mem_of_ne, of_eq_true, Eq.trans, Or, True, congr and 28 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.cospherical_DASK' line 2249 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, EuclideanGeometry.Cospherical, Insert.insert, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.D and 2 more
Proof uses: EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear, Eq.mpr, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Real.pi, id, congrArg, Imo2023P2.Imo2023q2Cfg.two_zsmul_oangle_ADK', Eq.mp, Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.prll_D_BC, affineSpan, Imo2023P2.Imo2023q2Cfg.F, And, propext, Set.mem_inter_iff, Imo2023P2.Imo2023q2Cfg.K'_set.eq_1 and 34 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.K'_in_K_set line 2269 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Imo2023P2.Imo2023q2Cfg.K_set
Proof uses: Eq.mpr, Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.F, id, congrArg, Imo2023P2.Imo2023q2Cfg.K_set.eq_1, Inter.inter, AffineSubspace, affineSpan, Insert.insert, Singleton.singleton, Imo2023P2.diag_inter_set.eq_1, And, propext, Set.mem_inter_iff, And.intro, EuclideanGeometry.Cospherical and 62 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AKC line 2304 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K_set, Sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASCF
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.KO_lt_Ω_radius line 2308 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K_set, LT.lt, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω and 1 more
Proof uses: Imo2023P2.Sphere.dist_le_radius_of_sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.C, Imo2023P2.Imo2023q2Cfg.sbtw_AKC, Imo2023P2.Imo2023q2Cfg.A_in_Ω, Imo2023P2.Imo2023q2Cfg.C_in_Ω
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_F line 2312 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F
Proof uses: LT.lt, EuclideanGeometry.Sphere.radius, Imo2023P2.Imo2023q2Cfg.Ω, Dist.dist, Imo2023P2.Imo2023q2Cfg.L, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.LO_gt_Ω_radius, Mathlib.Tactic.Contrapose.contrapose₃, Eq.mpr, Not, LE.le, id, implies_congr, Eq.refl, le_of_lt, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.K'_set, Inter.inter, AffineSubspace and 29 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_PDF line 2328 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Collinear, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.D, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Eq.mpr, id, congrArg, Set.insert_comm, Imo2023P2.Imo2023q2Cfg.L, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.collinear_LPS, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.collinear_BLE, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Or, Real.Angle, Real.pi, Real.Angle.sign_eq_zero_iff and 50 more
72 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_PDF line 2350 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Sbtw, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.F
Proof uses: EuclideanGeometry.sbtw_of_collinear_of_dist_center_lt_radius, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.collinear_PDF, Imo2023P2.Imo2023q2Cfg.P_in_Ω, Imo2023P2.Imo2023q2Cfg.DO_lt_Ω_radius, Imo2023P2.Imo2023q2Cfg.F_in_Ω, Imo2023P2.Imo2023q2Cfg.P_ne_F
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.F_ne_D line 2354 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Ne.symm, Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.sbtw_PDF
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFE line 2357 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.E
Proof uses: Imo2023P2.SphereOrder.rotate', Imo2023P2.sphereOrder_of_sbtw_diag_inter, Imo2023P2.Imo2023q2Cfg.D, Eq.mpr, EuclideanGeometry.Cospherical, Insert.insert, Set, Singleton.singleton, Exists, EuclideanGeometry.Sphere, LE.le, Metric.sphere, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.radius, id, congrArg, propext, EuclideanGeometry.cospherical_iff_exists_sphere, Exists.intro, Imo2023P2.Imo2023q2Cfg.Ω and 15 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_A line 2370 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Imo2023P2.SphereOrder.P₁_ne_P₂, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFE
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM line 2373 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.trans', Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.SphereOrder.rotate', Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFE, Imo2023P2.Imo2023q2Cfg.sphereOrder_AFME
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.P_ne_M line 2376 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.P₁_ne_P₄, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sphereOrder_PSFM line 2379 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Imo2023P2.SphereOrder, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.rotate', Imo2023P2.SphereOrder.trans', Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASFM, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.Z_set line 2382 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set
Proof uses: Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.exists_Z line 2384 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Exists, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set
Proof uses: Imo2023P2.SphereOrder.exists_diag_inter, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_PSFM
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_PZF line 2387 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Sbtw, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_PSFM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Z_ne_F line 2391 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Ne, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Sbtw.ne_right, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.sbtw_PZF
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_SZM line 2395 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Sbtw, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw', Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.sphereOrder_PSFM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Z_ne_S line 2399 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Ne, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sbtw_SZM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Z_ne_D line 2403 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Ne, Imo2023P2.Imo2023q2Cfg.D
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_BASM, Ne.symm, EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero, Eq.mp, congrArg, Real.Angle, Eq.symm, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_SZM, Sbtw.oangle_eq_left, Sbtw.symm and 2 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Z_ne_O line 2411 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Ne, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM, Ne.symm, EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero, Eq.mp, congrArg, Real.Angle, Eq.symm, Sbtw.oangle_eq_right, Sbtw.symm, Imo2023P2.Imo2023q2Cfg.sbtw_PZF, Sbtw.oangle_eq_left and 2 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_ne_D line 2419 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Ne, Imo2023P2.Imo2023q2Cfg.D
Proof uses: SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.SphereOrder.sign_oangle₂₃₄_ne_zero, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.sphereOrder_BAME, Ne.symm, EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero, Eq.mp, congrArg, Real.Angle, Eq.symm, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AXM, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_ADE, EuclideanGeometry.oangle_rotate_sign
30 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_SXD line 2428 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Collinear, Insert.insert, Imo2023P2.Imo2023q2Cfg.S, Singleton.singleton and 1 more
Proof uses: AffineSubspace, affineSpan, Imo2023P2.Imo2023q2Cfg.B, mem_affineSpan, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self, true_or, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_BXS, or_true, Imo2023P2.Imo2023q2Cfg.S_ne_B, Imo2023P2.Imo2023q2Cfg.sbtw_BDS, collinear_triple_of_mem_affineSpan_pair
45 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_SXD line 2437 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Sbtw, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.D
Proof uses: Or, Wbtw, Imo2023P2.Collinear.sbtw_or_wbtw_or_wbtw, Imo2023P2.Imo2023q2Cfg.collinear_SXD, False.elim, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, Wbtw.oangle_sign_eq_of_ne_left, Imo2023P2.Imo2023q2Cfg.X_ne_D, Mathlib.Tactic.Contrapose.contrapose₁, Imo2023P2.Imo2023q2Cfg.M, False, Eq.mpr, Not, True, Ne, id, implies_congr and 32 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.S_ne_O line 2464 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Ne, Imo2023P2.Imo2023q2Cfg.S, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: Ne.symm, Sbtw.ne_left, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sbtw_SOM
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_SOZ line 2467 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Collinear, Insert.insert, Imo2023P2.Imo2023q2Cfg.S, EuclideanGeometry.Sphere.center and 2 more
Proof uses: AffineSubspace, affineSpan, Imo2023P2.Imo2023q2Cfg.M, mem_affineSpan, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self, true_or, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_SOM, or_true, Imo2023P2.Imo2023q2Cfg.S_ne_M, Imo2023P2.Imo2023q2Cfg.sbtw_SZM, collinear_triple_of_mem_affineSpan_pair
45 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_SOZ line 2476 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Sbtw, Imo2023P2.Imo2023q2Cfg.S, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω
Proof uses: Or, Wbtw, Imo2023P2.Collinear.sbtw_or_wbtw_or_wbtw, Imo2023P2.Imo2023q2Cfg.collinear_SOZ, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.F, Wbtw.oangle_sign_eq_of_ne_left, Ne.symm, Imo2023P2.Imo2023q2Cfg.Z_ne_O, Mathlib.Tactic.Contrapose.contrapose₁, Imo2023P2.Imo2023q2Cfg.A, id, Not, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASFM, Eq.mp, Neg.neg and 22 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_MAE_eq_oangle_AMS line 2504 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.E, Imo2023P2.Imo2023q2Cfg.S
Proof uses: Imo2023P2.aux₆, Eq.mpr, SignType, Real.Angle.sign, id, congrArg, Eq.symm, EuclideanGeometry.oangle_rotate_sign, Neg.neg, EuclideanGeometry.oangle_swap₁₃_sign, Imo2023P2.SphereOrder.sign_oangle₁₂₃_eq_sign_oangle₃₄₁, Imo2023P2.Imo2023q2Cfg.sphereOrder_ASME, EuclideanGeometry.oangle_swap₂₃_sign, Ne, propext, Iff.ne, SignType.neg_eq_zero_iff, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero, HSMul.hSMul, EuclideanGeometry.Sphere.center and 29 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.DX_to_XS_eq_DA_to_MS line 2527 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, HDiv.hDiv, Dist.dist, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.S and 2 more
Proof uses: Ne, Eq.mpr, id, congrArg, propext, dist_ne_zero, Imo2023P2.Imo2023q2Cfg.X_ne_S, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_M, HMul.hMul, inferInstance, CommGroupWithZero, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval, Mathlib.Tactic.FieldSimp.NF.cons, Prod.mk, Int, Neg.neg, List.nil, Prod and 87 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 35 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.AE_paralell_SM line 2565 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, AffineSubspace.Parallel, affineSpan, Insert.insert, Set, Imo2023P2.Imo2023q2Cfg.A, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.E and 2 more
Proof uses: Eq.mpr, id, congrArg, Set.pair_comm, Imo2023P2.parallel_of_two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.h_E_ne_A, Ne.symm, Imo2023P2.Imo2023q2Cfg.M_ne_A, Imo2023P2.Imo2023q2Cfg.S_ne_M, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Neg.neg, EuclideanGeometry.oangle_rev, Imo2023P2.Imo2023q2Cfg.oangle_MAE_eq_oangle_AMS, Eq.symm, Eq.refl
46 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_OZF_eq_oangle_ADF line 2570 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, Real.Angle, EuclideanGeometry.oangle, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω and 3 more
Proof uses: EuclideanGeometry.oangle_eq_of_parallel, AffineSubspace, affineSpan, Insert.insert, Singleton.singleton, Mathlib.Tactic.Contrapose.contrapose₁, SignType, Real.Angle.sign, Imo2023P2.Imo2023q2Cfg.P, False, Eq.mpr, Not, True, Ne, id, implies_congr, not_false_eq_true, Eq.refl, Imo2023P2.SphereOrder.sign_oangle₁₂₃_ne_zero, Imo2023P2.Imo2023q2Cfg.M and 46 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 45 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.ZO_to_OS_eq_DA_to_MS line 2601 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Z_set, HDiv.hDiv, Dist.dist, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω and 4 more
Proof uses: Ne, Eq.mpr, id, congrArg, propext, dist_ne_zero, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_O, Imo2023P2.Imo2023q2Cfg.S_ne_M, HMul.hMul, inferInstance, CommGroupWithZero, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval, Mathlib.Tactic.FieldSimp.NF.cons, Prod.mk, Int, Neg.neg, List.nil, Prod and 109 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 34 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.DX_to_XS_eq_ZO_to_OS line 2638 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Z_set, HDiv.hDiv, Dist.dist, Imo2023P2.Imo2023q2Cfg.D and 3 more
Proof uses: Eq.mpr, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, id, congrArg, Imo2023P2.Imo2023q2Cfg.DX_to_XS_eq_DA_to_MS, Imo2023P2.Imo2023q2Cfg.ZO_to_OS_eq_DA_to_MS, Eq.refl
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.DS_to_XS_eq_ZS_to_OS line 2643 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Z_set, HDiv.hDiv, Dist.dist, Imo2023P2.Imo2023q2Cfg.D and 3 more
Proof uses: Eq.mpr, HAdd.hAdd, id, congrArg, Eq.symm, Wbtw.dist_add_dist, Sbtw.wbtw, Sbtw.symm, Imo2023P2.Imo2023q2Cfg.sbtw_SXD, Imo2023P2.Imo2023q2Cfg.sbtw_SOZ, Ne, propext, dist_ne_zero, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_O, Imo2023P2.Imo2023q2Cfg.X_ne_S, add_div, div_self, Imo2023P2.Imo2023q2Cfg.DX_to_XS_eq_ZO_to_OS, Eq.refl
56 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.DZ_to_XO_eq_ZS_to_OS line 2657 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Z_set, HDiv.hDiv, Dist.dist, Imo2023P2.Imo2023q2Cfg.D and 3 more
Proof uses: Eq.mpr, HMul.hMul, id, congrArg, Eq.symm, propext, mul_self_inj_of_nonneg, div_nonneg, dist_nonneg, Ne, dist_ne_zero, Ne.symm, Imo2023P2.Imo2023q2Cfg.S_ne_O, Imo2023P2.Imo2023q2Cfg.X_ne_O, Imo2023P2.Imo2023q2Cfg.X_ne_S, Imo2023P2.Imo2023q2Cfg.DS_to_XS_eq_ZS_to_OS, inferInstance, CommGroupWithZero, HPow.hPow, GroupWithZero and 103 more
122 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_OXS_eq_angle_ZDS line 2688 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Z_set, EuclideanGeometry.angle, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω and 2 more
Proof uses: Imo2023P2.angle_eq_of_cos_angle_eq, HMul.hMul, Dist.dist, HSub.hSub, HAdd.hAdd, Real.cos, EuclideanGeometry.law_cos, Ne, Eq.mpr, id, congrArg, propext, dist_ne_zero, Ne.symm, Imo2023P2.Imo2023q2Cfg.X_ne_O, Imo2023P2.Imo2023q2Cfg.X_ne_S, Imo2023P2.Imo2023q2Cfg.D_ne_S, Imo2023P2.Imo2023q2Cfg.Z_ne_D, HDiv.hDiv, Eq.mp and 119 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 42 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_OXS_eq_oangle_ZDS line 2732 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Z_set, Real.Angle, EuclideanGeometry.oangle, EuclideanGeometry.Sphere.center and 3 more
Proof uses: EuclideanGeometry.oangle_eq_of_angle_eq_of_sign_eq, Imo2023P2.Imo2023q2Cfg.angle_OXS_eq_angle_ZDS, Eq.mpr, SignType, Real.Angle.sign, id, congrArg, Eq.symm, EuclideanGeometry.oangle_rotate_sign, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_SXD, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_SOZ, Eq.refl
28 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.XO_paralell_FP line 2741 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, AffineSubspace.Parallel, affineSpan, Insert.insert, Singleton.singleton and 4 more
Proof uses: Imo2023P2.Imo2023q2Cfg.Z_set, Exists, Imo2023P2.Imo2023q2Cfg.exists_Z, AffineSubspace, Imo2023P2.Imo2023q2Cfg.D, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_PDF, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self, true_or, or_true, Imo2023P2.Imo2023q2Cfg.P_ne_F, Imo2023P2.Imo2023q2Cfg.sbtw_PZF and 33 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
def Imo2023P2.Imo2023q2Cfg.Y_set line 2764 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Set
Proof uses: Imo2023P2.diag_inter_set, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M
26 instances, coercions and auxiliary constants left out.
lemma Imo2023P2.Imo2023q2Cfg.exists_Y line 2766 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Exists, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set
Proof uses: Imo2023P2.SphereOrder.exists_diag_inter, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Y_ne_P line 2769 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, Ne, Imo2023P2.Imo2023q2Cfg.P
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Y_ne_A line 2772 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, Ne, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.SphereOrder.rotate, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM, Eq.mp, Imo2023P2.diag_inter_set, congrArg, Imo2023P2.diag_inter_set_rotate, Imo2023P2.Imo2023q2Cfg.Y_set.eq_1
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.Y_ne_F line 2776 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, Ne, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.SphereOrder.diag_inter_ne_P₁, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.SphereOrder.symm, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM, Eq.mp, Imo2023P2.diag_inter_set, congrArg, Imo2023P2.diag_inter_set_rotate, Imo2023P2.Imo2023q2Cfg.Y_set.eq_1
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_PYF line 2780 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, Sbtw, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AYM line 2784 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, Sbtw, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.M
Proof uses: Imo2023P2.SphereOrder.diag_inter_sbtw', Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F, Imo2023P2.Imo2023q2Cfg.sphereOrder_PAFM
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.angle_APY_eq line 2788 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.Y_set, EuclideanGeometry.angle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.P, HDiv.hDiv and 1 more
Proof uses: Eq.mpr, abs, Real.Angle.toReal, EuclideanGeometry.oangle, id, congrArg, EuclideanGeometry.angle_eq_abs_oangle_toReal, Ne.symm, Imo2023P2.Imo2023q2Cfg.P_ne_A, Imo2023P2.Imo2023q2Cfg.Y_ne_P, Or, Real.Angle, Neg.neg, propext, Real.Angle.abs_toReal_eq_pi_div_two_iff, HSMul.hSMul, Int, Eq.symm, Real.Angle.two_zsmul_eq_pi_iff, Imo2023P2.Imo2023q2Cfg.F and 8 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.collinear_AXY line 2798 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, Collinear, Insert.insert, Imo2023P2.Imo2023q2Cfg.A and 1 more
Proof uses: Imo2023P2.Imo2023q2Cfg.M, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AXM, Imo2023P2.Imo2023q2Cfg.sbtw_AYM, AffineSubspace, affineSpan, Collinear.mem_affineSpan_of_mem_of_ne, of_eq_true, Eq.trans, Or, True, congr, congrArg, eq_self, true_or, or_true, Ne.symm, Imo2023P2.Imo2023q2Cfg.M_ne_A, mem_affineSpan and 1 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.not_collinear_AXO line 2812 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Not, Collinear, Insert.insert, Imo2023P2.Imo2023q2Cfg.A and 3 more
Proof uses: Eq.mpr, id, congrArg, Set.insert_comm, SignType, Real.Angle.sign, EuclideanGeometry.oangle, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Imo2023P2.Imo2023q2Cfg.M, Real.Angle, Sbtw.oangle_eq_left, Imo2023P2.Imo2023q2Cfg.sbtw_AXM, Imo2023P2.Imo2023q2Cfg.F, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, EuclideanGeometry.oangle_rotate_sign, Imo2023P2.SphereOrder.sign_oangle₃₄₁_ne_zero, Imo2023P2.Imo2023q2Cfg.S and 1 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_AXO_eq_oangle_AYF line 2821 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A and 3 more
Proof uses: EuclideanGeometry.oangle_eq_of_parallel, Not, Collinear, Insert.insert, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.not_collinear_AXO, Mathlib.Tactic.Contrapose.contrapose₄, AffineSubspace, affineSpan, Eq.mpr, id, congrArg, Set.insert_comm, collinear_insert_of_mem_affineSpan_pair, left_mem_affineSpan_pair, Collinear.mem_affineSpan_of_mem_of_ne, Wbtw.collinear, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AOF, of_eq_true and 24 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_XOA_eq_oangle_YFA line 2844 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, Real.Angle, EuclideanGeometry.oangle, EuclideanGeometry.Sphere.center and 3 more
Proof uses: EuclideanGeometry.oangle_eq_of_parallel, Not, Collinear, Insert.insert, Singleton.singleton, Imo2023P2.Imo2023q2Cfg.not_collinear_AXO, Mathlib.Tactic.Contrapose.contrapose₄, AffineSubspace, affineSpan, Eq.mpr, id, congrArg, Set.pair_comm, Set.insert_comm, collinear_insert_of_mem_affineSpan_pair, Collinear.mem_affineSpan_of_mem_of_ne, Imo2023P2.Imo2023q2Cfg.collinear_AXY, of_eq_true, Eq.trans, Or and 22 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.AX_eq_AY_div_two line 2863 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, Dist.dist, Imo2023P2.Imo2023q2Cfg.A, HDiv.hDiv
Proof uses: Not, Collinear, Insert.insert, Singleton.singleton, EuclideanGeometry.Sphere.center, Imo2023P2.Imo2023q2Cfg.Ω, Imo2023P2.Imo2023q2Cfg.not_collinear_AXO, Imo2023P2.Imo2023q2Cfg.F, Eq.mpr, SignType, Real.Angle.sign, EuclideanGeometry.oangle, id, congrArg, Eq.symm, propext, EuclideanGeometry.oangle_sign_eq_zero_iff_collinear, Real.Angle, Imo2023P2.Imo2023q2Cfg.oangle_AXO_eq_oangle_AYF, HMul.hMul and 31 more
82 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.sbtw_AXY line 2888 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, Sbtw, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Or, Wbtw, Imo2023P2.Collinear.sbtw_or_wbtw_or_wbtw, Imo2023P2.Imo2023q2Cfg.collinear_AXY, False.elim, HAdd.hAdd, Dist.dist, Wbtw.dist_add_dist, Mathlib.Tactic.Contrapose.contrapose₁, And, HMul.hMul, False, Not, Eq.mpr, id, congrArg, propext, not_and_or, Or.inr, zero_eq_dist and 125 more
142 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_eq_midpoint_AY line 2916 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.Y_set, midpoint, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Dist.dist, HDiv.hDiv, Imo2023P2.Imo2023q2Cfg.AX_eq_AY_div_two, eq_midpoint_of_dist_eq_half, HAdd.hAdd, Wbtw.dist_add_dist, Sbtw.wbtw, Imo2023P2.Imo2023q2Cfg.sbtw_AXY, Eq.mpr, HSub.hSub, id, congrArg, Eq.mp, Eq.symm, propext, eq_sub_iff_add_eq', Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow, Nat.rawCast and 47 more
106 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.XP_eq_XA line 2927 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Dist.dist, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A
Proof uses: Imo2023P2.Imo2023q2Cfg.Y_set, Exists, Imo2023P2.Imo2023q2Cfg.exists_Y, EuclideanGeometry.angle, HDiv.hDiv, Real.pi, Imo2023P2.Imo2023q2Cfg.angle_APY_eq, EuclideanGeometry.Sphere.center, EuclideanGeometry.Sphere.ofDiameter, Imo2023P2.Imo2023q2Cfg.X_eq_midpoint_AY, Eq.mpr, id, congrArg, EuclideanGeometry.Sphere.radius, Eq.mp, EuclideanGeometry.Sphere, propext, EuclideanGeometry.mem_sphere', EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_ofDiameter, Eq.symm and 2 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_XPA_eq_oangle_PAX line 2939 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Real.Angle, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.A
Proof uses: EuclideanGeometry.oangle_eq_oangle_of_dist_eq, Imo2023P2.Imo2023q2Cfg.XP_eq_XA
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.two_zsmul_angle_SPM_eq line 2943 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.M and 1 more
Proof uses: Eq.mpr, Or, HDiv.hDiv, Neg.neg, id, congrArg, propext, Real.Angle.two_zsmul_eq_pi_iff, abs, Real.Angle.toReal, Eq.symm, Real.Angle.abs_toReal_eq_pi_div_two_iff, EuclideanGeometry.angle, EuclideanGeometry.angle_eq_abs_oangle_toReal, Ne.symm, Imo2023P2.Imo2023q2Cfg.P_ne_S, Imo2023P2.Imo2023q2Cfg.P_ne_M, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω and 3 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.two_zsmul_angle_APF_eq line 2950 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle, Imo2023P2.Imo2023q2Cfg.A, Imo2023P2.Imo2023q2Cfg.P, Imo2023P2.Imo2023q2Cfg.F and 1 more
Proof uses: Eq.mpr, Or, HDiv.hDiv, Neg.neg, id, congrArg, propext, Real.Angle.two_zsmul_eq_pi_iff, abs, Real.Angle.toReal, Eq.symm, Real.Angle.abs_toReal_eq_pi_div_two_iff, EuclideanGeometry.angle, EuclideanGeometry.angle_eq_abs_oangle_toReal, Ne.symm, Imo2023P2.Imo2023q2Cfg.P_ne_A, Imo2023P2.Imo2023q2Cfg.P_ne_F, Membership.mem, EuclideanGeometry.Sphere, Imo2023P2.Imo2023q2Cfg.Ω and 3 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.oangle_XPD_eq_oangle_PBD line 2957 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, Real.Angle, HSMul.hSMul, Int, EuclideanGeometry.oangle and 3 more
Proof uses: Eq.mpr, HAdd.hAdd, Imo2023P2.Imo2023q2Cfg.A, id, congrArg, Eq.symm, EuclideanGeometry.oangle_add, Ne.symm, Imo2023P2.Imo2023q2Cfg.P_ne_X, Imo2023P2.Imo2023q2Cfg.P_ne_A, Imo2023P2.Imo2023q2Cfg.P_ne_D, smul_add, Imo2023P2.Imo2023q2Cfg.oangle_XPA_eq_oangle_PAX, Imo2023P2.Imo2023q2Cfg.M, Sbtw.oangle_eq_right, Imo2023P2.Imo2023q2Cfg.sbtw_AXM, Imo2023P2.Imo2023q2Cfg.S, Imo2023P2.Imo2023q2Cfg.sbtw_BDS, HSub.hSub, propext and 33 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.PX_isTangentAt_ω line 2976 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, EuclideanGeometry.Sphere.IsTangentAt, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.P, affineSpan and 2 more
Proof uses: Imo2023P2.Sphere.IsTangentAt_of_two_zsmul_oangle_eq, Imo2023P2.Imo2023q2Cfg.D, Imo2023P2.Imo2023q2Cfg.B, Imo2023P2.Imo2023q2Cfg.P_in_ω, Imo2023P2.Imo2023q2Cfg.D_in_ω, Imo2023P2.Imo2023q2Cfg.B_in_ω, Imo2023P2.Imo2023q2Cfg.P_ne_D, Imo2023P2.Imo2023q2Cfg.h_P_ne_B, Imo2023P2.Imo2023q2Cfg.D_ne_B, Imo2023P2.Imo2023q2Cfg.P_ne_X, Imo2023P2.Imo2023q2Cfg.oangle_XPD_eq_oangle_PBD
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2023P2.Imo2023q2Cfg.X_in_tang_P_ω line 2981 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Membership.mem, Set, Imo2023P2.Imo2023q2Cfg.X_set, AffineSubspace, Imo2023P2.Imo2023q2Cfg.tang_P_ω
Proof uses: HAdd.hAdd, Subtype, Submodule, AffineSubspace.direction, Eq.mpr, id, congrArg, Imo2023P2.Imo2023q2Cfg.h_rank_tang_P_ω, Fact.out, Eq.refl, EuclideanGeometry.Sphere.orthRadius, Imo2023P2.Imo2023q2Cfg.ω, Imo2023P2.Imo2023q2Cfg.P, EuclideanGeometry.Sphere.IsTangentAt.eq_orthRadius_of_finrank_add_one_eq, Imo2023P2.Imo2023q2Cfg.h_tang_P_ω, Imo2023P2.Imo2023q2Cfg.ω_radius_ne_zero, affineSpan, Insert.insert, Singleton.singleton, Nat.succ and 8 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem Imo2023P2.Imo2023q2Cfg.result line 2994 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, Imo2023P2.Imo2023q2Cfg, Exists, And, Membership.mem, Set, Inter.inter, AffineSubspace, Imo2023P2.Imo2023q2Cfg.tang_P_ω and 11 more
Proof uses: Imo2023P2.Imo2023q2Cfg.X_set, Imo2023P2.Imo2023q2Cfg.exists_X, Exists.intro, And.intro, Imo2023P2.Imo2023q2Cfg.X_in_tang_P_ω, Set.mem_of_mem_inter_left, Imo2023P2.Imo2023q2Cfg.M, Imo2023P2.Imo2023q2Cfg.angle_BAX_eq_angle_XAC, Eq.mpr, id, congrArg, Eq.symm, Imo2023P2.Imo2023q2Cfg.angle_BAM_eq_angle_BAX, Imo2023P2.Imo2023q2Cfg.angle_BAM_acute
59 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
problem Imo2023P2.imo2023_p1 line 3013 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, Module.Oriented, Fin, EuclideanGeometry.Sphere, AffineSubspace, AffineIndependent, Nat.succ, Matrix.vecCons, Matrix.vecEmpty, Affine.Simplex.AcuteAngled, Affine.Simplex.mk and 27 more
Proof uses: Imo2023P2.Imo2023q2Cfg, Imo2023P2.Imo2023q2Cfg.mk, Imo2023P2.Imo2023q2Cfg.result
65 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.