url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
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
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.