internlm/Lean-Github · Datasets at Hugging Face

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/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
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
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
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
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
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
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
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
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
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
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
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)
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)
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
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
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
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
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
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)
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)
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)
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)
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 }
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 }
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 }
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
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
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
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
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‖
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‖
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
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‖
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
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
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
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
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
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
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
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
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
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
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_
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
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
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
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
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_
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
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
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
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_
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_
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)
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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_
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_
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)
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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
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
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
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
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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

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

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

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

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)

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)

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)

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)

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 }

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 }

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 }

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

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

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

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

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‖

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‖

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

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‖

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

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

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

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

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

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

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

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

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

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

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_

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

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

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

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

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_

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

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

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

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_

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_

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)

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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_

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_

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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)

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

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

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

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

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

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

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

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