url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
unfold cutSpace at h
|
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
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : x ∈ ⋂₀ (SetLike.coe '' H_)
Hi : Halfspace E
HiH : Hi ∈ H_
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
rw [Set.mem_sInter] at h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : x ∈ ⋂₀ (SetLike.coe '' H_)
Hi : Halfspace E
HiH : 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 : ∀ t ∈ SetLike.coe '' H_, x ∈ t
Hi : Halfspace E
HiH : Hi ∈ H_
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
specialize h Hi ⟨ Hi, HiH, rfl ⟩
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
h : ∀ t ∈ SetLike.coe '' H_, x ∈ t
Hi : Halfspace E
HiH : 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
Hi : Halfspace E
HiH : Hi ∈ H_
h : x ∈ ↑Hi
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
rw [Halfspace_mem] at h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi : Halfspace E
HiH : Hi ∈ H_
h : x ∈ ↑Hi
⊢ ↑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
Hi : Halfspace E
HiH : Hi ∈ H_
h : ↑Hi.f x ≤ Hi.α
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
exact h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi : Halfspace E
HiH : Hi ∈ H_
h : ↑Hi.f x ≤ Hi.α
⊢ ↑Hi.f x ≤ Hi.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
unfold cutSpace
|
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_
|
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 ∈ ⋂₀ (SetLike.coe '' H_)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
rw [Set.mem_sInter]
|
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 ∈ ⋂₀ (SetLike.coe '' H_)
|
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
⊢ ∀ t ∈ SetLike.coe '' H_, x ∈ t
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
rintro _ ⟨ Hi_, hHi_, rfl ⟩
|
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
⊢ ∀ t ∈ SetLike.coe '' H_, x ∈ t
|
case mpr.intro.intro
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
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x ∈ ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
specialize h Hi_ hHi_
|
case mpr.intro.intro
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
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x ∈ ↑Hi_
|
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : ↑Hi_.f x ≤ Hi_.α
⊢ x ∈ ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
rw [Halfspace_mem]
|
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : ↑Hi_.f x ≤ Hi_.α
⊢ x ∈ ↑Hi_
|
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : ↑Hi_.f x ≤ Hi_.α
⊢ ↑Hi_.f x ≤ Hi_.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
mem_cutSpace
|
[20, 1]
|
[38, 9]
|
exact h
|
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
h : ↑Hi_.f x ≤ Hi_.α
⊢ ↑Hi_.f x ≤ Hi_.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
rcases h with ⟨ x, hx ⟩
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
h : ∃ x, x ≠ 0
⊢ ∃ H_, cutSpace H_ = ∅
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
⊢ ∃ H_, cutSpace H_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
let xhat := (norm x)⁻¹ • x
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
⊢ ∃ H_, cutSpace H_ = ∅
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
⊢ ∃ H_, cutSpace H_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
let fval : NormedSpace.Dual ℝ E := InnerProductSpace.toDualMap ℝ _ xhat
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
⊢ ∃ H_, cutSpace H_ = ∅
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ∃ H_, cutSpace H_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
let f : {f : (NormedSpace.Dual ℝ E) // norm f = 1} := ⟨ fval , (by
change norm (innerSL ℝ ((norm x)⁻¹ • x)) = 1
have := @norm_smul_inv_norm ℝ _ E _ _ x hx
rw [IsROrC.ofReal_real_eq_id, id_eq] at this
rw [innerSL_apply_norm, this]
done
) ⟩
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ∃ H_, cutSpace H_ = ∅
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
⊢ ∃ H_, cutSpace H_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
refine ⟨ {Halfspace.mk f (-1), Halfspace.mk (-f) (-1)} , ?_ ⟩
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
⊢ ∃ H_, cutSpace H_ = ∅
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
⊢ cutSpace {{ f := f, α := -1 }, { f := -f, α := -1 }} = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
ext x
|
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
⊢ cutSpace {{ f := f, α := -1 }, { f := -f, α := -1 }} = ∅
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
⊢ x ∈ cutSpace {{ f := f, α := -1 }, { f := -f, α := -1 }} ↔ x ∈ ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
rw [Set.mem_empty_iff_false, iff_false, mem_cutSpace]
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
⊢ x ∈ cutSpace {{ f := f, α := -1 }, { f := -f, α := -1 }} ↔ x ∈ ∅
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
⊢ ¬∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
intro h
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
⊢ ¬∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
have h1 := h (Halfspace.mk f (-1)) (by simp)
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
⊢ False
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
have h2 := h (Halfspace.mk (-f) (-1)) (by simp)
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
⊢ False
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
h2 : ↑{ f := -f, α := -1 }.f x ≤ { f := -f, α := -1 }.α
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg] at h2
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
h2 : ↑{ f := -f, α := -1 }.f x ≤ { f := -f, α := -1 }.α
⊢ False
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
h2 : 1 ≤ ↑f x
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
change f.1 x ≤ -1 at h1
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
h2 : 1 ≤ ↑f x
⊢ False
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h2 : 1 ≤ ↑f x
h1 : ↑f x ≤ -1
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
linarith
|
case intro.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h2 : 1 ≤ ↑f x
h1 : ↑f x ≤ -1
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
change norm (innerSL ℝ ((norm x)⁻¹ • x)) = 1
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ‖fval‖ = 1
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
have := @norm_smul_inv_norm ℝ _ E _ _ x hx
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
this : ‖(↑‖x‖)⁻¹ • x‖ = 1
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
rw [IsROrC.ofReal_real_eq_id, id_eq] at this
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
this : ‖(↑‖x‖)⁻¹ • x‖ = 1
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
this : ‖‖x‖⁻¹ • x‖ = 1
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
rw [innerSL_apply_norm, this]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
this : ‖‖x‖⁻¹ • x‖ = 1
⊢ ‖(innerSL ℝ) (‖x‖⁻¹ • x)‖ = 1
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
simp
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
⊢ { f := f, α := -1 } ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
empty_cutSpace
|
[40, 1]
|
[61, 7]
|
simp
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x✝ : E
hx : x✝ ≠ 0
xhat : E := ‖x✝‖⁻¹ • x✝
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
f : { f // ‖f‖ = 1 } := { val := fval, property := ⋯ }
x : E
h : ∀ Hi ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := -1 }.f x ≤ { f := f, α := -1 }.α
⊢ { f := -f, α := -1 } ∈ {{ f := f, α := -1 }, { f := -f, α := -1 }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
refine ⟨ {Halfspace.mk f c, Halfspace.mk (-f) (-c)}, ?_ ⟩
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ ∃ H_, cutSpace H_ = {x | ↑f x = c}
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ cutSpace {{ f := f, α := c }, { f := -f, α := -c }} = {x | ↑f x = c}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
ext x
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
⊢ cutSpace {{ f := f, α := c }, { f := -f, α := -c }} = {x | ↑f x = c}
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ x ∈ cutSpace {{ f := f, α := c }, { f := -f, α := -c }} ↔ x ∈ {x | ↑f x = c}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
rw [mem_cutSpace, Set.mem_setOf]
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ x ∈ cutSpace {{ f := f, α := c }, { f := -f, α := -c }} ↔ x ∈ {x | ↑f x = c}
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) ↔ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
constructor
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) ↔ ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) → ↑f x = c
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ ↑f x = c → ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
intro h
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ (∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α) → ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
have h1 := h (Halfspace.mk f c) (by simp)
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
have h2 := h (Halfspace.mk (-f) (-c)) (by simp)
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
⊢ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg] at h2
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
⊢ ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : c ≤ ↑f x
⊢ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
change f.1 x ≤ c at h1
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
h2 : c ≤ ↑f x
⊢ ↑f x = c
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h2 : c ≤ ↑f x
h1 : ↑f x ≤ c
⊢ ↑f x = c
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
exact le_antisymm h1 h2
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h2 : c ≤ ↑f x
h1 : ↑f x ≤ c
⊢ ↑f x = c
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
simp
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
⊢ { f := f, α := c } ∈ {{ f := f, α := c }, { f := -f, α := -c }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
simp
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
h1 : ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
⊢ { f := -f, α := -c } ∈ {{ f := f, α := c }, { f := -f, α := -c }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
intro h Hi hHi
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
⊢ ↑f x = c → ∀ Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}, ↑Hi.f x ≤ Hi.α
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.mem_insert_iff] at hHi
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi ∈ {{ f := f, α := c }, { f := -f, α := -c }}
⊢ ↑Hi.f x ≤ Hi.α
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi = { f := f, α := c } ∨ Hi = { f := -f, α := -c }
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
rcases hHi with rfl | rfl
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
Hi : Halfspace E
hHi : Hi = { f := f, α := c } ∨ Hi = { f := -f, α := -c }
⊢ ↑Hi.f x ≤ Hi.α
|
case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
exact le_of_eq h
|
case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := f, α := c }.f x ≤ { f := f, α := c }.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg]
|
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ ↑{ f := -f, α := -c }.f x ≤ { f := -f, α := -c }.α
|
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ c ≤ ↑f x
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
hyperplane_cutSpace
|
[63, 1]
|
[85, 7]
|
exact le_of_eq h.symm
|
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : { f // ‖f‖ = 1 }
c : ℝ
x : E
h : ↑f x = c
⊢ c ≤ ↑f x
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
ext x
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
⊢ cutSpace (H_1 ∪ H_2) = cutSpace H_1 ∩ cutSpace H_2
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ x ∈ cutSpace (H_1 ∪ H_2) ↔ x ∈ cutSpace H_1 ∩ cutSpace H_2
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
rw [mem_cutSpace, Set.mem_inter_iff, mem_cutSpace, mem_cutSpace]
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ x ∈ cutSpace (H_1 ∪ H_2) ↔ x ∈ cutSpace H_1 ∩ cutSpace H_2
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) ↔ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
constructor
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) ↔ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) → (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ ((∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α) → ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
intro h
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ (∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α) → (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
⊢ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
constructor <;> intro Hi_ hH_ <;> exact h Hi_ (by simp only [Set.mem_union, hH_, true_or, or_true])
|
case h.mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
⊢ (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
simp only [Set.mem_union, hH_, true_or, or_true]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
Hi_ : Halfspace E
hH_ : Hi_ ∈ H_2
⊢ Hi_ ∈ H_1 ∪ H_2
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
intro h Hi hHi
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
⊢ ((∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α) → ∀ Hi ∈ H_1 ∪ H_2, ↑Hi.f x ≤ Hi.α
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∪ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
rw [Set.mem_union] at hHi
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∪ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∨ Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
rcases hHi with hHi | hHi
|
case h.mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1 ∨ Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1
⊢ ↑Hi.f x ≤ Hi.α
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
exact h.1 Hi hHi
|
case h.mpr.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_1
⊢ ↑Hi.f x ≤ Hi.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
inter_cutSpace
|
[87, 1]
|
[103, 7]
|
exact h.2 Hi hHi
|
case h.mpr.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_1 H_2 : Set (Halfspace E)
x : E
h : (∀ Hi ∈ H_1, ↑Hi.f x ≤ Hi.α) ∧ ∀ Hi ∈ H_2, ↑Hi.f x ≤ Hi.α
Hi : Halfspace E
hHi : Hi ∈ H_2
⊢ ↑Hi.f x ≤ Hi.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane.Finite
|
[109, 1]
|
[111, 107]
|
unfold orthoHyperplane
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
⊢ Set.Finite (orthoHyperplane x)
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
⊢ Set.Finite {{ f := pointDualLin x, α := 0 }, { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane.Finite
|
[109, 1]
|
[111, 107]
|
simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, and_true, Set.finite_singleton, Set.Finite.insert]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
⊢ Set.Finite {{ f := pointDualLin x, α := 0 }, { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
intro y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
⊢ ∀ (y : E), y ∈ cutSpace (orthoHyperplane x) ↔ ⟪↑x, y⟫_ℝ = 0
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (orthoHyperplane x) ↔ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
rw [mem_cutSpace]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (orthoHyperplane x) ↔ ⟪↑x, y⟫_ℝ = 0
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ (∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α) ↔ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
constructor
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ (∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α) ↔ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ (∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α) → ⟪↑x, y⟫_ℝ = 0
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ ⟪↑x, y⟫_ℝ = 0 → ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
intro h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ (∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α) → ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
have h1 := h (Halfspace.mk (pointDualLin x) 0)
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 :
{ f := pointDualLin x, α := 0 } ∈ orthoHyperplane x →
↑{ f := pointDualLin x, α := 0 }.f y ≤ { f := pointDualLin x, α := 0 }.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
simp only [pointDualLin, ne_eq, map_inv₀, IsROrC.conj_to_real, orthoHyperplane, Set.mem_singleton_iff,
Halfspace.mk.injEq, and_true, Set.mem_insert_iff, true_or, forall_true_left, InnerProductSpace.toDual_apply,
inner_smul_left] at h1
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 :
{ f := pointDualLin x, α := 0 } ∈ orthoHyperplane x →
↑{ f := pointDualLin x, α := 0 }.f y ≤ { f := pointDualLin x, α := 0 }.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
have h2 := h (Halfspace.mk (pointDualLin ⟨ -x.1, by rw [neg_ne_zero]; exact x.2 ⟩) 0)
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 :
{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 } ∈ orthoHyperplane x →
↑{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }.f y ≤
{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
simp only [pointDualLin, ne_eq, norm_neg, smul_neg, map_inv₀, IsROrC.conj_to_real, orthoHyperplane,
Set.mem_singleton_iff, Halfspace.mk.injEq, Subtype.mk.injEq, and_true, Set.mem_insert_iff,
or_true, forall_true_left, InnerProductSpace.toDual_apply, inner_neg_left, inner_smul_left] at h2
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 :
{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 } ∈ orthoHyperplane x →
↑{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }.f y ≤
{ f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }.α
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : -(‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ) ≤ 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
rw [neg_le, neg_zero] at h2
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : -(‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ) ≤ 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
have := le_antisymm h1 h2
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
this : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
rw [mul_eq_zero] at this
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
this : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
this : ‖↑x‖⁻¹ = 0 ∨ ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
cases' this with h3 h4
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
this : ‖↑x‖⁻¹ = 0 ∨ ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ‖↑x‖⁻¹ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h4 : ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
rw [neg_ne_zero]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
⊢ -↑x ≠ 0
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
⊢ ↑x ≠ 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
exact x.2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
⊢ ↑x ≠ 0
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
rw [inv_eq_zero, norm_eq_zero] at h3
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ‖↑x‖⁻¹ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ↑x = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
exfalso
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ↑x = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ↑x = 0
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
exact x.2 h3
|
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h3 : ↑x = 0
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
exact h4
|
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
h1 : ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ ≤ 0
h2 : 0 ≤ ‖↑x‖⁻¹ * ⟪↑x, y⟫_ℝ
h4 : ⟪↑x, y⟫_ℝ = 0
⊢ ⟪↑x, y⟫_ℝ = 0
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
intro h H hH
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
⊢ ⟪↑x, y⟫_ℝ = 0 → ∀ Hi ∈ orthoHyperplane x, ↑Hi.f y ≤ Hi.α
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H ∈ orthoHyperplane x
⊢ ↑H.f y ≤ H.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
unfold orthoHyperplane at hH
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H ∈ orthoHyperplane x
⊢ ↑H.f y ≤ H.α
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H ∈ {{ f := pointDualLin x, α := 0 }, { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }}
⊢ ↑H.f y ≤ H.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
simp only [ne_eq, Set.mem_singleton_iff, Halfspace.mk.injEq, and_true, Set.mem_insert_iff] at hH
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H ∈ {{ f := pointDualLin x, α := 0 }, { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }}
⊢ ↑H.f y ≤ H.α
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H = { f := pointDualLin x, α := 0 } ∨ H = { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }
⊢ ↑H.f y ≤ H.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplane_mem
|
[113, 1]
|
[145, 7]
|
cases' hH with H H <;>
simp only [H, pointDualLin, norm_neg, smul_neg, map_inv₀, IsROrC.conj_to_real, InnerProductSpace.toDual_apply,
inner_neg_left, inner_smul_left, neg_le, neg_zero, h, mul_zero, le_refl]
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : { x // x ≠ 0 }
y : E
h : ⟪↑x, y⟫_ℝ = 0
H : Halfspace E
hH : H = { f := pointDualLin x, α := 0 } ∨ H = { f := pointDualLin { val := -↑x, property := ⋯ }, α := 0 }
⊢ ↑H.f y ≤ H.α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
cutSpace_sUnion_orthoHyperplane
|
[147, 1]
|
[150, 81]
|
intro y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
⊢ ∀ (y : E), y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
cutSpace_sUnion_orthoHyperplane
|
[147, 1]
|
[150, 81]
|
unfold cutSpace
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ ⋂₀ (SetLike.coe '' ⋃₀ (orthoHyperplane '' X)) ↔ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ ⋂₀ (SetLike.coe '' ↑i)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
cutSpace_sUnion_orthoHyperplane
|
[147, 1]
|
[150, 81]
|
rw [Set.sUnion_eq_iUnion, Set.image_iUnion, Set.sInter_iUnion, Set.mem_iInter]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ ⋂₀ (SetLike.coe '' ⋃₀ (orthoHyperplane '' X)) ↔ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ ⋂₀ (SetLike.coe '' ↑i)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
intro y
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
⊢ ∀ (y : E), y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
rw [cutSpace_sUnion_orthoHyperplane]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ y ∈ cutSpace (⋃₀ (orthoHyperplane '' X)) ↔ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i) ↔ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
constructor
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i) ↔ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i) → ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0) → ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
intro h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i) → ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
⊢ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
intro x hx
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
⊢ ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
x : { x // x ≠ 0 }
hx : x ∈ X
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
simp at h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
x : { x // x ≠ 0 }
hx : x ∈ X
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
x : { x // x ≠ 0 }
hx : x ∈ X
h :
∀ (a : Set (Halfspace E)) (x : E) (x_1 : ¬x = 0),
{ val := x, property := x_1 } ∈ X → orthoHyperplane { val := x, property := x_1 } = a → y ∈ cutSpace a
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
specialize h (orthoHyperplane x) x.1 x.2 hx rfl
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
x : { x // x ≠ 0 }
hx : x ∈ X
h :
∀ (a : Set (Halfspace E)) (x : E) (x_1 : ¬x = 0),
{ val := x, property := x_1 } ∈ X → orthoHyperplane { val := x, property := x_1 } = a → y ∈ cutSpace a
⊢ ⟪↑x, y⟫_ℝ = 0
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
x : { x // x ≠ 0 }
hx : x ∈ X
h : y ∈ cutSpace (orthoHyperplane x)
⊢ ⟪↑x, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
exact (orthoHyperplane_mem x y).mp h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
x : { x // x ≠ 0 }
hx : x ∈ X
h : y ∈ cutSpace (orthoHyperplane x)
⊢ ⟪↑x, y⟫_ℝ = 0
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
intro h
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
⊢ (∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0) → ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
⊢ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
simp
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
⊢ ∀ (i : ↑(orthoHyperplane '' X)), y ∈ cutSpace ↑i
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
⊢ ∀ (a : Set (Halfspace E)) (x : E) (x_1 : ¬x = 0),
{ val := x, property := x_1 } ∈ X → orthoHyperplane { val := x, property := x_1 } = a → y ∈ cutSpace a
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
rintro a x1 x2 hx rfl
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
⊢ ∀ (a : Set (Halfspace E)) (x : E) (x_1 : ¬x = 0),
{ val := x, property := x_1 } ∈ X → orthoHyperplane { val := x, property := x_1 } = a → y ∈ cutSpace a
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
x1 : E
x2 : ¬x1 = 0
hx : { val := x1, property := x2 } ∈ X
⊢ y ∈ cutSpace (orthoHyperplane { val := x1, property := x2 })
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
rw [orthoHyperplane_mem]
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
x1 : E
x2 : ¬x1 = 0
hx : { val := x1, property := x2 } ∈ X
⊢ y ∈ cutSpace (orthoHyperplane { val := x1, property := x2 })
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
x1 : E
x2 : ¬x1 = 0
hx : { val := x1, property := x2 } ∈ X
⊢ ⟪↑{ val := x1, property := x2 }, y⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
orthoHyperplanes_mem
|
[152, 1]
|
[168, 7]
|
exact h _ hx
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set { x // x ≠ 0 }
y : E
h : ∀ x ∈ X, ⟪↑x, y⟫_ℝ = 0
x1 : E
x2 : ¬x1 = 0
hx : { val := x1, property := x2 } ∈ X
⊢ ⟪↑{ val := x1, property := x2 }, y⟫_ℝ = 0
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Cutspace.lean
|
Submodule_cut_finite
|
[175, 1]
|
[182, 7]
|
apply Set.Finite.sUnion ?_ (fun t ht => by
rcases ht with ⟨ x, _, rfl ⟩
exact orthoHyperplane.Finite _)
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (Submodule_cut p)
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
p : Subspace ℝ E
⊢ Set.Finite (orthoHyperplane '' (Subtype.val ⁻¹' Set.range (Subtype.val ∘ ⇑(FiniteDimensional.finBasis ℝ ↥pᗮ))))
|
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