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https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cut_finite | [175, 1] | [182, 7] | apply Set.Finite.image | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (orthoHyperplane '' (Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)))) | case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cut_finite | [175, 1] | [182, 7] | apply Set.Finite.preimage (Set.injOn_of_injective Subtype.val_injective _) | case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))) | case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cut_finite | [175, 1] | [182, 7] | apply Set.finite_range | case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cut_finite | [175, 1] | [182, 7] | rcases ht with ⟨ x, _, rfl ⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
t : Set (Halfspace E)
ht : t ∈ orthoHyperplane '' (Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)))
⊢ Set.Finite t | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : { x // x ≠ 0 }
left✝ : x ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))
⊢ Set.Finite (orthoHyperplane x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cut_finite | [175, 1] | [182, 7] | exact orthoHyperplane.Finite _ | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : { x // x ≠ 0 }
left✝ : x ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))
⊢ Set.Finite (orthoHyperplane x) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | use Submodule_cut p | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ ∃ H_, Set.Finite H_ ∧ ↑p = cutSpace H_ | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Submodule_cut p) ∧ ↑p = cutSpace (Submodule_cut p) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | use Submodule_cut_finite p | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Submodule_cut p) ∧ ↑p = cutSpace (Submodule_cut p) | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ ↑p = cutSpace (Submodule_cut p) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | ext x | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ ↑p = cutSpace (Submodule_cut p) | case right.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ ↑p ↔ x ∈ cutSpace (Submodule_cut p) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | constructor | case right.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ ↑p ↔ x ∈ cutSpace (Submodule_cut p) | case right.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ ↑p → x ∈ cutSpace (Submodule_cut p)
case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ cutSpace (Submodule_cut p) → x ∈ ↑p |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rintro hx Hi_ ⟨ H, ⟨ _, ⟨ v, ⟨ i, hi ⟩, rfl ⟩ , hHHalfpair ⟩, rfl ⟩ | case right.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ ↑p → x ∈ cutSpace (Submodule_cut p) | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
H : Halfspace E
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
hHHalfpair : H ∈ orthoHyperplane v
⊢ x ∈ ↑H |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rw [Halfspace_mem] | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
H : Halfspace E
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
hHHalfpair : H ∈ orthoHyperplane v
⊢ x ∈ ↑H | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
H : Halfspace E
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
hHHalfpair : H ∈ orthoHyperplane v
⊢ ↑H.f x ≤ H.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | revert hHHalfpair H | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
H : Halfspace E
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
hHHalfpair : H ∈ orthoHyperplane v
⊢ ↑H.f x ≤ H.α | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
⊢ ∀ H ∈ orthoHyperplane v, ↑H.f x ≤ H.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rw [← mem_cutSpace, orthoHyperplane_mem, ← hi, Submodule.inner_left_of_mem_orthogonal hx] | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
⊢ ∀ H ∈ orthoHyperplane v, ↑H.f x ≤ H.α | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
⊢ (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i ∈ pᗮ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | exact Submodule.coe_mem ((FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ }) i) | case right.h.mp.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hx : x ∈ ↑p
v : { x // x ≠ 0 }
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
hi : (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i = ↑v
⊢ (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)) i ∈ pᗮ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rintro hHi_ | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
⊢ x ∈ cutSpace (Submodule_cut p) → x ∈ ↑p | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : x ∈ cutSpace (Submodule_cut p)
⊢ x ∈ ↑p |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rw [Submodule_cut, orthoHyperplanes_mem] at hHi_ | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : x ∈ cutSpace (Submodule_cut p)
⊢ x ∈ ↑p | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
⊢ x ∈ ↑p |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rw [SetLike.mem_coe, ← Submodule.orthogonal_orthogonal p] | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
⊢ x ∈ ↑p | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
⊢ x ∈ pᗮᗮ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | intro y hy | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
⊢ x ∈ pᗮᗮ | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
⊢ ⟪y, x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | have : ∀ i, inner (Subtype.val (FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ } i)) x = (0:ℝ) := by
intro i
let v : E := (FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ }) i
let v' : { x // x ≠ 0 } := ⟨ v, fun hv => (Basis.ne_zero (FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ }) i) (Submodule.coe_eq_zero.mp hv) ⟩
exact hHi_ v' ⟨ i, rfl ⟩ | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
⊢ ⟪y, x⟫_ℝ = 0 | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
this : ∀ (i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)), ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0
⊢ ⟪y, x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | rw [← Submodule.mem_orthogonal_Basis] at this | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
this : ∀ (i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)), ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0
⊢ ⟪y, x⟫_ℝ = 0 | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
this : x ∈ pᗮᗮ
⊢ ⟪y, x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | exact this _ hy | case right.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
this : x ∈ pᗮᗮ
⊢ ⟪y, x⟫_ℝ = 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | intro i | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
⊢ ∀ (i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)), ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | let v : E := (FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ }) i | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
v : E := ↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i)
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | let v' : { x // x ≠ 0 } := ⟨ v, fun hv => (Basis.ne_zero (FiniteDimensional.finBasis ℝ { x // x ∈ pᗮ }) i) (Submodule.coe_eq_zero.mp hv) ⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
v : E := ↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i)
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
v : E := ↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i)
v' : { x // x ≠ 0 } := { val := v, property := ⋯ }
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Cutspace.lean | Submodule_cutspace | [184, 1] | [207, 7] | exact hHi_ v' ⟨ i, rfl ⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
x : E
hHi_ : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ)), ⟪↑x_1, x⟫_ℝ = 0
y : E
hy : y ∈ pᗮ
i : Fin (FiniteDimensional.finrank ℝ ↥pᗮ)
v : E := ↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i)
v' : { x // x ≠ 0 } := { val := v, property := ⋯ }
⊢ ⟪↑((FiniteDimensional.finBasis ℝ ↥pᗮ) i), x⟫_ℝ = 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rintro x1 x2 ε ⟨ hxseg, hne ⟩ hε | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ ∀ (x1 x2 : E) (ε : ℝ),
x ∈ openSegment ℝ x1 x2 ∧ ¬(x1 = x ∧ x2 = x) →
0 < ε →
∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hne : ¬(x1 = x ∧ x2 = x)
hε : 0 < ε
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | push_neg at hne | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hne : ¬(x1 = x ∧ x2 = x)
hε : 0 < ε
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hε : 0 < ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have hxseg' := hxseg | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hε : 0 < ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [openSegment_eq_image', Set.mem_image] at hxseg | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : x ∈ openSegment ℝ x1 x2
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : ∃ x_1 ∈ Set.Ioo 0 1, x1 + x_1 • (x2 - x1) = x
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rcases hxseg with ⟨ t, ht, htt ⟩ | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hxseg : ∃ x_1 ∈ Set.Ioo 0 1, x1 + x_1 • (x2 - x1) = x
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | let v := x2 - x1 | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | let t1 := (-(min t (ε/norm v)/2)) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | let t2 := ((min (1-t) (ε/norm v))/2) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | use t1 • v + x | case intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ ∃ x1' x2',
x ∈ openSegment ℝ x1' x2' ∧ segment ℝ x1' x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(x1' = x ∧ x2' = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ ∃ x2',
x ∈ openSegment ℝ (t1 • v + x) x2' ∧
segment ℝ (t1 • v + x) x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ x2' = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | use t2 • v + x | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ ∃ x2',
x ∈ openSegment ℝ (t1 • v + x) x2' ∧
segment ℝ (t1 • v + x) x2' ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ x2' = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have hx12 : x1 ≠ x2 := by
intro h
rw [←h, openSegment_same] at hxseg'
exact (h.symm ▸ hne) (Set.eq_of_mem_singleton hxseg').symm (Set.eq_of_mem_singleton hxseg').symm | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have ht1pos: 0 < min t (ε / ‖x2 - x1‖) := lt_min ht.1 <| div_pos hε <| norm_sub_pos_iff.mpr (Ne.symm hx12) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have ht2pos: 0 < min (1 - t) (ε / ‖x2 - x1‖) :=
lt_min (by linarith [ht.2]) <| div_pos hε <| norm_sub_pos_iff.mpr (Ne.symm hx12) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have ht1 : t1 < 0 := neg_lt_zero.mpr <| half_pos ht1pos | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have ht2 : 0 < t2 := half_pos ht2pos | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have ht12 : 0 < t2 - t1 := sub_pos.mpr <| lt_trans ht1 ht2 | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | constructor | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) ∧
segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x)
case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | constructor | case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε ∧ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h.right.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε
case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ¬(t1 • v + x = x ∧ t2 • v + x = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | intro h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
⊢ x1 ≠ x2 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
h : x1 = x2
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [←h, openSegment_same] at hxseg' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
h : x1 = x2
⊢ False | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ {x1}
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
h : x1 = x2
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact (h.symm ▸ hne) (Set.eq_of_mem_singleton hxseg').symm (Set.eq_of_mem_singleton hxseg').symm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ {x1}
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
h : x1 = x2
⊢ False | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [ht.2] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
⊢ 0 < 1 - t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [openSegment_eq_image', Set.mem_image] | case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x ∈ openSegment ℝ (t1 • v + x) (t2 • v + x) | case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ∃ x_1 ∈ Set.Ioo 0 1, t1 • v + x + x_1 • (t2 • v + x - (t1 • v + x)) = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | refine ⟨ (-t1/(t2 - t1)), ?_, ?_ ⟩ | case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ∃ x_1 ∈ Set.Ioo 0 1, t1 • v + x + x_1 • (t2 • v + x - (t1 • v + x)) = x | case h.left.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ -t1 / (t2 - t1) ∈ Set.Ioo 0 1
case h.left.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t1 • v + x + (-t1 / (t2 - t1)) • (t2 • v + x - (t1 • v + x)) = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | constructor | case h.left.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ -t1 / (t2 - t1) ∈ Set.Ioo 0 1 | case h.left.refine_1.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 / (t2 - t1)
case h.left.refine_1.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ -t1 / (t2 - t1) < 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [div_pos_iff] | case h.left.refine_1.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 / (t2 - t1) | case h.left.refine_1.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 ∧ 0 < t2 - t1 ∨ -t1 < 0 ∧ t2 - t1 < 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | left | case h.left.refine_1.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 ∧ 0 < t2 - t1 ∨ -t1 < 0 ∧ t2 - t1 < 0 | case h.left.refine_1.left.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 ∧ 0 < t2 - t1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact ⟨ neg_pos_of_neg ht1, ht12 ⟩ | case h.left.refine_1.left.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < -t1 ∧ 0 < t2 - t1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [div_lt_one_iff] | case h.left.refine_1.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ -t1 / (t2 - t1) < 1 | case h.left.refine_1.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < t2 - t1 ∧ -t1 < t2 - t1 ∨ t2 - t1 = 0 ∨ t2 - t1 < 0 ∧ t2 - t1 < -t1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | left | case h.left.refine_1.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < t2 - t1 ∧ -t1 < t2 - t1 ∨ t2 - t1 = 0 ∨ t2 - t1 < 0 ∧ t2 - t1 < -t1 | case h.left.refine_1.right.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < t2 - t1 ∧ -t1 < t2 - t1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact ⟨ ht12, neg_lt_sub_iff_lt_add.mpr <| lt_add_of_le_of_pos (le_refl _) ht2 ⟩ | case h.left.refine_1.right.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < t2 - t1 ∧ -t1 < t2 - t1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [smul_sub (-t1 / (t2 - t1)), smul_add (-t1 / (t2 - t1)), smul_smul, smul_add, smul_smul,
add_sub_add_comm, sub_self, add_zero, ←sub_smul, ←mul_sub, div_mul_cancel _ ?_, add_comm,
← add_assoc, ← add_smul, neg_add_self, zero_smul, zero_add] | case h.left.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t1 • v + x + (-t1 / (t2 - t1)) • (t2 • v + x - (t1 • v + x)) = x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t2 - t1 ≠ 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact Ne.symm (ne_of_lt ht12) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t2 - t1 ≠ 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [Set.subset_inter_iff] | case h.right.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∩ Metric.ball x ε | case h.right.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∧ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | constructor | case h.right.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 ∧ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2
case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | have := @convex_openSegment ℝ _ _ _ _ x1 x2 | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
this : Convex ℝ (openSegment ℝ x1 x2)
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [convex_iff_segment_subset] at this | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
this : Convex ℝ (openSegment ℝ x1 x2)
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
this : ∀ ⦃x : E⦄, x ∈ openSegment ℝ x1 x2 → ∀ ⦃y : E⦄, y ∈ openSegment ℝ x1 x2 → segment ℝ x y ⊆ openSegment ℝ x1 x2
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | apply this <;> clear this <;> rw [←htt] <;>
rw [@add_comm _ _ x1, ←add_assoc, ← add_smul, @add_comm _ _ _ t, openSegment_eq_image'] | case h.right.left.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
this : ∀ ⦃x : E⦄, x ∈ openSegment ℝ x1 x2 → ∀ ⦃y : E⦄, y ∈ openSegment ℝ x1 x2 → segment ℝ x y ⊆ openSegment ℝ x1 x2
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ openSegment ℝ x1 x2 | case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (t + t1) • v + x1 ∈ (fun θ => x1 + θ • (x2 - x1)) '' Set.Ioo 0 1
case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (t + t2) • v + x1 ∈ (fun θ => x1 + θ • (x2 - x1)) '' Set.Ioo 0 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact ⟨ t + t1,
⟨ lt_of_le_of_lt' (by linarith [min_le_left t (ε/norm v)] : t -t/2 ≤ t -(min t (ε /norm v)/2))
(by linarith [ht.1]), lt_trans (add_lt_of_neg_right t ht1) ht.2 ⟩,
by unfold_let v; simp only [ge_iff_le]; rw [add_comm, @add_comm _ _ t t1, sub_eq_neg_add] ⟩ | case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (t + t1) • v + x1 ∈ (fun θ => x1 + θ • (x2 - x1)) '' Set.Ioo 0 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [min_le_left t (ε/norm v)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t - t / 2 ≤ t - min t (ε / ‖v‖) / 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [ht.1] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < t - t / 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | unfold_let v | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (fun θ => x1 + θ • (x2 - x1)) (t + t1) = (t + t1) • v + x1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (fun θ => x1 + θ • (x2 - x1)) (t + t1) = (t + t1) • (x2 - x1) + x1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | simp only [ge_iff_le] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (fun θ => x1 + θ • (x2 - x1)) (t + t1) = (t + t1) • (x2 - x1) + x1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x1 + (t + t1) • (x2 - x1) = (t + t1) • (x2 - x1) + x1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [add_comm, @add_comm _ _ t t1, sub_eq_neg_add] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x1 + (t + t1) • (x2 - x1) = (t + t1) • (x2 - x1) + x1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | refine ⟨ t + t2,
⟨ lt_trans ht.1 (by linarith [ht2pos] : t < t + (min (1 - t) (ε / ‖x2 - x1‖) / 2)), ?_ ⟩,
by simp only [ge_iff_le] ;rw [add_comm] ⟩ | case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (t + t2) • v + x1 ∈ (fun θ => x1 + θ • (x2 - x1)) '' Set.Ioo 0 1 | case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t + t2 < 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact lt_of_lt_of_le' (by linarith [ht.2]) (by linarith [min_le_left (1 - t) ((ε / ‖x2 - x1‖))]
: t + min (1 - t) (ε / ‖x2 - x1‖) / 2 ≤ t + ((1 - t) / 2)) | case h.right.left.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t + t2 < 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [ht2pos] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t < t + min (1 - t) (ε / ‖x2 - x1‖) / 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | simp only [ge_iff_le] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ (fun θ => x1 + θ • (x2 - x1)) (t + t2) = (t + t2) • v + x1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x1 + (t + t2) • (x2 - x1) = (t + t2) • v + x1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [add_comm] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ x1 + (t + t2) • (x2 - x1) = (t + t2) • v + x1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [ht.2] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t + (1 - t) / 2 < 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith [min_le_left (1 - t) ((ε / ‖x2 - x1‖))] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t + min (1 - t) (ε / ‖x2 - x1‖) / 2 ≤ t + (1 - t) / 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | clear ht hxseg' hne | case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε | case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [← half_lt_self_iff] at hε | case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε | case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | apply (convex_iff_segment_subset.mp <| convex_ball x ε ) <;> rw [Metric.mem_ball] <;> norm_num <;> unfold_let | case h.right.left.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ segment ℝ (t1 • v + x) (t2 • v + x) ⊆ Metric.ball x ε | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖-(min t (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε
case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min (1 - t) (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | simp | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖-(min t (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε
case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min (1 - t) (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min t (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε
case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min (1 - t) (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | all_goals {
rw [norm_smul, Real.norm_eq_abs, abs_of_pos (by linarith), ← min_div_div_right (by linarith), Monotone.map_min fun _ _ => (mul_le_mul_right (norm_sub_pos_iff.mpr (Ne.symm hx12))).mpr] apply min_lt_of_right_lt ; rw [@div_mul_comm _ _ _ 2, mul_comm, div_mul_div_cancel _ (Ne.symm (ne_of_lt (norm_sub_pos_iff.mpr (Ne.symm hx12))))] ; exact hε
} | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min t (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε
case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min (1 - t) (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [norm_smul, Real.norm_eq_abs, abs_of_pos (by linarith), ← min_div_div_right (by linarith), Monotone.map_min fun _ _ => (mul_le_mul_right (norm_sub_pos_iff.mpr (Ne.symm hx12))).mpr] | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ‖(min (1 - t) (ε / ‖x2 - x1‖) / 2) • (x2 - x1)‖ < ε | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ min ((1 - t) / 2 * ‖x2 - x1‖) (ε / ‖x2 - x1‖ / 2 * ‖x2 - x1‖) < ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | apply min_lt_of_right_lt | case h.right.left.right.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ min ((1 - t) / 2 * ‖x2 - x1‖) (ε / ‖x2 - x1‖ / 2 * ‖x2 - x1‖) < ε | case h.right.left.right.a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ε / ‖x2 - x1‖ / 2 * ‖x2 - x1‖ < ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rw [@div_mul_comm _ _ _ 2, mul_comm, div_mul_div_cancel _ (Ne.symm (ne_of_lt (norm_sub_pos_iff.mpr (Ne.symm hx12))))] | case h.right.left.right.a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ε / ‖x2 - x1‖ / 2 * ‖x2 - x1‖ < ε | case h.right.left.right.a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ε / 2 < ε |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact hε | case h.right.left.right.a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ε / 2 < ε | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 < min (1 - t) (ε / ‖x2 - x1‖) / 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : ε / 2 < ε
t : ℝ
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ 0 ≤ 2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | push_neg | case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ ¬(t1 • v + x = x ∧ t2 • v + x = x) | case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t1 • v + x = x → t2 • v + x ≠ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | intro h1 | case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
⊢ t1 • v + x = x → t2 • v + x ≠ x | case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
⊢ t2 • v + x ≠ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | rcases (em (x1 = x)) with (rfl | hx1x)
<;> norm_num
<;> intro h
<;> rw [sub_eq_zero] at h
<;> rcases h with h | rfl | case h.right.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
⊢ t2 • v + x ≠ x | case h.right.right.inl.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x1 = x1 → x2 ≠ x1
hxseg' : x1 ∈ openSegment ℝ x1 x2
htt : x1 + t • (x2 - x1) = x1
h1 : t1 • v + x1 = x1
h : t2 = 0
⊢ False
case h.right.right.inl.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x2 = x2 → x2 ≠ x2
hxseg' : x2 ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x2
h1 : t1 • v + x2 = x2
⊢ False
case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact (ne_of_lt ht2) h.symm | case h.right.right.inl.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x1 = x1 → x2 ≠ x1
hxseg' : x1 ∈ openSegment ℝ x1 x2
htt : x1 + t • (x2 - x1) = x1
h1 : t1 • v + x1 = x1
h : t2 = 0
⊢ False
case h.right.right.inl.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x2 = x2 → x2 ≠ x2
hxseg' : x2 ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x2
h1 : t1 • v + x2 = x2
⊢ False
case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False | case h.right.right.inl.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x2 = x2 → x2 ≠ x2
hxseg' : x2 ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x2
h1 : t1 • v + x2 = x2
⊢ False
case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | simp at hne | case h.right.right.inl.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
hne : x2 = x2 → x2 ≠ x2
hxseg' : x2 ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x2
h1 : t1 • v + x2 = x2
⊢ False
case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False | case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | exact (ne_of_lt ht2) h.symm | case h.right.right.inr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
ε : ℝ
hε : 0 < ε
hne : x1 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x1 x2
t : ℝ
ht : t ∈ Set.Ioo 0 1
htt : x1 + t • (x2 - x1) = x
v : E := x2 - x1
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x1 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x1‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x1‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x1 = x
h : t2 = 0
⊢ False
case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False | case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | hxSegBallInterSeg | [16, 1] | [124, 7] | simp at hx12 | case h.right.right.inr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x2 : E
ε : ℝ
hε : 0 < ε
t : ℝ
ht : t ∈ Set.Ioo 0 1
hne : x2 = x → x2 ≠ x
hxseg' : x ∈ openSegment ℝ x2 x2
htt : x2 + t • (x2 - x2) = x
v : E := x2 - x2
t1 : ℝ := -(min t (ε / ‖v‖) / 2)
t2 : ℝ := min (1 - t) (ε / ‖v‖) / 2
hx12 : x2 ≠ x2
ht1pos : 0 < min t (ε / ‖x2 - x2‖)
ht2pos : 0 < min (1 - t) (ε / ‖x2 - x2‖)
ht1 : t1 < 0
ht2 : 0 < t2
ht12 : 0 < t2 - t1
h1 : t1 • v + x = x
hx1x : ¬x2 = x
⊢ False | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope.I_mem | [130, 1] | [135, 7] | rintro Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
⊢ ∀ (Hi_ : Halfspace E), Hi_ ∈ I H_ x ↔ Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
⊢ Hi_ ∈ I H_ x ↔ Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope.I_mem | [130, 1] | [135, 7] | unfold I | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
⊢ Hi_ ∈ I H_ x ↔ Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
⊢ Hi_ ∈ {Hi_ | Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_} ↔ Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope.I_mem | [130, 1] | [135, 7] | rw [Set.mem_setOf] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
⊢ Hi_ ∈ {Hi_ | Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_} ↔ Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope.I_sub | [137, 1] | [141, 7] | unfold Hpolytope.I | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
⊢ I H_ x ⊆ H_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
⊢ {Hi_ | Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_} ⊆ H_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope.I_sub | [137, 1] | [141, 7] | simp only [Set.sep_subset] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
⊢ {Hi_ | Hi_ ∈ H_ ∧ x ∈ frontier ↑Hi_} ⊆ H_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rintro x hxH | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
⊢ ∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} |
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