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/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
. simp
|
case cons.false
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if false = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if true = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
|
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if true = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
|
https://github.com/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
. simp; apply Nat.le_succ_of_le; assumption
|
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if true = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
|
no goals
|
https://github.com/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
simp
|
case cons.false
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if false = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
|
no goals
|
https://github.com/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
simp
|
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if true = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
|
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (backtrack ds) ≤ Nat.succ (List.length ds)
|
https://github.com/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
apply Nat.le_succ_of_le
|
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (backtrack ds) ≤ Nat.succ (List.length ds)
|
case cons.true.h
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (backtrack ds) ≤ List.length ds
|
https://github.com/benjaminfjones/reckonlean.git
|
8768f7342ba226cfc2d7b92e47432f1da66eff25
|
ReckonLean/Dpll.lean
|
length_backtrack
|
[195, 1]
|
[204, 50]
|
assumption
|
case cons.true.h
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (backtrack ds) ≤ List.length ds
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_neg
|
[21, 1]
|
[25, 7]
|
change (⟨-f.1, _ ⟩: {f : (NormedSpace.Dual ℝ E) // norm f = 1}).1 = -f.1
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ ↑(-f) = -↑f
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ ↑{ val := -↑f, property := ⋯ } = -↑f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_neg
|
[21, 1]
|
[25, 7]
|
simp
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ ↑{ val := -↑f, property := ⋯ } = -↑f
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
intro f
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ ∀ (f : { f // ‖f‖ = 1 }), Function.Surjective ⇑↑f
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ Function.Surjective ⇑↑f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
apply LinearMap.surjective_of_ne_zero
|
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ Function.Surjective ⇑↑f
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ ↑↑f ≠ 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
intro h
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
⊢ ↑↑f ≠ 0
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑↑f = 0
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
rw [← ContinuousLinearMap.coe_zero, ContinuousLinearMap.coe_inj] at h
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑↑f = 0
⊢ False
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑f = 0
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
have := h ▸ f.2
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑f = 0
⊢ False
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑f = 0
this : ‖0‖ = 1
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
unitSphereDual_surj
|
[27, 1]
|
[35, 7]
|
simp only [norm_zero, zero_ne_one] at this
|
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : { f // ‖f‖ = 1 }
h : ↑f = 0
this : ‖0‖ = 1
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_mem
|
[146, 1]
|
[149, 6]
|
intro x
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ∀ (x : E), x ∈ ↑H_ ↔ ↑H_.f x ≤ H_.α
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
⊢ x ∈ ↑H_ ↔ ↑H_.f x ≤ H_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_mem
|
[146, 1]
|
[149, 6]
|
rw [H_.h]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
⊢ x ∈ ↑H_ ↔ ↑H_.f x ≤ H_.α
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
⊢ x ∈ ⇑↑H_.f ⁻¹' {x | x ≤ H_.α} ↔ ↑H_.f x ≤ H_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_mem
|
[146, 1]
|
[149, 6]
|
rfl
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
⊢ x ∈ ⇑↑H_.f ⁻¹' {x | x ≤ H_.α} ↔ ↑H_.f x ≤ H_.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_convex
|
[151, 1]
|
[153, 63]
|
rw [H_.h]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Convex ℝ ↑H_
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Convex ℝ (⇑↑H_.f ⁻¹' {x | x ≤ H_.α})
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_convex
|
[151, 1]
|
[153, 63]
|
exact convex_halfspace_le (LinearMap.isLinear H_.f.1.1) H_.α
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Convex ℝ (⇑↑H_.f ⁻¹' {x | x ≤ H_.α})
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_closed
|
[155, 1]
|
[157, 53]
|
rw [H_.h]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ IsClosed ↑H_
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ IsClosed (⇑↑H_.f ⁻¹' {x | x ≤ H_.α})
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_closed
|
[155, 1]
|
[157, 53]
|
exact IsClosed.preimage (H_.f.1.cont) isClosed_Iic
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ IsClosed (⇑↑H_.f ⁻¹' {x | x ≤ H_.α})
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
apply affineSpan_eq_top_of_nonempty_interior
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ affineSpan ℝ ↑H_ = ⊤
|
case hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Set.Nonempty (interior ((convexHull ℝ) ↑H_))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
apply Set.Nonempty.mono (?_ : H_.f.1 ⁻¹' (Metric.ball (H_.α -1) (1/2)) ⊆ (interior ((convexHull ℝ) H_.S)))
|
case hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Set.Nonempty (interior ((convexHull ℝ) ↑H_))
|
case hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Set.Nonempty (⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2))
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ interior ((convexHull ℝ) (Halfspace.S H_))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
rw [IsOpen.subset_interior_iff (IsOpen.preimage (?_) Metric.isOpen_ball)]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ interior ((convexHull ℝ) (Halfspace.S H_))
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ (convexHull ℝ) (Halfspace.S H_)
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
apply subset_trans ?_ (subset_convexHull ℝ (SetLike.coe H_))
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ (convexHull ℝ) (Halfspace.S H_)
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
intro x hx
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2) ⊆ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : x ∈ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2)
⊢ x ∈ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
rw [Set.mem_preimage, Real.ball_eq_Ioo, Set.mem_Ioo] at hx
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : x ∈ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2)
⊢ x ∈ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : H_.α - 1 - 1 / 2 < ↑H_.f x ∧ ↑H_.f x < H_.α - 1 + 1 / 2
⊢ x ∈ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
rw [Halfspace_mem H_]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : H_.α - 1 - 1 / 2 < ↑H_.f x ∧ ↑H_.f x < H_.α - 1 + 1 / 2
⊢ x ∈ ↑H_
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : H_.α - 1 - 1 / 2 < ↑H_.f x ∧ ↑H_.f x < H_.α - 1 + 1 / 2
⊢ ↑H_.f x ≤ H_.α
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
linarith
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : H_.α - 1 - 1 / 2 < ↑H_.f x ∧ ↑H_.f x < H_.α - 1 + 1 / 2
⊢ ↑H_.f x ≤ H_.α
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
exact H_.f.1.cont
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Continuous ⇑↑H_.f
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
cases' unitSphereDual_surj H_.f (H_.α -1) with x hx
|
case hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
⊢ Set.Nonempty (⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2))
|
case hs.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ Set.Nonempty (⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
use x
|
case hs.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ Set.Nonempty (⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2))
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ x ∈ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
rw [Set.mem_preimage, Metric.mem_ball, dist_sub_eq_dist_add_right, hx, sub_add_cancel, dist_self]
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ x ∈ ⇑↑H_.f ⁻¹' Metric.ball (H_.α - 1) (1 / 2)
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ 0 < 1 / 2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_span
|
[159, 1]
|
[177, 7]
|
linarith
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Halfspace E
x : E
hx : ↑H_.f x = H_.α - 1
⊢ 0 < 1 / 2
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.S
|
[182, 1]
|
[187, 7]
|
ext y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
⊢ ↑(Halfspace_translation x H_) = (fun x_1 => x_1 + x) '' ↑H_
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
y : E
⊢ y ∈ ↑(Halfspace_translation x H_) ↔ y ∈ (fun x_1 => x_1 + x) '' ↑H_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.S
|
[182, 1]
|
[187, 7]
|
rw [Halfspace_translation, Halfspace_mem, Set.image_add_right, Set.mem_preimage, ← sub_eq_add_neg,
Halfspace_mem, ContinuousLinearMap.map_sub, sub_le_iff_le_add]
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
y : E
⊢ y ∈ ↑(Halfspace_translation x H_) ↔ y ∈ (fun x_1 => x_1 + x) '' ↑H_
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
mem_Halfspace_translation
|
[189, 1]
|
[193, 7]
|
intro y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
⊢ ∀ (y : E), y ∈ ↑(Halfspace_translation x H_) ↔ y - x ∈ ↑H_
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
y : E
⊢ y ∈ ↑(Halfspace_translation x H_) ↔ y - x ∈ ↑H_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
mem_Halfspace_translation
|
[189, 1]
|
[193, 7]
|
rw [Halfspace_translation.S, Set.image_add_right, Set.mem_preimage, sub_eq_add_neg]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H_ : Halfspace E
y : E
⊢ y ∈ ↑(Halfspace_translation x H_) ↔ y - x ∈ ↑H_
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
intro H1 H2 h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ Function.Injective fun x_1 => Halfspace_translation x x_1
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
⊢ H1 = H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
rw [SetLike.ext_iff]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
⊢ H1 = H2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
⊢ ∀ (x : E), x ∈ H1 ↔ x ∈ H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
intro y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
⊢ ∀ (x : E), x ∈ H1 ↔ x ∈ H2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
y : E
⊢ y ∈ H1 ↔ y ∈ H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
rw [SetLike.ext_iff] at h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : (fun x_1 => Halfspace_translation x x_1) H1 = (fun x_1 => Halfspace_translation x x_1) H2
y : E
⊢ y ∈ H1 ↔ y ∈ H2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : ∀ (x_1 : E), x_1 ∈ (fun x_2 => Halfspace_translation x x_2) H1 ↔ x_1 ∈ (fun x_2 => Halfspace_translation x x_2) H2
y : E
⊢ y ∈ H1 ↔ y ∈ H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
specialize h (y + x)
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
h : ∀ (x_1 : E), x_1 ∈ (fun x_2 => Halfspace_translation x x_2) H1 ↔ x_1 ∈ (fun x_2 => Halfspace_translation x x_2) H2
y : E
⊢ y ∈ H1 ↔ y ∈ H2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
y : E
h : y + x ∈ (fun x_1 => Halfspace_translation x x_1) H1 ↔ y + x ∈ (fun x_1 => Halfspace_translation x x_1) H2
⊢ y ∈ H1 ↔ y ∈ H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
rw [← SetLike.mem_coe, ← SetLike.mem_coe, mem_Halfspace_translation, mem_Halfspace_translation, add_sub_cancel] at h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
y : E
h : y + x ∈ (fun x_1 => Halfspace_translation x x_1) H1 ↔ y + x ∈ (fun x_1 => Halfspace_translation x x_1) H2
⊢ y ∈ H1 ↔ y ∈ H2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
y : E
h : y ∈ ↑H1 ↔ y ∈ ↑H2
⊢ y ∈ H1 ↔ y ∈ H2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace_translation.injective
|
[195, 1]
|
[203, 10]
|
exact h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
H1 H2 : Halfspace E
y : E
h : y ∈ ↑H1 ↔ y ∈ ↑H2
⊢ y ∈ H1 ↔ y ∈ H2
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
have := ContinuousLinearMap.frontier_preimage Hi_.f.1 (unitSphereDual_surj Hi_.f) (Set.Iic Hi_.α)
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' Set.Iic Hi_.α) = ⇑↑Hi_.f ⁻¹' frontier (Set.Iic Hi_.α)
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
simp only [ne_eq, LinearMap.coe_toContinuousLinearMap', Set.nonempty_Ioi, frontier_Iic'] at this
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' Set.Iic Hi_.α) = ⇑↑Hi_.f ⁻¹' frontier (Set.Iic Hi_.α)
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' Set.Iic Hi_.α) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
change frontier ( Hi_.f.1 ⁻¹' {x | x ≤ Hi_.α}) = Hi_.f.1 ⁻¹' {Hi_.α} at this
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' Set.Iic Hi_.α) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α}) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
rw [Hi_.h, this]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α}) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ frontier ↑Hi_ = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α}) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ ⇑↑Hi_.f ⁻¹' {Hi_.α} = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
clear this
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
this : frontier (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α}) = ⇑↑Hi_.f ⁻¹' {Hi_.α}
⊢ ⇑↑Hi_.f ⁻¹' {Hi_.α} = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ ⇑↑Hi_.f ⁻¹' {Hi_.α} = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
unfold Set.preimage
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ ⇑↑Hi_.f ⁻¹' {Hi_.α} = {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ {x | ↑Hi_.f x ∈ {Hi_.α}} = {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
frontierHalfspace_Hyperplane
|
[205, 1]
|
[213, 7]
|
simp only [ne_eq, Set.mem_singleton_iff]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ {x | ↑Hi_.f x ∈ {Hi_.α}} = {x | ↑Hi_.f x = Hi_.α}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_convex
|
[215, 1]
|
[218, 7]
|
exact @convex_hyperplane ℝ E ℝ _ _ _ _ _ _ Hi_.f.1 (LinearMap.isLinear Hi_.f.1) Hi_.α
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
Hi_ : Halfspace E
⊢ Convex ℝ {x | ↑Hi_.f x = Hi_.α}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
intro s hs a ha
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
⊢ ∀ (s : Fin n → E),
Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α} →
∀ (a : Fin n → ℝ),
Finset.sum Finset.univ a = 1 → (Finset.affineCombination ℝ Finset.univ s) a ∈ {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
⊢ (Finset.affineCombination ℝ Finset.univ s) a ∈ {x | ↑Hi_.f x = Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
rw [Finset.affineCombination_eq_linear_combination _ _ _ ha, Set.mem_setOf, map_sum]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
⊢ (Finset.affineCombination ℝ Finset.univ s) a ∈ {x | ↑Hi_.f x = Hi_.α}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
⊢ (Finset.sum Finset.univ fun x => ↑Hi_.f (a x • s x)) = Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
have hg : (fun i => Hi_.f.1 (a i • s i)) = fun i => a i * Hi_.α := by
ext i
rw [Set.range_subset_iff] at hs
specialize hs i
rw [Set.mem_setOf] at hs
rw [ContinuousLinearMap.map_smulₛₗ, smul_eq_mul, RingHom.id_apply, hs]
done
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
⊢ (Finset.sum Finset.univ fun x => ↑Hi_.f (a x • s x)) = Hi_.α
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
hg : (fun i => ↑Hi_.f (a i • s i)) = fun i => a i * Hi_.α
⊢ (Finset.sum Finset.univ fun x => ↑Hi_.f (a x • s x)) = Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
rw [hg, ←Finset.sum_mul, ha, one_mul]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
hg : (fun i => ↑Hi_.f (a i • s i)) = fun i => a i * Hi_.α
⊢ (Finset.sum Finset.univ fun x => ↑Hi_.f (a x • s x)) = Hi_.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
ext i
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
⊢ (fun i => ↑Hi_.f (a i • s i)) = fun i => a i * Hi_.α
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
rw [Set.range_subset_iff] at hs
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : Set.range s ⊆ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : ∀ (y : Fin n), s y ∈ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
specialize hs i
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
hs : ∀ (y : Fin n), s y ∈ {x | ↑Hi_.f x = Hi_.α}
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
hs : s i ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
rw [Set.mem_setOf] at hs
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
hs : s i ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
hs : ↑Hi_.f (s i) = Hi_.α
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Hyperplane_affineClosed
|
[220, 1]
|
[234, 7]
|
rw [ContinuousLinearMap.map_smulₛₗ, smul_eq_mul, RingHom.id_apply, hs]
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
Hi_ : Halfspace E
s : Fin n → E
a : Fin n → ℝ
ha : Finset.sum Finset.univ a = 1
i : Fin n
hs : ↑Hi_.f (s i) = Hi_.α
⊢ ↑Hi_.f (a i • s i) = a i * Hi_.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_raw
|
[236, 1]
|
[240, 45]
|
rcases H_' with ⟨ ⟨ f, hf ⟩, C ⟩
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ∃ H_, ((∀ (x : ↥p), ↑H_.f ↑x = ↑H_'.f x) ∧ ‖↑H_.f‖ = ‖↑H_'.f‖) ∧ H_.α = H_'.α
|
case mk.mk
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
C : ℝ
f : NormedSpace.Dual ℝ ↥p
hf : ‖f‖ = 1
⊢ ∃ H_,
((∀ (x : ↥p), ↑H_.f ↑x = ↑{ f := { val := f, property := hf }, α := C }.f x) ∧
‖↑H_.f‖ = ‖↑{ f := { val := f, property := hf }, α := C }.f‖) ∧
H_.α = { f := { val := f, property := hf }, α := C }.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_raw
|
[236, 1]
|
[240, 45]
|
choose g hg using Real.exists_extension_norm_eq p f
|
case mk.mk
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
C : ℝ
f : NormedSpace.Dual ℝ ↥p
hf : ‖f‖ = 1
⊢ ∃ H_,
((∀ (x : ↥p), ↑H_.f ↑x = ↑{ f := { val := f, property := hf }, α := C }.f x) ∧
‖↑H_.f‖ = ‖↑{ f := { val := f, property := hf }, α := C }.f‖) ∧
H_.α = { f := { val := f, property := hf }, α := C }.α
|
case mk.mk
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
C : ℝ
f : NormedSpace.Dual ℝ ↥p
hf : ‖f‖ = 1
g : E →L[ℝ] ℝ
hg : (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖
⊢ ∃ H_,
((∀ (x : ↥p), ↑H_.f ↑x = ↑{ f := { val := f, property := hf }, α := C }.f x) ∧
‖↑H_.f‖ = ‖↑{ f := { val := f, property := hf }, α := C }.f‖) ∧
H_.α = { f := { val := f, property := hf }, α := C }.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_raw
|
[236, 1]
|
[240, 45]
|
exact ⟨ ⟨ ⟨ g, hg.2 ▸ hf ⟩, C ⟩, hg, rfl ⟩
|
case mk.mk
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
C : ℝ
f : NormedSpace.Dual ℝ ↥p
hf : ‖f‖ = 1
g : E →L[ℝ] ℝ
hg : (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖
⊢ ∃ H_,
((∀ (x : ↥p), ↑H_.f ↑x = ↑{ f := { val := f, property := hf }, α := C }.f x) ∧
‖↑H_.f‖ = ‖↑{ f := { val := f, property := hf }, α := C }.f‖) ∧
H_.α = { f := { val := f, property := hf }, α := C }.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_f
|
[247, 1]
|
[250, 62]
|
unfold val
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ∀ (x : ↥p), ↑((fun H_ x => H_) (Classical.choose ⋯) ⋯).f ↑x = ↑H_'.f x
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_f
|
[247, 1]
|
[250, 62]
|
exact (Classical.choose_spec (Halfspace.val_raw p H_')).1.1
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ∀ (x : ↥p), ↑((fun H_ x => H_) (Classical.choose ⋯) ⋯).f ↑x = ↑H_'.f x
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_C
|
[252, 1]
|
[255, 60]
|
unfold val
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ (val p H_').α = H_'.α
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ((fun H_ x => H_) (Classical.choose ⋯) ⋯).α = H_'.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_C
|
[252, 1]
|
[255, 60]
|
exact (Classical.choose_spec (Halfspace.val_raw p H_')).2
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ((fun H_ x => H_) (Classical.choose ⋯) ⋯).α = H_'.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
have := Halfspace.val_f p H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ ↑(val p H_') ∩ ↑p = Subtype.val '' ↑H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
⊢ ↑(val p H_') ∩ ↑p = Subtype.val '' ↑H_'
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
apply subset_antisymm <;> intro x <;> rw [Set.mem_inter_iff, Set.mem_image]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
⊢ ↑(val p H_') ∩ ↑p = Subtype.val '' ↑H_'
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ x ∈ ↑(val p H_') ∧ x ∈ ↑p → ∃ x_1 ∈ ↑H_', ↑x_1 = x
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ (∃ x_1 ∈ ↑H_', ↑x_1 = x) → x ∈ ↑(val p H_') ∧ x ∈ ↑p
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
rintro ⟨ hxH_', hxp ⟩
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ x ∈ ↑(val p H_') ∧ x ∈ ↑p → ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
refine ⟨ ⟨ x, hxp ⟩, ?_, rfl ⟩
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ { val := x, property := hxp } ∈ ↑H_'
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
rw [Halfspace_mem, ← (this ⟨ x, hxp ⟩), ← Halfspace.val_C p H_']
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ { val := x, property := hxp } ∈ ↑H_'
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ↑(val p H_').f ↑{ val := x, property := hxp } ≤ (val p H_').α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
exact hxH_'
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ↑(val p H_').f ↑{ val := x, property := hxp } ≤ (val p H_').α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
rintro ⟨ ⟨ x', hx'p ⟩, hx'H_', rfl ⟩
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ (∃ x_1 ∈ ↑H_', ↑x_1 = x) → x ∈ ↑(val p H_') ∧ x ∈ ↑p
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_') ∧ ↑{ val := x', property := hx'p } ∈ ↑p
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
refine ⟨ ?_, hx'p ⟩
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_') ∧ ↑{ val := x', property := hx'p } ∈ ↑p
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
rw [Halfspace_mem, ← (this ⟨ x', hx'p ⟩), ← Halfspace.val_C p H_'] at hx'H_'
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : ↑(val p H_').f ↑{ val := x', property := hx'p } ≤ (val p H_').α
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq
|
[257, 1]
|
[271, 7]
|
exact hx'H_'
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : ↑(val p H_').f ↑{ val := x', property := hx'p } ≤ (val p H_').α
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
intro H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
⊢ ∀ (H_' : Halfspace ↥p), (fun H_ => ↑(val p H_) ∩ ↑p) H_' = (fun H_ => Subtype.val '' H_) ↑H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ (fun H_ => ↑(val p H_) ∩ ↑p) H_' = (fun H_ => Subtype.val '' H_) ↑H_'
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
have := Halfspace.val_f p H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
⊢ (fun H_ => ↑(val p H_) ∩ ↑p) H_' = (fun H_ => Subtype.val '' H_) ↑H_'
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
⊢ (fun H_ => ↑(val p H_) ∩ ↑p) H_' = (fun H_ => Subtype.val '' H_) ↑H_'
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
apply subset_antisymm <;> intro x <;> rw [Set.mem_inter_iff, Set.mem_image]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
⊢ (fun H_ => ↑(val p H_) ∩ ↑p) H_' = (fun H_ => Subtype.val '' H_) ↑H_'
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ x ∈ ↑(val p H_') ∧ x ∈ ↑p → ∃ x_1 ∈ ↑H_', ↑x_1 = x
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ (∃ x_1 ∈ ↑H_', ↑x_1 = x) → x ∈ ↑(val p H_') ∧ x ∈ ↑p
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
rintro ⟨ hxH_', hxp ⟩
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ x ∈ ↑(val p H_') ∧ x ∈ ↑p → ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
refine ⟨ ⟨ x, hxp ⟩, ?_, rfl ⟩
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ∃ x_1 ∈ ↑H_', ↑x_1 = x
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ { val := x, property := hxp } ∈ ↑H_'
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
rw [Halfspace_mem, ← (this ⟨ x, hxp ⟩), ← Halfspace.val_C p H_']
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ { val := x, property := hxp } ∈ ↑H_'
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ↑(val p H_').f ↑{ val := x, property := hxp } ≤ (val p H_').α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
exact hxH_'
|
case a.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
hxH_' : x ∈ ↑(val p H_')
hxp : x ∈ ↑p
⊢ ↑(val p H_').f ↑{ val := x, property := hxp } ≤ (val p H_').α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
rintro ⟨ ⟨ x', hx'p ⟩, hx'H_', rfl ⟩
|
case a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x : E
⊢ (∃ x_1 ∈ ↑H_', ↑x_1 = x) → x ∈ ↑(val p H_') ∧ x ∈ ↑p
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_') ∧ ↑{ val := x', property := hx'p } ∈ ↑p
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
refine ⟨ ?_, hx'p ⟩
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_') ∧ ↑{ val := x', property := hx'p } ∈ ↑p
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
rw [Halfspace_mem, ← (this ⟨ x', hx'p ⟩), ← Halfspace.val_C p H_'] at hx'H_'
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : { val := x', property := hx'p } ∈ ↑H_'
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : ↑(val p H_').f ↑{ val := x', property := hx'p } ≤ (val p H_').α
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Halfspace.lean
|
Halfspace.val_eq'
|
[273, 1]
|
[288, 7]
|
exact hx'H_'
|
case a.intro.mk.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
p : Subspace ℝ E
inst✝ : CompleteSpace ↥p
H_' : Halfspace ↥p
this : ∀ (x : ↥p), ↑(val p H_').f ↑x = ↑H_'.f x
x' : E
hx'p : x' ∈ p
hx'H_' : ↑(val p H_').f ↑{ val := x', property := hx'p } ≤ (val p H_').α
⊢ ↑{ val := x', property := hx'p } ∈ ↑(val p H_')
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Convex_cutSpace
|
[7, 1]
|
[10, 29]
|
apply convex_sInter
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ Convex ℝ (cutSpace H_)
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ ∀ s ∈ SetLike.coe '' H_, Convex ℝ s
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Convex_cutSpace
|
[7, 1]
|
[10, 29]
|
rintro _ ⟨ Hi_, _, rfl ⟩
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ ∀ s ∈ SetLike.coe '' H_, Convex ℝ s
|
case h.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ Convex ℝ ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Convex_cutSpace
|
[7, 1]
|
[10, 29]
|
exact Halfspace_convex Hi_
|
case h.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ Convex ℝ ↑Hi_
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Closed_cutSpace
|
[12, 1]
|
[18, 21]
|
apply isClosed_sInter
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ IsClosed (cutSpace H_)
|
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ ∀ t ∈ SetLike.coe '' H_, IsClosed t
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Closed_cutSpace
|
[12, 1]
|
[18, 21]
|
rintro _ ⟨ Hi_, _, rfl ⟩
|
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
⊢ ∀ t ∈ SetLike.coe '' H_, IsClosed t
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Closed_cutSpace
|
[12, 1]
|
[18, 21]
|
rw [Hi_.h]
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed ↑Hi_
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α})
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Closed_cutSpace
|
[12, 1]
|
[18, 21]
|
apply IsClosed.preimage (Hi_.f.1.cont)
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed (⇑↑Hi_.f ⁻¹' {x | x ≤ Hi_.α})
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed {x | x ≤ Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Closed_cutSpace
|
[12, 1]
|
[18, 21]
|
exact isClosed_Iic
|
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ IsClosed {x | x ≤ Hi_.α}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
constructor <;> intro h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
⊢ x ∈ cutSpace H_ ↔ ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : x ∈ cutSpace H_
⊢ ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
⊢ x ∈ cutSpace H_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
intro Hi HiH
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : x ∈ cutSpace H_
⊢ ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : x ∈ cutSpace H_
Hi : Halfspace E
HiH : Hi ∈ H_
⊢ ↑Hi.f x ≤ Hi.α
|
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