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
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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stringlengths 73
2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | rw [Submodule.mem_orthogonal] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ v ∈ Kᗮ ↔ ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0 | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) ↔ ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ v ∈ Kᗮ ↔ ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | constructor | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) ↔ ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0 | case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) → ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0) → ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) ↔ ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | intro h i | case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) → ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0 | case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0
i : ι
⊢ ⟪↑(b i), v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ u ∈ K, ⟪u, v⟫_𝕜 = 0) → ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | exact h _ (Submodule.coe_mem (b i)) | case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0
i : ι
⊢ ⟪↑(b i), v⟫_𝕜 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0
i : ι
⊢ ⟪↑(b i), v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | intro h x hx | case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0) → ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0 | case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : x ∈ K
⊢ ⟪x, v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
⊢ (∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0) → ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | rw [Basis.mem_submodule_iff b] at hx | case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : x ∈ K
⊢ ⟪x, v⟫_𝕜 = 0 | case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : ∃ c, x = Finsupp.sum c fun i x => x • ↑(b i)
⊢ ⟪x, v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : x ∈ K
⊢ ⟪x, v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | rcases hx with ⟨ a, rfl ⟩ | case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : ∃ c, x = Finsupp.sum c fun i x => x • ↑(b i)
⊢ ⟪x, v⟫_𝕜 = 0 | case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ⟪Finsupp.sum a fun i x => x • ↑(b i), v⟫_𝕜 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
x : E
hx : ∃ c, x = Finsupp.sum c fun i x => x • ↑(b i)
⊢ ⟪x, v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | rw [Finsupp.sum_inner] | case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ⟪Finsupp.sum a fun i x => x • ↑(b i), v⟫_𝕜 = 0 | case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ (Finsupp.sum a fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ⟪Finsupp.sum a fun i x => x • ↑(b i), v⟫_𝕜 = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | apply Finset.sum_eq_zero | case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ (Finsupp.sum a fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) = 0 | case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ∀ x ∈ a.support, (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) x (a x) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ (Finsupp.sum a fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | intro i _ | case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ∀ x ∈ a.support, (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) x (a x) = 0 | case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
i : ι
a✝ : i ∈ a.support
⊢ (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) i (a i) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
⊢ ∀ x ∈ a.support, (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) x (a x) = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Submodule.mem_orthogonal_Basis | [130, 1] | [146, 7] | simp only [h i, smul_eq_mul, mul_zero] | case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
i : ι
a✝ : i ∈ a.support
⊢ (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) i (a i) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
b : Basis ι 𝕜 ↥K
v : E
h : ∀ (i : ι), ⟪↑(b i), v⟫_𝕜 = 0
a : ι →₀ 𝕜
i : ι
a✝ : i ∈ a.support
⊢ (fun i a => (starRingEnd 𝕜) a • ⟪↑(b i), v⟫_𝕜) i (a i) = 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | AffineMap.preimage_convexHull | [148, 1] | [154, 7] | have h1 := Set.image_preimage_eq_of_subset hs | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
⊢ ⇑f ⁻¹' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f ⁻¹' s) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
⊢ ⇑f ⁻¹' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f ⁻¹' s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
⊢ ⇑f ⁻¹' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f ⁻¹' s)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | AffineMap.preimage_convexHull | [148, 1] | [154, 7] | ext x | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
⊢ ⇑f ⁻¹' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f ⁻¹' s) | case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
x : E
⊢ x ∈ ⇑f ⁻¹' (convexHull 𝕜) s ↔ x ∈ (convexHull 𝕜) (⇑f ⁻¹' s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
⊢ ⇑f ⁻¹' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f ⁻¹' s)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | AffineMap.preimage_convexHull | [148, 1] | [154, 7] | rw [Set.mem_preimage, ← Function.Injective.mem_set_image hf, AffineMap.toFun_eq_coe, AffineMap.image_convexHull, h1] | case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
x : E
⊢ x ∈ ⇑f ⁻¹' (convexHull 𝕜) s ↔ x ∈ (convexHull 𝕜) (⇑f ⁻¹' s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set F
f : E →ᵃ[𝕜] F
hf : Function.Injective f.toFun
hs : s ⊆ Set.range ⇑f
h1 : ⇑f '' (⇑f ⁻¹' s) = s
x : E
⊢ x ∈ ⇑f ⁻¹' (convexHull 𝕜) s ↔ x ∈ (convexHull 𝕜) (⇑f ⁻¹' s)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | induction n with
| zero => exact i.elim0
| succ n ih =>
match i with
| 0 => exact List.mem_cons_self _ _
| (mk (Nat.succ m) h) =>
have : m < n := by omega
let m' : Fin n := ⟨m, this⟩
unfold Nat.fin_list_range
apply List.mem_cons_of_mem
simp only [List.mem_map]
use m'
use ih m'
rfl | n : ℕ
i : Fin n
⊢ i ∈ Nat.fin_list_range n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
i : Fin n
⊢ i ∈ Nat.fin_list_range n
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | exact i.elim0 | case zero
i : Fin Nat.zero
⊢ i ∈ Nat.fin_list_range Nat.zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : Fin Nat.zero
⊢ i ∈ Nat.fin_list_range Nat.zero
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | match i with
| 0 => exact List.mem_cons_self _ _
| (mk (Nat.succ m) h) =>
have : m < n := by omega
let m' : Fin n := ⟨m, this⟩
unfold Nat.fin_list_range
apply List.mem_cons_of_mem
simp only [List.mem_map]
use m'
use ih m'
rfl | case succ
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
⊢ i ∈ Nat.fin_list_range (Nat.succ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
⊢ i ∈ Nat.fin_list_range (Nat.succ n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | exact List.mem_cons_self _ _ | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
⊢ 0 ∈ Nat.fin_list_range (Nat.succ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
⊢ 0 ∈ Nat.fin_list_range (Nat.succ n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | have : m < n := by omega | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n) | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | let m' : Fin n := ⟨m, this⟩ | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n) | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | unfold Nat.fin_list_range | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n) | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ 0 :: List.map succ (Nat.fin_list_range n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ Nat.fin_list_range (Nat.succ n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | apply List.mem_cons_of_mem | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ 0 :: List.map succ (Nat.fin_list_range n) | case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ List.map succ (Nat.fin_list_range n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ 0 :: List.map succ (Nat.fin_list_range n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | simp only [List.mem_map] | case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ List.map succ (Nat.fin_list_range n) | case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ ∃ a ∈ Nat.fin_list_range n, succ a = { val := Nat.succ m, isLt := h } | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ { val := Nat.succ m, isLt := h } ∈ List.map succ (Nat.fin_list_range n)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | use m' | case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ ∃ a ∈ Nat.fin_list_range n, succ a = { val := Nat.succ m, isLt := h } | case h
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ m' ∈ Nat.fin_list_range n ∧ succ m' = { val := Nat.succ m, isLt := h } | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ ∃ a ∈ Nat.fin_list_range n, succ a = { val := Nat.succ m, isLt := h }
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | use ih m' | case h
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ m' ∈ Nat.fin_list_range n ∧ succ m' = { val := Nat.succ m, isLt := h } | case right
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ succ m' = { val := Nat.succ m, isLt := h } | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ m' ∈ Nat.fin_list_range n ∧ succ m' = { val := Nat.succ m, isLt := h }
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | rfl | case right
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ succ m' = { val := Nat.succ m, isLt := h } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
this : m < n
m' : Fin n := { val := m, isLt := this }
⊢ succ m' = { val := Nat.succ m, isLt := h }
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Pre.lean | Fin.mem_fin_list_range | [189, 1] | [204, 7] | omega | n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
⊢ m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : ∀ (i : Fin n), i ∈ Nat.fin_list_range n
i : Fin (Nat.succ n)
m : ℕ
h : Nat.succ m < Nat.succ n
⊢ m < n
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual.α | [22, 1] | [23, 43] | rfl | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ (pointDual p).α = ‖↑p‖⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ (pointDual p).α = ‖↑p‖⁻¹
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual.h | [25, 1] | [26, 107] | rfl | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ ↑(pointDual p) = ⇑((InnerProductSpace.toDual ℝ E) (‖↑p‖⁻¹ • ↑p)) ⁻¹' {x | x ≤ ‖↑p‖⁻¹} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ ↑(pointDual p) = ⇑((InnerProductSpace.toDual ℝ E) (‖↑p‖⁻¹ • ↑p)) ⁻¹' {x | x ≤ ‖↑p‖⁻¹}
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | rw [pointDual.h, map_smulₛₗ, map_inv₀, IsROrC.conj_to_real, Set.preimage_setOf_eq,
Set.mem_setOf_eq, map_zero, ← one_div] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 ∈ ↑(pointDual p) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 ≤ 1 / ‖↑p‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 ∈ ↑(pointDual p)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | apply le_of_lt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 ≤ 1 / ‖↑p‖ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 / ‖↑p‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 ≤ 1 / ‖↑p‖
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | rw [div_pos_iff] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 / ‖↑p‖ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖ ∨ 1 < 0 ∧ ‖↑p‖ < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 / ‖↑p‖
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | left | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖ ∨ 1 < 0 ∧ ‖↑p‖ < 0 | case a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖ ∨ 1 < 0 ∧ ‖↑p‖ < 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | exact ⟨ zero_lt_one, by rw [norm_pos_iff]; exact p.2 ⟩ | case a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < 1 ∧ 0 < ‖↑p‖
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | rw [norm_pos_iff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < ‖↑p‖ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ ↑p ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ 0 < ‖↑p‖
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_origin | [28, 1] | [36, 7] | exact p.2 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ ↑p ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
⊢ ↑p ≠ 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_pointDual | [38, 1] | [43, 7] | rw [pointDual.h, Set.mem_preimage, InnerProductSpace.toDual_apply, Set.mem_setOf,
inner_smul_left, IsROrC.conj_to_real, ← mul_le_mul_left (by rw [norm_pos_iff]; exact p.2 : 0 < norm p.1),
← mul_assoc, mul_inv_cancel (norm_ne_zero_iff.mpr p.2), one_mul] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ x ∈ ↑(pointDual p) ↔ ⟪↑p, x⟫_ℝ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ x ∈ ↑(pointDual p) ↔ ⟪↑p, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_pointDual | [38, 1] | [43, 7] | rw [norm_pos_iff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ 0 < ‖↑p‖ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ ↑p ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ 0 < ‖↑p‖
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_pointDual | [38, 1] | [43, 7] | exact p.2 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ ↑p ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p : { p // p ≠ 0 }
x : E
⊢ ↑p ≠ 0
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | pointDual_comm | [45, 1] | [48, 7] | rw [mem_pointDual, mem_pointDual, real_inner_comm] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p q : { p // p ≠ 0 }
⊢ ↑p ∈ ↑(pointDual q) ↔ ↑q ∈ ↑(pointDual p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
p q : { p // p ≠ 0 }
⊢ ↑p ∈ ↑(pointDual q) ↔ ↑q ∈ ↑(pointDual p)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_closed | [54, 1] | [59, 27] | apply isClosed_sInter | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ IsClosed (polarDual X) | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), IsClosed t | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ IsClosed (polarDual X)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_closed | [54, 1] | [59, 27] | intro Hi_s h | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), IsClosed t | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ IsClosed Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), IsClosed t
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_closed | [54, 1] | [59, 27] | rw [Set.mem_image] at h | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ IsClosed Hi_s | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ IsClosed Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ IsClosed Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_closed | [54, 1] | [59, 27] | rcases h with ⟨ Hi_, _, rfl ⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ IsClosed Hi_s | case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ IsClosed ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ IsClosed Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_closed | [54, 1] | [59, 27] | exact Halfspace_closed _ | case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ IsClosed ↑Hi_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ IsClosed ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_convex | [61, 1] | [66, 27] | apply convex_sInter | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ Convex ℝ (polarDual X) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), Convex ℝ s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ Convex ℝ (polarDual X)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_convex | [61, 1] | [66, 27] | intro Hi_s h | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), Convex ℝ s | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ Convex ℝ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ ∀ s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), Convex ℝ s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_convex | [61, 1] | [66, 27] | rw [Set.mem_image] at h | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ Convex ℝ Hi_s | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ Convex ℝ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ Convex ℝ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_convex | [61, 1] | [66, 27] | rcases h with ⟨ Hi_, _, rfl ⟩ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ Convex ℝ Hi_s | case h.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ Convex ℝ ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ Convex ℝ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_convex | [61, 1] | [66, 27] | exact Halfspace_convex _ | case h.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ Convex ℝ ↑Hi_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
left✝ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ Convex ℝ ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | intro Hi_s h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ 0 ∈ polarDual X | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ 0 ∈ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
⊢ 0 ∈ polarDual X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | rw [Set.mem_image] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ 0 ∈ Hi_s | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ 0 ∈ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ 0 ∈ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | rcases h with ⟨ Hi_, h, rfl ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ 0 ∈ Hi_s | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ 0 ∈ ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Hi_s : Set E
h : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ 0 ∈ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | rw [Set.mem_image] at h | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ 0 ∈ ↑Hi_ | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ 0 ∈ ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ 0 ∈ ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | rcases h with ⟨ p, _, rfl ⟩ | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ 0 ∈ ↑Hi_ | case intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
p : { p // p ≠ 0 }
left✝ : p ∈ Subtype.val ⁻¹' X
⊢ 0 ∈ ↑(pointDual p) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
Hi_ : Halfspace E
h : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ 0 ∈ ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_origin | [68, 1] | [75, 27] | exact pointDual_origin p | case intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
p : { p // p ≠ 0 }
left✝ : p ∈ Subtype.val ⁻¹' X
⊢ 0 ∈ ↑(pointDual p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
p : { p // p ≠ 0 }
left✝ : p ∈ Subtype.val ⁻¹' X
⊢ 0 ∈ ↑(pointDual p)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | unfold polarDual | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ polarDual X ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ polarDual X ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [Set.mem_sInter] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | constructor | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) → ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1) → ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) ↔ ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | intro h x hx | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) → ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1 | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
⊢ ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t) → ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | cases' (em (x = 0)) with hx0 hx0 | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
⊢ ⟪x, v⟫_ℝ ≤ 1 | case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
⊢ ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | specialize h (SetLike.coe <| pointDual ⟨ x, hx0 ⟩) ?_ | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1 | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑(pointDual { val := x, property := hx0 }) ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : v ∈ ↑(pointDual { val := x, property := hx0 })
⊢ ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [mem_pointDual] at h | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : v ∈ ↑(pointDual { val := x, property := hx0 })
⊢ ⟪x, v⟫_ℝ ≤ 1 | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : ⟪↑{ val := x, property := hx0 }, v⟫_ℝ ≤ 1
⊢ ⟪x, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : v ∈ ↑(pointDual { val := x, property := hx0 })
⊢ ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | exact h | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : ⟪↑{ val := x, property := hx0 }, v⟫_ℝ ≤ 1
⊢ ⟪x, v⟫_ℝ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v x : E
hx : x ∈ X
hx0 : ¬x = 0
h : ⟪↑{ val := x, property := hx0 }, v⟫_ℝ ≤ 1
⊢ ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [hx0, inner_zero_left] | case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1 | case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ ⟪x, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | exact zero_le_one | case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : x = 0
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | apply Set.mem_image_of_mem | case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑(pointDual { val := x, property := hx0 }) ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)) | case mp.inr.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ pointDual { val := x, property := hx0 } ∈ pointDual '' (Subtype.val ⁻¹' X) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑(pointDual { val := x, property := hx0 }) ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | apply Set.mem_image_of_mem | case mp.inr.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ pointDual { val := x, property := hx0 } ∈ pointDual '' (Subtype.val ⁻¹' X) | case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ { val := x, property := hx0 } ∈ Subtype.val ⁻¹' X | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ pointDual { val := x, property := hx0 } ∈ pointDual '' (Subtype.val ⁻¹' X)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [Set.mem_preimage] | case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ { val := x, property := hx0 } ∈ Subtype.val ⁻¹' X | case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑{ val := x, property := hx0 } ∈ X | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ { val := x, property := hx0 } ∈ Subtype.val ⁻¹' X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | exact hx | case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑{ val := x, property := hx0 } ∈ X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.h.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ↑{ val := x, property := hx0 } ∈ X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | intro h Hi_s hHi_s | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1) → ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ v ∈ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ (∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1) → ∀ t ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X)), v ∈ t
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [Set.mem_image] at hHi_s | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ v ∈ Hi_s | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ v ∈ Hi_s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : Hi_s ∈ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' X))
⊢ v ∈ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rcases hHi_s with ⟨ Hi_, hHi_, rfl ⟩ | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ v ∈ Hi_s | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ v ∈ ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_s : Set E
hHi_s : ∃ x ∈ pointDual '' (Subtype.val ⁻¹' X), ↑x = Hi_s
⊢ v ∈ Hi_s
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [Set.mem_image] at hHi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ v ∈ ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ v ∈ ↑Hi_ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : Hi_ ∈ pointDual '' (Subtype.val ⁻¹' X)
⊢ v ∈ ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rcases hHi_ with ⟨ p, hp, rfl ⟩ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ v ∈ ↑Hi_ | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
⊢ v ∈ ↑(pointDual p) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
Hi_ : Halfspace E
hHi_ : ∃ x ∈ Subtype.val ⁻¹' X, pointDual x = Hi_
⊢ v ∈ ↑Hi_
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | specialize h p.1 hp | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
⊢ v ∈ ↑(pointDual p) | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ v ∈ ↑(pointDual p) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
h : ∀ x ∈ X, ⟪x, v⟫_ℝ ≤ 1
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
⊢ v ∈ ↑(pointDual p)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | rw [mem_pointDual] | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ v ∈ ↑(pointDual p) | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ ⟪↑p, v⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ v ∈ ↑(pointDual p)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual | [77, 1] | [110, 7] | exact h | case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ ⟪↑p, v⟫_ℝ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
p : { p // p ≠ 0 }
hp : p ∈ Subtype.val ⁻¹' X
h : ⟪↑p, v⟫_ℝ ≤ 1
⊢ ⟪↑p, v⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | mem_polarDual' | [112, 1] | [115, 7] | simp_rw [mem_polarDual, real_inner_comm] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ polarDual X ↔ ∀ x ∈ X, ⟪v, x⟫_ℝ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
v : E
⊢ v ∈ polarDual X ↔ ∀ x ∈ X, ⟪v, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | rw [Set.subset_def, Set.subset_def] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ X ⊆ polarDual Y → Y ⊆ polarDual X | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ (∀ x ∈ X, x ∈ polarDual Y) → ∀ x ∈ Y, x ∈ polarDual X | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ X ⊆ polarDual Y → Y ⊆ polarDual X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | intro h y hy | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ (∀ x ∈ X, x ∈ polarDual Y) → ∀ x ∈ Y, x ∈ polarDual X | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ y ∈ polarDual X | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ (∀ x ∈ X, x ∈ polarDual Y) → ∀ x ∈ Y, x ∈ polarDual X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | rw [mem_polarDual] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ y ∈ polarDual X | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ ∀ x ∈ X, ⟪x, y⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ y ∈ polarDual X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | intro x hx | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ ∀ x ∈ X, ⟪x, y⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪x, y⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
⊢ ∀ x ∈ X, ⟪x, y⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | rw [real_inner_comm] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪x, y⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪y, x⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪x, y⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | specialize h x hx | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪y, x⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : x ∈ polarDual Y
⊢ ⟪y, x⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
h : ∀ x ∈ X, x ∈ polarDual Y
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
⊢ ⟪y, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | rw [mem_polarDual] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : x ∈ polarDual Y
⊢ ⟪y, x⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ∀ x_1 ∈ Y, ⟪x_1, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : x ∈ polarDual Y
⊢ ⟪y, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | specialize h y hy | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ∀ x_1 ∈ Y, ⟪x_1, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ⟪y, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ∀ x_1 ∈ Y, ⟪x_1, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm_half | [117, 1] | [128, 7] | exact h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ⟪y, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
y : E
hy : y ∈ Y
x : E
hx : x ∈ X
h : ⟪y, x⟫_ℝ ≤ 1
⊢ ⟪y, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | polarDual_comm | [130, 1] | [133, 7] | constructor <;> exact fun h => polarDual_comm_half _ _ h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ X ⊆ polarDual Y ↔ Y ⊆ polarDual X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X Y : Set E
⊢ X ⊆ polarDual Y ↔ Y ⊆ polarDual X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | apply subset_antisymm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ polarDual (polarDual X) = X | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ polarDual (polarDual X) ⊆ X
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ X ⊆ polarDual (polarDual X) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ polarDual (polarDual X) = X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | intro x hx | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ polarDual (polarDual X) ⊆ X | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∈ polarDual (polarDual X)
⊢ x ∈ X | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ polarDual (polarDual X) ⊆ X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | contrapose! hx | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∈ polarDual (polarDual X)
⊢ x ∈ X | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ x ∉ polarDual (polarDual X) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∈ polarDual (polarDual X)
⊢ x ∈ X
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | rw [mem_polarDual] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ x ∉ polarDual (polarDual X) | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ¬∀ x_1 ∈ polarDual X, ⟪x_1, x⟫_ℝ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ x ∉ polarDual (polarDual X)
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | push_neg | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ¬∀ x_1 ∈ polarDual X, ⟪x_1, x⟫_ℝ ≤ 1 | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ¬∀ x_1 ∈ polarDual X, ⟪x_1, x⟫_ℝ ≤ 1
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | rcases geometric_hahn_banach_point_closed hXcv hXcl hx with ⟨ f, α, h, hX ⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ | case a.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | use (α⁻¹) • (InnerProductSpace.toDual ℝ E).symm f | case a.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f ∈ polarDual X ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ ∃ x_1 ∈ polarDual X, 1 < ⟪x_1, x⟫_ℝ
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | rw [mem_polarDual'] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f ∈ polarDual X ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f ∈ polarDual X ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | have hαneg : 0 < -α := (neg_pos.mpr ((ContinuousLinearMap.map_zero f) ▸ (hX 0 hX0))) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | constructor <;> intros <;> (try apply le_of_lt) <;> rw [real_inner_smul_left,
InnerProductSpace.toDual_symm_apply, ←neg_lt_neg_iff, ←neg_mul, mul_comm, neg_inv, ← division_def] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ | case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ -1 < f x✝ / -α
case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ f x / -α < -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ (∀ x ∈ X, ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ ≤ 1) ∧
1 < ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x⟫_ℝ
TACTIC:
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/Polar.lean | doublePolarDual_self | [135, 1] | [159, 7] | try apply le_of_lt | case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x✝⟫_ℝ ≤ 1 | case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x✝⟫_ℝ < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x✝⟫_ℝ ≤ 1
TACTIC:
|
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