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https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | origin_Hpolytope | [90, 1] | [115, 7] | rw [inner_zero_right] | case refine_2.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : { x // x ≠ 0 }
a✝ : x ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)
⊢ ⟪↑x, 0⟫_ℝ = 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | intro f c | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ ∀ (f : { f // ‖f‖ = 1 }) (c : ℝ), ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = {x | ↑f x = c} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = {x | ↑f x = c} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | refine ⟨ {Halfspace.mk f c, Halfspace.mk (-f) (-c)},
(by simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.finite_singleton, Set.Finite.insert]) , ?_ ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = {x | ↑f x = c} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ Hpolytope ⋯ = {x | ↑f x = c} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | ext x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ Hpolytope ⋯ = {x | ↑f x = c} | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ {x | ↑f x = c} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | rw [mem_Hpolytope, Set.mem_setOf] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ {x | ↑f x = c} | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) ↔ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | constructor | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) ↔ ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) → ↑f x = c
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ ↑f x = c → ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.finite_singleton, Set.Finite.insert] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ Set.Finite {{ f := f, α := c }, { f := -f, α := -c }} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | intro h | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) → ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | have h1 := h (Halfspace.mk f c) (by simp) | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | have h2 := h (Halfspace.mk (-f) (-c)) (by simp) | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
⊢ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg] at h2 | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
⊢ ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : c ≤ ↑f x
⊢ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | change f.1 x ≤ c at h1 | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : c ≤ ↑f x
⊢ ↑f x = c | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h2 : c ≤ ↑f x
h1 : ↑f x ≤ c
⊢ ↑f x = c |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | exact le_antisymm h1 h2 | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h2 : c ≤ ↑f x
h1 : ↑f x ≤ c
⊢ ↑f x = c | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ { f := f, α := c } ∈ {{ f := f, α := c }, { f := -f, α := -c }} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ { f := -f, α := -c } ∈ {{ f := f, α := c }, { f := -f, α := -c }} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | intro h Hi hHi | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ ↑f x = c → ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}
⊢ ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.mem_insert_iff] at hHi | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}
⊢ ↑Hi.f x ≤ Hi.α | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi = { f := f, α := c } ∨ Hi = { f := -f, α := -c }
⊢ ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | rcases hHi with rfl | rfl | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi = { f := f, α := c } ∨ Hi = { f := -f, α := -c }
⊢ ↑Hi.f x ≤ Hi.α | case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | exact le_of_eq h | case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg] | case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α | case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ c ≤ ↑f x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | hyperplane_Hpolytope | [117, 1] | [142, 7] | exact le_of_eq h.symm | case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ c ≤ ↑f x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | ext x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
⊢ Hpolytope ⋯ = Hpolytope hH_1 ∩ Hpolytope hH_2 | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ Hpolytope hH_1 ∩ Hpolytope hH_2 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | rw [mem_Hpolytope, Set.mem_inter_iff, mem_Hpolytope, mem_Hpolytope] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ Hpolytope hH_1 ∩ Hpolytope hH_2 | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) ↔ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | constructor | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) ↔ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) → (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ ((∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α) → ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | intro h | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) → (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
⊢ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | constructor <;> intro Hi_ hH_ <;> exact h Hi_ (by simp only [Set.mem_union, hH_, true_or, or_true]) | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
⊢ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | simp only [Set.mem_union, hH_, true_or, or_true] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
Hi_ : Halfspace E
hH_ : Hi_ ∈ H_2
⊢ Hi_ ∈ H_1 ∪ H_2 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | intro h Hi hHi | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
⊢ ((∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α) → ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∪ H_2
⊢ ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | rw [Set.mem_union] at hHi | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∪ H_2
⊢ ↑Hi.f x ≤ Hi.α | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∨ Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | rcases hHi with hHi | hHi | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∨ Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α | case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1
⊢ ↑Hi.f x ≤ Hi.α
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | exact h.1 Hi hHi | case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1
⊢ ↑Hi.f x ≤ Hi.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | inter_Hpolytope | [144, 1] | [160, 7] | exact h.2 Hi hHi | case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
hH_1 : Set.Finite H_1
hH_2 : Set.Finite H_2
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Vpolytope_translation | [162, 1] | [166, 7] | rw [Vpolytope, convexHull_add, convexHull_singleton] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
S : Set E
hS : Set.Finite S
x : E
⊢ Vpolytope ⋯ = Vpolytope hS + {x} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
S : Set E
hS : Set.Finite S
x : E
⊢ (convexHull ℝ) S + {x} = Vpolytope hS + {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Vpolytope_translation | [162, 1] | [166, 7] | rfl | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
S : Set E
hS : Set.Finite S
x : E
⊢ (convexHull ℝ) S + {x} = Vpolytope hS + {x} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [Hpolytope, Hpolytope, Set.sInter_image, Set.sInter_image] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
⊢ Hpolytope ⋯ = Hpolytope hH_ + {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
⊢ ⋂ x_1 ∈ Halfspace_translation x '' H_, ↑x_1 = (⋂ x ∈ H_, ↑x) + {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | ext y | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
⊢ ⋂ x_1 ∈ Halfspace_translation x '' H_, ↑x_1 = (⋂ x ∈ H_, ↑x) + {x} | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ y ∈ ⋂ x_1 ∈ Halfspace_translation x '' H_, ↑x_1 ↔ y ∈ (⋂ x ∈ H_, ↑x) + {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [Set.mem_iInter, Set.add_singleton] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ y ∈ ⋂ x_1 ∈ Halfspace_translation x '' H_, ↑x_1 ↔ y ∈ (⋂ x ∈ H_, ↑x) + {x} | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ (i : Halfspace E), y ∈ ⋂ (_ : i ∈ Halfspace_translation x '' H_), ↑i) ↔ y ∈ (fun x_1 => x_1 + x) '' ⋂ x ∈ H_, ↑x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | simp only [Set.mem_iInter, SetLike.mem_coe, Set.image_add_right, Set.mem_preimage] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ (i : Halfspace E), y ∈ ⋂ (_ : i ∈ Halfspace_translation x '' H_), ↑i) ↔ y ∈ (fun x_1 => x_1 + x) '' ⋂ x ∈ H_, ↑x | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ Halfspace_translation x '' H_, y ∈ i) ↔ ∀ i ∈ H_, y + -x ∈ i |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | constructor | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ Halfspace_translation x '' H_, y ∈ i) ↔ ∀ i ∈ H_, y + -x ∈ i | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ Halfspace_translation x '' H_, y ∈ i) → ∀ i ∈ H_, y + -x ∈ i
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ H_, y + -x ∈ i) → ∀ i ∈ Halfspace_translation x '' H_, y ∈ i |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | intro h Hi_ hHi_ | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ Halfspace_translation x '' H_, y ∈ i) → ∀ i ∈ H_, y + -x ∈ i | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ Halfspace_translation x '' H_, y ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ y + -x ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | specialize h (Halfspace_translation x Hi_) (Set.mem_image_of_mem _ hHi_) | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ Halfspace_translation x '' H_, y ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ y + -x ∈ Hi_ | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : y ∈ Halfspace_translation x Hi_
⊢ y + -x ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [← SetLike.mem_coe, mem_Halfspace_translation, sub_eq_add_neg] at h | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : y ∈ Halfspace_translation x Hi_
⊢ y + -x ∈ Hi_ | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : y + -x ∈ ↑Hi_
⊢ y + -x ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | exact h | case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : y + -x ∈ ↑Hi_
⊢ y + -x ∈ Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | intro h Hi_ hHi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
⊢ (∀ i ∈ H_, y + -x ∈ i) → ∀ i ∈ Halfspace_translation x '' H_, y ∈ i | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | specialize h (Halfspace_translation (-x) Hi_) (?_) | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
⊢ y ∈ Hi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
⊢ Halfspace_translation (-x) Hi_ ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [Set.mem_image] at hHi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
⊢ Halfspace_translation (-x) Hi_ ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : ∃ x_1 ∈ H_, Halfspace_translation x x_1 = Hi_
⊢ Halfspace_translation (-x) Hi_ ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rcases hHi_ with ⟨ Hi_', hHi_', rfl ⟩ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_ : Halfspace E
hHi_ : ∃ x_1 ∈ H_, Halfspace_translation x x_1 = Hi_
⊢ Halfspace_translation (-x) Hi_ ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
⊢ Halfspace_translation (-x) (Halfspace_translation x Hi_') ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | have : Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_':= by
rw [SetLike.ext_iff]
intro z
rw [← SetLike.mem_coe, ← SetLike.mem_coe, mem_Halfspace_translation, mem_Halfspace_translation,
sub_neg_eq_add, add_sub_cancel]
done | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
⊢ Halfspace_translation (-x) (Halfspace_translation x Hi_') ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
this : Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_'
⊢ Halfspace_translation (-x) (Halfspace_translation x Hi_') ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [this] | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
this : Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_'
⊢ Halfspace_translation (-x) (Halfspace_translation x Hi_') ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
this : Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_'
⊢ Hi_' ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | assumption | case h.mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
this : Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_'
⊢ Hi_' ∈ H_
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [← SetLike.mem_coe, mem_Halfspace_translation, add_sub_cancel, SetLike.mem_coe] at h | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y + -x ∈ Halfspace_translation (-x) Hi_
⊢ y ∈ Hi_ | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y ∈ Hi_
⊢ y ∈ Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | exact h | case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Halfspace_translation x '' H_
h : y ∈ Hi_
⊢ y ∈ Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [SetLike.ext_iff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
⊢ Halfspace_translation (-x) (Halfspace_translation x Hi_') = Hi_' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
⊢ ∀ (x_1 : E), x_1 ∈ Halfspace_translation (-x) (Halfspace_translation x Hi_') ↔ x_1 ∈ Hi_' |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | intro z | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
⊢ ∀ (x_1 : E), x_1 ∈ Halfspace_translation (-x) (Halfspace_translation x Hi_') ↔ x_1 ∈ Hi_' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
z : E
⊢ z ∈ Halfspace_translation (-x) (Halfspace_translation x Hi_') ↔ z ∈ Hi_' |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polytope.lean | Hpolytope_translation | [168, 1] | [195, 7] | rw [← SetLike.mem_coe, ← SetLike.mem_coe, mem_Halfspace_translation, mem_Halfspace_translation,
sub_neg_eq_add, add_sub_cancel] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
h : ∀ i ∈ H_, y + -x ∈ i
Hi_' : Halfspace E
hHi_' : Hi_' ∈ H_
z : E
⊢ z ∈ Halfspace_translation (-x) (Halfspace_translation x Hi_') ↔ z ∈ Hi_' | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Finite.translation | [25, 1] | [28, 30] | rw [Set.add_singleton] | α : Type
inst✝ : AddGroup α
S : Set α
hS : Set.Finite S
x : α
⊢ Set.Finite (S + {x}) | α : Type
inst✝ : AddGroup α
S : Set α
hS : Set.Finite S
x : α
⊢ Set.Finite ((fun x_1 => x_1 + x) '' S) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Finite.translation | [25, 1] | [28, 30] | exact Set.Finite.image _ hS | α : Type
inst✝ : AddGroup α
S : Set α
hS : Set.Finite S
x : α
⊢ Set.Finite ((fun x_1 => x_1 + x) '' S) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | rw [Set.add_singleton, Set.mem_image] | α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ s ∈ S + {x} ↔ s - x ∈ S | α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ (∃ x_1 ∈ S, x_1 + x = s) ↔ s - x ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | constructor | α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ (∃ x_1 ∈ S, x_1 + x = s) ↔ s - x ∈ S | case mp
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ (∃ x_1 ∈ S, x_1 + x = s) → s - x ∈ S
case mpr
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ s - x ∈ S → ∃ x_1 ∈ S, x_1 + x = s |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | rintro ⟨y, hy, rfl⟩ | case mp
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ (∃ x_1 ∈ S, x_1 + x = s) → s - x ∈ S | case mp.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
hy : y ∈ S
⊢ y + x - x ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | rw [add_sub_cancel] | case mp.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
hy : y ∈ S
⊢ y + x - x ∈ S | case mp.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
hy : y ∈ S
⊢ y ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | exact hy | case mp.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
hy : y ∈ S
⊢ y ∈ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | intro h | case mpr
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
⊢ s - x ∈ S → ∃ x_1 ∈ S, x_1 + x = s | case mpr
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
h : s - x ∈ S
⊢ ∃ x_1 ∈ S, x_1 + x = s |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | exact ⟨s - x, h, by rw [sub_add_cancel]⟩ | case mpr
α : Type
inst✝ : AddGroup α
S : Set α
x s : α
h : s - x ∈ S
⊢ ∃ x_1 ∈ S, x_1 + x = s | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.mem_translation | [30, 1] | [41, 7] | rw [sub_add_cancel] | α : Type
inst✝ : AddGroup α
S : Set α
x s : α
h : s - x ∈ S
⊢ s - x + x = s | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | ext y | α : Type
inst✝ : AddGroup α
S : Set α
x : α
⊢ S - {x} = S + {-x} | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ y ∈ S - {x} ↔ y ∈ S + {-x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | simp only [sub_singleton, mem_image, add_singleton, image_add_right, neg_neg, mem_preimage] | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ y ∈ S - {x} ↔ y ∈ S + {-x} | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ (∃ x_1 ∈ S, x_1 - x = y) ↔ y + x ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | refine ⟨ ?_, fun h => ⟨y + x, h, by rw [add_sub_cancel]⟩ ⟩ | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ (∃ x_1 ∈ S, x_1 - x = y) ↔ y + x ∈ S | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ (∃ x_1 ∈ S, x_1 - x = y) → y + x ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | rintro ⟨z, hz, rfl⟩ | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ (∃ x_1 ∈ S, x_1 - x = y) → y + x ∈ S | case h.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x z : α
hz : z ∈ S
⊢ z - x + x ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | rw [sub_add_cancel] | case h.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x z : α
hz : z ∈ S
⊢ z - x + x ∈ S | case h.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x z : α
hz : z ∈ S
⊢ z ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | exact hz | case h.intro.intro
α : Type
inst✝ : AddGroup α
S : Set α
x z : α
hz : z ∈ S
⊢ z ∈ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.sub_eq_neg_add | [46, 1] | [53, 11] | rw [add_sub_cancel] | α : Type
inst✝ : AddGroup α
S : Set α
x y : α
h : y + x ∈ S
⊢ y + x - x = y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.neg_add_cancel_right' | [55, 1] | [59, 7] | ext y | α : Type
inst✝ : AddGroup α
S : Set α
x : α
⊢ S - {x} + {x} = S | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ y ∈ S - {x} + {x} ↔ y ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.neg_add_cancel_right' | [55, 1] | [59, 7] | simp only [sub_singleton, add_singleton, mem_image, exists_exists_and_eq_and, sub_add_cancel, exists_eq_right] | case h
α : Type
inst✝ : AddGroup α
S : Set α
x y : α
⊢ y ∈ S - {x} + {x} ↔ y ∈ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | interior_eq_compl_closure_compl | [61, 1] | [64, 7] | rw [← compl_compl s, compl_compl sᶜ, interior_compl] | α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ interior s = (closure sᶜ)ᶜ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | ext x | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
⊢ ⋂₀ ((fun x => x ∩ t) '' s) = ⋂₀ s ∩ t | case h
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ x ∈ ⋂₀ ((fun x => x ∩ t) '' s) ↔ x ∈ ⋂₀ s ∩ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | simp only [mem_sInter, mem_inter_iff, mem_singleton_iff, and_imp] | case h
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ x ∈ ⋂₀ ((fun x => x ∩ t) '' s) ↔ x ∈ ⋂₀ s ∩ t | case h
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1) ↔ (∀ t ∈ s, x ∈ t) ∧ x ∈ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | constructor | case h
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1) ↔ (∀ t ∈ s, x ∈ t) ∧ x ∈ t | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1) → (∀ t ∈ s, x ∈ t) ∧ x ∈ t
case h.mpr
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t → ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | intro h | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1) → (∀ t ∈ s, x ∈ t) ∧ x ∈ t | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | have : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s := by
rw [mem_image]
exact ⟨Nonempty.some hs, hs.some_mem, rfl⟩ | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | refine ⟨ ?_, (h (hs.some ∩ t) this).2⟩ | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
⊢ ∀ t ∈ s, x ∈ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | intro y hy | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
⊢ ∀ t ∈ s, x ∈ t | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
⊢ x ∈ y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | have : y ∩ t ∈ (fun x => x ∩ t) '' s := by
rw [mem_image]
exact ⟨y, hy, rfl⟩ | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
⊢ x ∈ y | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this✝ : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
this : y ∩ t ∈ (fun x => x ∩ t) '' s
⊢ x ∈ y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | exact (h (y ∩ t) this).1 | case h.mp
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this✝ : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
this : y ∩ t ∈ (fun x => x ∩ t) '' s
⊢ x ∈ y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | rw [mem_image] | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
⊢ Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
⊢ ∃ x ∈ s, x ∩ t = Nonempty.some hs ∩ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | exact ⟨Nonempty.some hs, hs.some_mem, rfl⟩ | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
⊢ ∃ x ∈ s, x ∩ t = Nonempty.some hs ∩ t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | rw [mem_image] | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
⊢ y ∩ t ∈ (fun x => x ∩ t) '' s | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
⊢ ∃ x ∈ s, x ∩ t = y ∩ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | exact ⟨y, hy, rfl⟩ | α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1
this : Nonempty.some hs ∩ t ∈ (fun x => x ∩ t) '' s
y : Set α
hy : y ∈ s
⊢ ∃ x ∈ s, x ∩ t = y ∩ t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | rintro h y ⟨ z, hz, rfl ⟩ | case h.mpr
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
⊢ (∀ t ∈ s, x ∈ t) ∧ x ∈ t → ∀ t_1 ∈ (fun x => x ∩ t) '' s, x ∈ t_1 | case h.mpr.intro.intro
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : (∀ t ∈ s, x ∈ t) ∧ x ∈ t
z : Set α
hz : z ∈ s
⊢ x ∈ (fun x => x ∩ t) z |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.sInter_inter_comm | [66, 1] | [85, 7] | exact mem_inter (h.1 z hz) h.2 | case h.mpr.intro.intro
α : Type u_1
s : Set (Set α)
hs : Set.Nonempty s
t : Set α
x : α
h : (∀ t ∈ s, x ∈ t) ∧ x ∈ t
z : Set α
hz : z ∈ s
⊢ x ∈ (fun x => x ∩ t) z | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | refine subset_antisymm (image_sInter_subset S f) ?_ | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
⊢ f '' ⋂₀ S = ⋂ s ∈ S, f '' s | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
⊢ ⋂ s ∈ S, f '' s ⊆ f '' ⋂₀ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | intro y hy | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
⊢ ⋂ s ∈ S, f '' s ⊆ f '' ⋂₀ S | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
y : β
hy : y ∈ ⋂ s ∈ S, f '' s
⊢ y ∈ f '' ⋂₀ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | simp_all | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
y : β
hy : y ∈ ⋂ s ∈ S, f '' s
⊢ y ∈ f '' ⋂₀ S | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
y : β
hy : ∀ i ∈ S, ∃ x ∈ i, f x = y
⊢ ∃ x, (∀ t ∈ S, x ∈ t) ∧ f x = y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | rcases hy hS.some hS.some_mem with ⟨x, _hxInhSsome_, rfl⟩ | α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
y : β
hy : ∀ i ∈ S, ∃ x ∈ i, f x = y
⊢ ∃ x, (∀ t ∈ S, x ∈ t) ∧ f x = y | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
⊢ ∃ x_1, (∀ t ∈ S, x_1 ∈ t) ∧ f x_1 = f x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | refine ⟨x, ?_, rfl⟩ | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
⊢ ∃ x_1, (∀ t ∈ S, x_1 ∈ t) ∧ f x_1 = f x | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
⊢ ∀ t ∈ S, x ∈ t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | intro s hsInS | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
⊢ ∀ t ∈ S, x ∈ t | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
⊢ x ∈ s |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | rcases hy s hsInS with ⟨z, hzIns, hfzEqfx⟩ | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
⊢ x ∈ s | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
z : α
hzIns : z ∈ s
hfzEqfx : f z = f x
⊢ x ∈ s |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | convert hzIns | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
z : α
hzIns : z ∈ s
hfzEqfx : f z = f x
⊢ x ∈ s | case h.e'_4
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
z : α
hzIns : z ∈ s
hfzEqfx : f z = f x
⊢ x = z |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Set.Nonempty.image_sInter | [87, 1] | [99, 7] | exact hf hfzEqfx.symm | case h.e'_4
α : Type u_1
β : Type u_2
S : Set (Set α)
hS : Set.Nonempty S
f : α → β
hf : Function.Injective f
x : α
_hxInhSsome_ : x ∈ Nonempty.some hS
hy : ∀ i ∈ S, ∃ x_1 ∈ i, f x_1 = f x
s : Set α
hsInS : s ∈ S
z : α
hzIns : z ∈ s
hfzEqfx : f z = f x
⊢ x = z | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | AffineEquiv.coe_VSubconst | [120, 1] | [120, 172] | rfl | 𝕜 E P : Type
inst✝³ : Field 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : AddTorsor E P
x : P
⊢ ⇑(VSubconst 𝕜 x) = fun x_1 => x_1 -ᵥ x | no goals |
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