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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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