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/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
refine ⟨ Subtype.val ⁻¹' (S - {s}), ?_, ?_ ⟩
|
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ ∃ S',
Set.Finite S' ∧
(convexHull ℝ) S' = ⇑(Submodule.subtype (AffineSubspace.direction SpanS)) ⁻¹' (convexHull ℝ) (S - {↑s'})
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite (Subtype.val ⁻¹' (S - {s}))
case refine_2
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ (convexHull ℝ) (Subtype.val ⁻¹' (S - {s})) =
⇑(Submodule.subtype (AffineSubspace.direction SpanS)) ⁻¹' (convexHull ℝ) (S - {↑s'})
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
apply Set.Finite.preimage (Set.injOn_of_injective Subtype.val_injective _)
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite (Subtype.val ⁻¹' (S - {s}))
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite (S - {s})
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [Set.sub_singleton]
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite (S - {s})
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite ((fun x => x - s) '' S)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact hS.image _
|
case refine_1
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Finite ((fun x => x - s) '' S)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [← Submodule.coeSubtype, ← LinearMap.coe_toAffineMap, ← AffineMap.preimage_convexHull]
|
case refine_2
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ (convexHull ℝ) (Subtype.val ⁻¹' (S - {s})) =
⇑(Submodule.subtype (AffineSubspace.direction SpanS)) ⁻¹' (convexHull ℝ) (S - {↑s'})
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective (LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS))).toFun
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
all_goals (try rw [AffineMap.toFun_eq_coe])
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective (LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS))).toFun
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
all_goals rw [LinearMap.coe_toAffineMap, Submodule.coeSubtype]
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective Subtype.val
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range Subtype.val
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact Subtype.val_injective
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective Subtype.val
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range Subtype.val
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range Subtype.val
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [Subtype.range_coe_subtype]
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range Subtype.val
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ {x | x ∈ AffineSubspace.direction SpanS}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact AffineSubspace.direction_subset_subset (subset_affineSpan ℝ S)
(subset_trans (Set.singleton_subset_iff.mpr hs) (subset_affineSpan ℝ S))
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ {x | x ∈ AffineSubspace.direction SpanS}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(try rw [AffineMap.toFun_eq_coe])
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective (LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS))).toFun
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
try rw [AffineMap.toFun_eq_coe]
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective (LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS))).toFun
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [AffineMap.toFun_eq_coe]
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective (LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS))).toFun
|
case refine_2.hf
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Function.Injective ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [LinearMap.coe_toAffineMap, Submodule.coeSubtype]
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range ⇑(LinearMap.toAffineMap (Submodule.subtype (AffineSubspace.direction SpanS)))
|
case refine_2.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ S - {s} ⊆ Set.range Subtype.val
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
apply AffineSubspace.direction_nontrivial_of_nontrivial
|
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
⊢ Nontrivial ↥(AffineSubspace.direction SpanS)
|
case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
⊢ Nontrivial ↥SpanS
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact affineSpan_nontrivial ℝ (Set.nontrivial_coe_sort.mpr hSnontrivial)
|
case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
⊢ Nontrivial ↥SpanS
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(apply Set.Nonempty.image)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
apply Set.Nonempty.image
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(try (change Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)))
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
try (change Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1))
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(change Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1))
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
change Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(try apply Set.Nonempty.image)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
try apply Set.Nonempty.image
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
apply Set.Nonempty.image
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(by_contra h)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
by_contra h
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(rw [Set.not_nonempty_iff_eq_empty] at h)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [Set.not_nonempty_iff_eq_empty] at h
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(rw [Hpolytope, h, Set.image_empty, Set.sInter_empty] at hHV)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [Hpolytope, h, Set.image_empty, Set.sInter_empty] at hHV
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
(exact IsCompact.ne_univ (Compact_Vpolytope hS'Fin) hHV.symm)
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact IsCompact.ne_univ (Compact_Vpolytope hS'Fin) hHV.symm
|
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
rw [Set.not_nontrivial_iff] at hStrivial
|
case left.inr
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : ¬Set.Nontrivial S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
|
case left.inr
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact Hpolytope_of_Vpolytope_subsingleton _ hStrivial
|
case left.inr
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/MainTheorem.lean
|
MainTheoremOfPolytopes
|
[586, 1]
|
[686, 7]
|
exact Vpolytope_of_Hpolytope
|
case right
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
⊢ ∀ {H_ : Set (Halfspace E)} (hH_ : Set.Finite H_),
IsCompact (Hpolytope hH_) → ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Closed_Vpolytope
|
[14, 1]
|
[15, 72]
|
exact Set.Finite.isClosed_convexHull hS
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
S : Set E
hS : Set.Finite S
⊢ IsClosed (Vpolytope hS)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Convex_Hpolytope
|
[24, 1]
|
[28, 29]
|
apply convex_sInter
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
⊢ Convex ℝ (Hpolytope hH_)
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
⊢ ∀ s ∈ SetLike.coe '' H_, Convex ℝ s
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Convex_Hpolytope
|
[24, 1]
|
[28, 29]
|
rintro _ ⟨ Hi_, _, rfl ⟩
|
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
⊢ ∀ 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)
hH_ : Set.Finite H_
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ Convex ℝ ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Convex_Hpolytope
|
[24, 1]
|
[28, 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)
hH_ : Set.Finite H_
Hi_ : Halfspace E
left✝ : Hi_ ∈ H_
⊢ Convex ℝ ↑Hi_
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Closed_Hpolytope
|
[30, 1]
|
[37, 21]
|
apply isClosed_sInter
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H : Set (Halfspace E)
hH_ : Set.Finite H
⊢ IsClosed (Hpolytope hH_)
|
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H : Set (Halfspace E)
hH_ : Set.Finite H
⊢ ∀ t ∈ SetLike.coe '' H, IsClosed t
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Closed_Hpolytope
|
[30, 1]
|
[37, 21]
|
rintro _ ⟨ Hi_, _, rfl ⟩
|
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H : Set (Halfspace E)
hH_ : Set.Finite H
⊢ ∀ 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)
hH_ : Set.Finite H
Hi_ : Halfspace E
left✝ : Hi_ ∈ H
⊢ IsClosed ↑Hi_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Closed_Hpolytope
|
[30, 1]
|
[37, 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)
hH_ : Set.Finite H
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)
hH_ : Set.Finite H
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/Polytope.lean
|
Closed_Hpolytope
|
[30, 1]
|
[37, 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)
hH_ : Set.Finite H
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)
hH_ : Set.Finite H
Hi_ : Halfspace E
left✝ : Hi_ ∈ H
⊢ IsClosed {x | x ≤ Hi_.α}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Closed_Hpolytope
|
[30, 1]
|
[37, 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)
hH_ : Set.Finite H
Hi_ : Halfspace E
left✝ : Hi_ ∈ H
⊢ IsClosed {x | x ≤ Hi_.α}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Hpolytope_same
|
[39, 1]
|
[42, 6]
|
unfold Hpolytope
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_1 hH_2 : Set.Finite H_
⊢ Hpolytope hH_1 = Hpolytope hH_2
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_1 hH_2 : Set.Finite H_
⊢ ⋂₀ (SetLike.coe '' H_) = ⋂₀ (SetLike.coe '' H_)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
Hpolytope_same
|
[39, 1]
|
[42, 6]
|
rfl
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_1 hH_2 : Set.Finite H_
⊢ ⋂₀ (SetLike.coe '' H_) = ⋂₀ (SetLike.coe '' H_)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 9]
|
constructor <;> intro h
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
⊢ x ∈ Hpolytope hH_ ↔ ∀ 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)
hH_ : Set.Finite H_
x : E
h : x ∈ Hpolytope hH_
⊢ ∀ 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)
hH_ : Set.Finite H_
x : E
h : ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
⊢ x ∈ Hpolytope hH_
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 9]
|
intro Hi HiH
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
h : x ∈ Hpolytope hH_
⊢ ∀ 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)
hH_ : Set.Finite H_
x : E
h : x ∈ Hpolytope hH_
Hi : Halfspace E
HiH : Hi ∈ H_
⊢ ↑Hi.f x ≤ Hi.α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 9]
|
unfold Hpolytope at h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
h : x ∈ Hpolytope hH_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 9]
|
exact h
|
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 9]
|
unfold Hpolytope
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
h : ∀ Hi ∈ H_, ↑Hi.f x ≤ Hi.α
⊢ x ∈ Hpolytope hH_
|
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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)
hH_ : Set.Finite H_
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/Polytope.lean
|
mem_Hpolytope
|
[44, 1]
|
[62, 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)
hH_ : Set.Finite H_
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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
have h := exists_ne (0:E)
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
h : ∃ y, y ≠ 0
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
rcases h with ⟨ x, hx ⟩
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
h : ∃ y, y ≠ 0
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
let xhat := (norm x)⁻¹ • x
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
let fval : NormedSpace.Dual ℝ E := InnerProductSpace.toDualMap ℝ _ xhat
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial E
x : E
hx : x ≠ 0
xhat : E := ‖x‖⁻¹ • x
fval : NormedSpace.Dual ℝ E := (InnerProductSpace.toDualMap ℝ E) xhat
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
refine ⟨ {Halfspace.mk f (-1), Halfspace.mk (-f) (-1)} ,
(by simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.finite_singleton, Set.Finite.insert]) , ?_ ⟩
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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 := ⋯ }
⊢ Hpolytope ⋯ = ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
ext x
|
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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 := ⋯ }
⊢ Hpolytope ⋯ = ∅
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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 ∈ Hpolytope ⋯ ↔ x ∈ ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
rw [Set.mem_empty_iff_false, iff_false, mem_Hpolytope]
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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 ∈ Hpolytope ⋯ ↔ x ∈ ∅
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
intro h
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
have h1 := h (Halfspace.mk f (-1)) (by simp)
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
have h2 := h (Halfspace.mk (-f) (-1)) (by simp)
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
rw [unitSphereDual_neg, ContinuousLinearMap.neg_apply, neg_le, neg_neg] at h2
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
change f.1 x ≤ -1 at h1
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
linarith
|
case intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
change norm (innerSL ℝ ((norm x)⁻¹ • x)) = 1
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
have := @norm_smul_inv_norm ℝ _ E _ _ x hx
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
rw [IsROrC.ofReal_real_eq_id, id_eq] at this
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
rw [innerSL_apply_norm, this]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
simp only [Set.mem_singleton_iff, Halfspace.mk.injEq, Set.finite_singleton, Set.Finite.insert]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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 := ⋯ }
⊢ Set.Finite {{ f := f, α := -1 }, { f := -f, α := -1 }}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
simp
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
empty_Hpolytope
|
[64, 1]
|
[88, 7]
|
simp
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : Nontrivial 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/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
refine ⟨ ⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range (FiniteDimensional.finBasis ℝ E))), ?_, ?_ ⟩
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = {0}
|
case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E))))
case refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Hpolytope ?refine_1 = {0}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
apply Set.Finite.sUnion ?_ (fun t ht => by
rcases ht with ⟨ x, _, rfl ⟩
exact orthoHyperplane.Finite _)
|
case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E))))
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
apply Set.Finite.image
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)))
|
case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
apply Set.Finite.preimage (Set.injOn_of_injective Subtype.val_injective _)
|
case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E))
|
case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (Set.range ⇑(FiniteDimensional.finBasis ℝ E))
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
exact Set.finite_range _
|
case hs
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Finite (Set.range ⇑(FiniteDimensional.finBasis ℝ E))
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
rcases ht with ⟨ x, _, rfl ⟩
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
t : Set (Halfspace E)
ht : t ∈ orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E))
⊢ Set.Finite t
|
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : { x // x ≠ 0 }
left✝ : x ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)
⊢ Set.Finite (orthoHyperplane x)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
exact orthoHyperplane.Finite _
|
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : { x // x ≠ 0 }
left✝ : x ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)
⊢ Set.Finite (orthoHyperplane x)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
ext x
|
case refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
⊢ Hpolytope ⋯ = {0}
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ {0}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
rw [Set.mem_singleton_iff]
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x ∈ {0}
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
change x ∈ cutSpace ( (⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ↑(FiniteDimensional.finBasis ℝ E))))) ↔ x = 0
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ Hpolytope ⋯ ↔ x = 0
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ cutSpace (⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)))) ↔ x = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
rw [orthoHyperplanes_mem]
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ cutSpace (⋃₀ (orthoHyperplane '' (Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)))) ↔ x = 0
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0) ↔ x = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
constructor
|
case refine_2.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0) ↔ x = 0
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0) → x = 0
case refine_2.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x = 0 → ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
intro h
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0) → x = 0
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
⊢ x = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
apply InnerProductSpace.ext_inner_left_basis (FiniteDimensional.finBasis ℝ E)
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
⊢ x = 0
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
⊢ ∀ (i : Fin (FiniteDimensional.finrank ℝ E)),
⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = ⟪(FiniteDimensional.finBasis ℝ E) i, 0⟫_ℝ
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
intro i
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
⊢ ∀ (i : Fin (FiniteDimensional.finrank ℝ E)),
⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = ⟪(FiniteDimensional.finBasis ℝ E) i, 0⟫_ℝ
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
i : Fin (FiniteDimensional.finrank ℝ E)
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = ⟪(FiniteDimensional.finBasis ℝ E) i, 0⟫_ℝ
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
rw [inner_zero_right]
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
i : Fin (FiniteDimensional.finrank ℝ E)
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = ⟪(FiniteDimensional.finBasis ℝ E) i, 0⟫_ℝ
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
i : Fin (FiniteDimensional.finrank ℝ E)
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
simp only [Set.mem_preimage, Set.mem_range, forall_exists_index, Subtype.forall] at h
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
h : ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
i : Fin (FiniteDimensional.finrank ℝ E)
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = 0
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
i : Fin (FiniteDimensional.finrank ℝ E)
h :
∀ (a : E),
a ≠ 0 → ∀ (x_1 : Fin (FiniteDimensional.finrank ℝ E)), (FiniteDimensional.finBasis ℝ E) x_1 = a → ⟪a, x⟫_ℝ = 0
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = 0
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
exact h (FiniteDimensional.finBasis ℝ E i) (Basis.ne_zero (FiniteDimensional.finBasis ℝ E) i) i rfl
|
case refine_2.h.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
i : Fin (FiniteDimensional.finrank ℝ E)
h :
∀ (a : E),
a ≠ 0 → ∀ (x_1 : Fin (FiniteDimensional.finrank ℝ E)), (FiniteDimensional.finBasis ℝ E) x_1 = a → ⟪a, x⟫_ℝ = 0
⊢ ⟪(FiniteDimensional.finBasis ℝ E) i, x⟫_ℝ = 0
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polytope.lean
|
origin_Hpolytope
|
[90, 1]
|
[115, 7]
|
rintro rfl x _
|
case refine_2.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x = 0 → ∀ x_1 ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E), ⟪↑x_1, x⟫_ℝ = 0
|
case refine_2.h.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
x : { x // x ≠ 0 }
a✝ : x ∈ Subtype.val ⁻¹' Set.range ⇑(FiniteDimensional.finBasis ℝ E)
⊢ ⟪↑x, 0⟫_ℝ = 0
|
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