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train · 219k rows

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/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rwa [mem_attachFin] at this	case intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this✝² : Set.Sized r ↑𝒞 this✝¹ : (∂ ^[i] 𝒞).card = k.choose (r - i) A B : Finset (Fin n) hA : A ⊆ range'k ∧ A.card = r HB₁ : { ofColex := B } < { ofColex := A } HB₂ : B.card = r t : Fin n ht : t ∈ B this✝ : { ofColex := image Fin.val B } < { ofColex := image Fin.val A } b✝ : ℕ a : Fin n ha : a ∈ A q : ↑a = b✝ this : a ∈ range'k ⊢ ↑a ∈ range k	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rwa [toColex_image_lt_toColex_image Fin.val_strictMono]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this✝ : Set.Sized r ↑𝒞 this : (∂ ^[i] 𝒞).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ 𝒞 HB₁ : { ofColex := B } < { ofColex := A } HB₂ : B.card = r t : Fin n ht : t ∈ B ⊢ { ofColex := image Fin.val B } < { ofColex := image Fin.val A }	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rw [this, card_powersetCard, card_attachFin, card_range]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this✝ : Set.Sized r ↑𝒞 this : ∂ ^[i] 𝒞 = powersetCard (r - i) range'k ⊢ (∂ ^[i] 𝒞).card = k.choose (r - i)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rintro ⟨A, Ah, BsubA, card_sdiff_i⟩	case a.mp α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B : Finset (Fin n) ⊢ (∃ s ∈ 𝒞, B ⊆ s ∧ (s \ B).card = i) → B ⊆ range'k ∧ B.card = r - i	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ∈ 𝒞 BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B ⊆ range'k ∧ B.card = r - i
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rw [mem_powersetCard] at Ah	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ∈ 𝒞 BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B ⊆ range'k ∧ B.card = r - i	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B ⊆ range'k ∧ B.card = r - i
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	refine' ⟨BsubA.trans Ah.1, _⟩	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B ⊆ range'k ∧ B.card = r - i	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B.card = r - i
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	symm	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ B.card = r - i	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ r - i = B.card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rw [Nat.sub_eq_iff_eq_add hir, ← Ah.2, ← card_sdiff_i, ← card_union_of_disjoint disjoint_sdiff, union_sdiff_of_subset BsubA]	case a.mp.intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B A : Finset (Fin n) Ah : A ⊆ range'k ∧ A.card = r BsubA : B ⊆ A card_sdiff_i : (A \ B).card = i ⊢ r - i = B.card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rw [card_sdiff BsubC, hcard, hB, Nat.sub_sub_self hir]	case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B : Finset (Fin n) hBk : B ⊆ range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B ⊆ C hCrange : C ⊆ range'k hcard : C.card = r ⊢ (C \ B).card = i	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.lovasz_form	[334, 1]	[382, 91]	rwa [hB, card_attachFin, card_range, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left]	case a.mpr.intro.refine_1 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n r k i : ℕ 𝒜 𝒞✝ : Finset (Finset (Fin n)) hir : i ≤ r hrk : r ≤ k hkn : k ≤ n h₁ : Set.Sized r ↑𝒜 h₂ : k.choose r ≤ 𝒜.card range'k : Finset (Fin n) := (range k).attachFin ⋯ 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : 𝒞.card = k.choose r this : Set.Sized r ↑𝒞 B : Finset (Fin n) hBk : B ⊆ range'k hB : B.card = r - i ⊢ i + B.card ≤ range'k.card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	cases' Nat.eq_zero_or_pos r with b h1r	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 ⊢ 𝒜.card ≤ (n - 1).choose (r - 1)	case inl α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ 𝒜.card ≤ (n - 1).choose (r - 1) case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 ⊢ 𝒜.card ≤ (n - 1).choose (r - 1)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	refine' le_of_not_lt fun size ↦ _	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 ⊢ 𝒜.card ≤ (n - 1).choose (r - 1)	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have : Disjoint 𝒜 (∂^[n - 2 * r] 𝒜ᶜˢ) := disjoint_right.2 fun A hAbar hA ↦ by simp [mem_shadow_iterate_iff_exists_sdiff, mem_compls] at hAbar obtain ⟨C, hC, hAC, _⟩ := hAbar exact h𝒜 hA hC (disjoint_of_subset_left hAC disjoint_compl_right)	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have : r ≤ n := h₃.trans (Nat.div_le_self n 2)	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this : r ≤ n ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have : 1 ≤ n := ‹1 ≤ r›.trans ‹r ≤ n›	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this : r ≤ n ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card := by rwa [card_compls, choose_symm_of_eq_add (tsub_add_tsub_cancel ‹r ≤ n› ‹1 ≤ r›).symm]	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have h𝒜bar : (𝒜ᶜˢ : Set (Finset (Fin n))).Sized (n - r) := by simpa using h₂.compls	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have : n - 2 * r ≤ n - r := by rw [tsub_le_tsub_iff_left ‹r ≤ n›] exact Nat.le_mul_of_pos_left _ zero_lt_two	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have kk := lovasz_form ‹n - 2 * r ≤ n - r› ((tsub_le_tsub_iff_left ‹1 ≤ n›).2 h1r) tsub_le_self h𝒜bar z.le	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have q : n - r - (n - 2 * r) = r := by rw [tsub_right_comm, Nat.sub_sub_self, two_mul] apply Nat.add_sub_cancel rw [mul_comm, ← Nat.le_div_iff_mul_le' zero_lt_two] exact h₃	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [q] at kk	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	have : n.choose r < (𝒜 ∪ ∂^[n - 2 * r] 𝒜ᶜˢ).card := by rw [card_union_of_disjoint ‹_›] convert lt_of_le_of_lt (add_le_add_left kk _) (add_lt_add_right size _) using 1 convert Nat.choose_succ_succ _ _ using 3 all_goals rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos]	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	apply this.not_le	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ False	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ≤ n.choose r
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	convert Set.Sized.card_le _	case inr α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ≤ n.choose r	case h.e'_4.h.e'_1 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ n = Fintype.card (Fin n) case inr.convert_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Fintype (Fin n) case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑(𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [coe_union, Set.sized_union]	case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑(𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ)	case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑𝒜 ∧ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	refine' ⟨‹_›, _⟩	case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑𝒜 ∧ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)	case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	convert h𝒜bar.shadow_iterate	case inr.convert_5 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)	case h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r = n - r - (n - 2 * r)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [q]	case h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r = n - r - (n - 2 * r)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	convert Nat.zero_le _	case inl α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ 𝒜.card ≤ (n - 1).choose (r - 1)	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ 𝒜.card = 0
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [Finset.card_eq_zero, eq_empty_iff_forall_not_mem]	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ 𝒜.card = 0	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ ∀ (x : Finset (Fin n)), x ∉ 𝒜
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	refine' fun A HA ↦ h𝒜 HA HA _	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 ⊢ ∀ (x : Finset (Fin n)), x ∉ 𝒜	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 A : Finset (Fin n) HA : A ∈ 𝒜 ⊢ Disjoint A A
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [disjoint_self_iff_empty, ← Finset.card_eq_zero, ← b]	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 A : Finset (Fin n) HA : A ∈ 𝒜 ⊢ Disjoint A A	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 A : Finset (Fin n) HA : A ∈ 𝒜 ⊢ A.card = r
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	exact h₂ HA	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 b : r = 0 A : Finset (Fin n) HA : A ∈ 𝒜 ⊢ A.card = r	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	simp [mem_shadow_iterate_iff_exists_sdiff, mem_compls] at hAbar	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card A : Finset (Fin n) hAbar : A ∈ ∂ ^[n - 2 * r] 𝒜ᶜˢ hA : A ∈ 𝒜 ⊢ False	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card A : Finset (Fin n) hA : A ∈ 𝒜 hAbar : ∃ s, sᶜ ∈ 𝒜 ∧ A ⊆ s ∧ (s \ A).card = n - 2 * r ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	obtain ⟨C, hC, hAC, _⟩ := hAbar	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card A : Finset (Fin n) hA : A ∈ 𝒜 hAbar : ∃ s, sᶜ ∈ 𝒜 ∧ A ⊆ s ∧ (s \ A).card = n - 2 * r ⊢ False	case intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card A : Finset (Fin n) hA : A ∈ 𝒜 C : Finset (Fin n) hC : Cᶜ ∈ 𝒜 hAC : A ⊆ C right✝ : (C \ A).card = n - 2 * r ⊢ False
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	exact h𝒜 hA hC (disjoint_of_subset_left hAC disjoint_compl_right)	case intro.intro.intro α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card A : Finset (Fin n) hA : A ∈ 𝒜 C : Finset (Fin n) hC : Cᶜ ∈ 𝒜 hAC : A ⊆ C right✝ : (C \ A).card = n - 2 * r ⊢ False	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rwa [card_compls, choose_symm_of_eq_add (tsub_add_tsub_cancel ‹r ≤ n› ‹1 ≤ r›).symm]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n ⊢ (n - 1).choose (n - r) < 𝒜ᶜˢ.card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	simpa using h₂.compls	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card ⊢ Set.Sized (n - r) ↑𝒜ᶜˢ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [tsub_le_tsub_iff_left ‹r ≤ n›]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ ⊢ n - 2 * r ≤ n - r	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ ⊢ r ≤ 2 * r
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	exact Nat.le_mul_of_pos_left _ zero_lt_two	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝ : r ≤ n this : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ ⊢ r ≤ 2 * r	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [tsub_right_comm, Nat.sub_sub_self, two_mul]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ n - r - (n - 2 * r) = r	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r + r - r = r α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ 2 * r ≤ n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	apply Nat.add_sub_cancel	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r + r - r = r α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ 2 * r ≤ n	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ 2 * r ≤ n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [mul_comm, ← Nat.le_div_iff_mul_le' zero_lt_two]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ 2 * r ≤ n	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r ≤ n / 2
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	exact h₃	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ r ≤ n / 2	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [card_union_of_disjoint ‹_›]	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n.choose r < 𝒜.card + (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	convert lt_of_le_of_lt (add_le_add_left kk _) (add_lt_add_right size _) using 1	α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n.choose r < 𝒜.card + (∂ ^[n - 2 * r] 𝒜ᶜˢ).card	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n.choose r = (n - 1).choose (r - 1) + (n - 1).choose r
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	convert Nat.choose_succ_succ _ _ using 3	case h.e'_3 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n.choose r = (n - 1).choose (r - 1) + (n - 1).choose r	case h.e'_2.h.e'_1 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n = (n - 1).succ case h.e'_2.h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ r = (r - 1).succ case h.e'_3.h.e'_6.h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ r = (r - 1).succ
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	all_goals rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos]	case h.e'_2.h.e'_1 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ n = (n - 1).succ case h.e'_2.h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ r = (r - 1).succ case h.e'_3.h.e'_6.h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ r = (r - 1).succ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos]	case h.e'_3.h.e'_6.h.e'_2 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝¹ : r ≤ n this✝ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r ⊢ r = (r - 1).succ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/KruskalKatona.lean	Finset.EKR	[386, 1]	[436, 9]	rw [Fintype.card_fin]	case h.e'_4.h.e'_1 α : Type u_1 inst✝ : LinearOrder α s U V : Finset α n : ℕ 𝒜 : Finset (Finset (Fin n)) r : ℕ h𝒜 : (↑𝒜).Intersecting h₂ : Set.Sized r ↑𝒜 h₃ : r ≤ n / 2 h1r : r > 0 size : (n - 1).choose (r - 1) < 𝒜.card this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ) this✝² : r ≤ n this✝¹ : 1 ≤ n z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ this✝ : n - 2 * r ≤ n - r kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card q : n - r - (n - 2 * r) = r this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ⊢ n = Fintype.card (Fin n)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Algebra/Order/Ring/Canonical.lean	mul_tsub_one	[7, 1]	[7, 84]	rw [mul_tsub, mul_one]	α : Type u_1 inst✝⁴ : CanonicallyOrderedCommSemiring α inst✝³ : Sub α inst✝² : OrderedSub α inst✝¹ : IsTotal α fun x x_1 => x ≤ x_1 inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 a b : α ⊢ a * (b - 1) = a * b - a	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Algebra/Order/Ring/Canonical.lean	tsub_one_mul	[8, 1]	[8, 84]	rw [tsub_mul, one_mul]	α : Type u_1 inst✝⁴ : CanonicallyOrderedCommSemiring α inst✝³ : Sub α inst✝² : OrderedSub α inst✝¹ : IsTotal α fun x x_1 => x ≤ x_1 inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 a b : α ⊢ (a - 1) * b = a * b - b	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Probability/Independence/Basic.lean	ProbabilityTheory.iIndep_comap_iff	[11, 1]	[12, 42]	simp_rw [← generateFrom_singleton]	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => MeasurableSpace.comap (fun x => x ∈ s i) ⊤) μ ↔ iIndepSet s μ	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => generateFrom {s i}) μ ↔ iIndepSet s μ
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Probability/Independence/Basic.lean	ProbabilityTheory.iIndep_comap_iff	[11, 1]	[12, 42]	rfl	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => generateFrom {s i}) μ ↔ iIndepSet s μ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	Real.closedBall_eq_segment	[26, 1]	[27, 96]	rw [closedBall_eq_Icc, segment_eq_Icc ((sub_le_self _ hε).trans $ le_add_of_nonneg_right hε)]	ε r : ℝ hε : 0 ≤ ε ⊢ closedBall r ε = segment ℝ (r - ε) (r + ε)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	classical lift x to affineSpan ℝ s using subset_affineSpan _ _ hx set x : affineSpan ℝ s := x with hx clear_value x subst hx obtain ⟨ε, hε, hεx⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs _ hx) set f : Finset (Fin $ finrank ℝ $ vectorSpan ℝ s) → vectorSpan ℝ s := fun u ↦ (ε / ∑ i, ‖finBasis ℝ (vectorSpan ℝ s) i‖) • ∑ i, if i ∈ u then finBasis ℝ (vectorSpan ℝ s) i else -finBasis ℝ (vectorSpan ℝ s) i with hf sorry	E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E x : E hs : IsOpen (Subtype.val ⁻¹' s) hx : x ∈ s ⊢ ∃ t, x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	lift x to affineSpan ℝ s using subset_affineSpan _ _ hx	E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E x : E hs : IsOpen (Subtype.val ⁻¹' s) hx : x ∈ s ⊢ ∃ t, x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	set x : affineSpan ℝ s := x with hx	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x✝ : ↥(affineSpan ℝ s) x : ↥(affineSpan ℝ s) := x✝ hx✝ : ↑x ∈ s hx : x = x✝ ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	clear_value x	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x✝ : ↥(affineSpan ℝ s) x : ↥(affineSpan ℝ s) := x✝ hx✝ : ↑x ∈ s hx : x = x✝ ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x✝ x : ↥(affineSpan ℝ s) hx✝ : ↑x ∈ s hx : x = x✝ ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	subst hx	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x✝ x : ↥(affineSpan ℝ s) hx✝ : ↑x ∈ s hx : x = x✝ ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	obtain ⟨ε, hε, hεx⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs _ hx)	case intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ε : ℝ hε : 0 < ε hεx : closedBall x ε ⊆ Subtype.val ⁻¹' s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	set f : Finset (Fin $ finrank ℝ $ vectorSpan ℝ s) → vectorSpan ℝ s := fun u ↦ (ε / ∑ i, ‖finBasis ℝ (vectorSpan ℝ s) i‖) • ∑ i, if i ∈ u then finBasis ℝ (vectorSpan ℝ s) i else -finBasis ℝ (vectorSpan ℝ s) i with hf	case intro.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ε : ℝ hε : 0 < ε hεx : closedBall x ε ⊆ Subtype.val ⁻¹' s ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	case intro.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ε : ℝ hε : 0 < ε hεx : closedBall x ε ⊆ Subtype.val ⁻¹' s f : Finset (Fin (finrank ℝ ↥(vectorSpan ℝ s))) → ↥(vectorSpan ℝ s) := fun u => (ε / ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), ‖(finBasis ℝ ↥(vectorSpan ℝ s)) i‖) • ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), if i ∈ u then (finBasis ℝ ↥(vectorSpan ℝ s)) i else -(finBasis ℝ ↥(vectorSpan ℝ s)) i hf : f = fun u => (ε / ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), ‖(finBasis ℝ ↥(vectorSpan ℝ s)) i‖) • ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), if i ∈ u then (finBasis ℝ ↥(vectorSpan ℝ s)) i else -(finBasis ℝ ↥(vectorSpan ℝ s)) i ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	IsOpen.exists_mem_intrinsicInterior_convexHull_finset	[52, 1]	[68, 8]	sorry	case intro.intro.intro E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E hs : IsOpen (Subtype.val ⁻¹' s) x : ↥(affineSpan ℝ s) hx : ↑x ∈ s ε : ℝ hε : 0 < ε hεx : closedBall x ε ⊆ Subtype.val ⁻¹' s f : Finset (Fin (finrank ℝ ↥(vectorSpan ℝ s))) → ↥(vectorSpan ℝ s) := fun u => (ε / ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), ‖(finBasis ℝ ↥(vectorSpan ℝ s)) i‖) • ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), if i ∈ u then (finBasis ℝ ↥(vectorSpan ℝ s)) i else -(finBasis ℝ ↥(vectorSpan ℝ s)) i hf : f = fun u => (ε / ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), ‖(finBasis ℝ ↥(vectorSpan ℝ s)) i‖) • ∑ i : Fin (finrank ℝ ↥(vectorSpan ℝ s)), if i ∈ u then (finBasis ℝ ↥(vectorSpan ℝ s)) i else -(finBasis ℝ ↥(vectorSpan ℝ s)) i ⊢ ∃ t, ↑x ∈ intrinsicInterior ℝ ((convexHull ℝ) ↑t) ∧ (convexHull ℝ) ↑t ⊆ s	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	ConvexOn.continuousOn	[104, 1]	[115, 8]	classical sorry	E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E f : E → ℝ hf : ConvexOn ℝ s f ⊢ ContinuousOn f (intrinsicInterior ℝ s)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/ConvexContinuous.lean	ConvexOn.continuousOn	[104, 1]	[115, 8]	sorry	E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : FiniteDimensional ℝ E s : Set E f : E → ℝ hf : ConvexOn ℝ s f ⊢ ContinuousOn f (intrinsicInterior ℝ s)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Probability/Independence/Basic.lean	ProbabilityTheory.iIndep_comap_iff	[11, 1]	[12, 42]	simp_rw [← generateFrom_singleton]	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => MeasurableSpace.comap (fun x => x ∈ s i) ⊤) μ ↔ iIndepSet s μ	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => generateFrom {s i}) μ ↔ iIndepSet s μ
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Probability/Independence/Basic.lean	ProbabilityTheory.iIndep_comap_iff	[11, 1]	[12, 42]	rfl	Ω : Type u_1 ι : Type u_2 β : ι → Type u_3 inst✝ : MeasurableSpace Ω π : ι → Set (Set Ω) μ : Measure Ω S : Finset ι s : ι → Set Ω f : (i : ι) → Ω → β i ⊢ iIndep (fun i => generateFrom {s i}) μ ↔ iIndepSet s μ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.length_rtake	[10, 1]	[11, 65]	rw [Nat.sub_sub_eq_min, min_comm]	α : Type u_1 l✝ l' l₀ l₁ l₂ : List α a b : α m n✝ n : ℕ l : List α ⊢ l.length - (l.length - n) = min n l.length	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.take_reverse	[13, 1]	[14, 54]	rw [rtake_eq_reverse_take_reverse, reverse_reverse]	α : Type u_1 l✝ l' l₀ l₁ l₂ : List α a b : α m n✝ n : ℕ l : List α ⊢ take n l.reverse = (l.rtake n).reverse	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.rtake_reverse	[16, 1]	[17, 54]	rw [rtake_eq_reverse_take_reverse, reverse_reverse]	α : Type u_1 l✝ l' l₀ l₁ l₂ : List α a b : α m n✝ n : ℕ l : List α ⊢ l.reverse.rtake n = (take n l).reverse	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.rtake_rtake	[19, 1]	[20, 95]	rw [rtake_eq_reverse_take_reverse, ← take_reverse, take_take, rtake_eq_reverse_take_reverse]	α : Type u_1 l✝ l' l₀ l₁ l₂ : List α a b : α m✝ n✝ n m : ℕ l : List α ⊢ (l.rtake m).rtake n = l.rtake (min n m)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.take_prefix_take	[37, 1]	[38, 91]	rw [prefix_iff_eq_take, length_take, take_take, min_right_comm, min_eq_left h, take_min]	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ h : m ≤ n ⊢ take m l <+: take n l	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.IsPrefix.take	[48, 1]	[49, 44]	rw [hl.eq_take]	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <+: l' h : l.length ≤ n ⊢ l <+: take n l'	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <+: l' h : l.length ≤ n ⊢ take l.length l' <+: take n l'
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.IsPrefix.take	[48, 1]	[49, 44]	exact take_prefix_take h	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <+: l' h : l.length ≤ n ⊢ take l.length l' <+: take n l'	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.IsSuffix.rtake	[51, 1]	[52, 47]	rw [hl.eq_rtake]	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <:+ l' h : l.length ≤ n ⊢ l <:+ l'.rtake n	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <:+ l' h : l.length ≤ n ⊢ l'.rtake l.length <:+ l'.rtake n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.IsSuffix.rtake	[51, 1]	[52, 47]	exact rtake_suffix_rtake h	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l <:+ l' h : l.length ≤ n ⊢ l'.rtake l.length <:+ l'.rtake n	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	induction' l₀ with a l₀ ih generalizing n l₂	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l₀ <+ l₂ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n	case nil α : Type u_1 l l' l₁ : List α a b : α m : ℕ l₂ : List α n : ℕ hl : [] <+ l₂ h₀ : [].length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, [] <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n case cons α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α n : ℕ hl : a :: l₀ <+ l₂ h₀ : (a :: l₀).length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	cases n	case cons α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α n : ℕ hl : a :: l₀ <+ l₂ h₀ : (a :: l₀).length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n	case cons.zero α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α hl : a :: l₀ <+ l₂ h₀ : (a :: l₀).length ≤ 0 h₂ : 0 ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = 0 case cons.succ α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α hl : a :: l₀ <+ l₂ n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 h₂ : n✝ + 1 ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n✝ + 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	cases' l₂ with b l₂	case cons.succ α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α hl : a :: l₀ <+ l₂ n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 h₂ : n✝ + 1 ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n✝ + 1	case cons.succ.nil α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 hl : a :: l₀ <+ [] h₂ : n✝ + 1 ≤ [].length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ [] ∧ l₁.length = n✝ + 1 case cons.succ.cons α : Type u_1 l l' l₁ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 b : α l₂ : List α hl : a :: l₀ <+ b :: l₂ h₂ : n✝ + 1 ≤ (b :: l₂).length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ b :: l₂ ∧ l₁.length = n✝ + 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	obtain ⟨l₁, h₀₁, h₁₂, rfl⟩ := ih sorry (Nat.succ_le_succ_iff.1 h₀) (Nat.le_of_succ_le h₂)	case cons.succ.cons α : Type u_1 l l' l₁ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 b : α l₂ : List α hl : a :: l₀ <+ b :: l₂ h₂ : n✝ + 1 ≤ (b :: l₂).length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ b :: l₂ ∧ l₁.length = n✝ + 1	case cons.succ.cons.intro.intro.intro α : Type u_1 l l' l₁✝ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n b : α l₂ : List α hl : a :: l₀ <+ b :: l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₁₂ : l₁ <+ b :: l₂ h₀ : (a :: l₀).length ≤ l₁.length + 1 h₂ : l₁.length + 1 ≤ (b :: l₂).length ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	rw [cons_sublist_cons_iff'] at hl	case cons.succ.cons.intro.intro.intro α : Type u_1 l l' l₁✝ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n b : α l₂ : List α hl : a :: l₀ <+ b :: l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₁₂ : l₁ <+ b :: l₂ h₀ : (a :: l₀).length ≤ l₁.length + 1 h₂ : l₁.length + 1 ≤ (b :: l₂).length ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1	case cons.succ.cons.intro.intro.intro α : Type u_1 l l' l₁✝ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n b : α l₂ : List α hl : a :: l₀ <+ l₂ ∨ a = b ∧ l₀ <+ l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₁₂ : l₁ <+ b :: l₂ h₀ : (a :: l₀).length ≤ l₁.length + 1 h₂ : l₁.length + 1 ≤ (b :: l₂).length ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	obtain hl \| ⟨rfl, hl⟩ := hl	case cons.succ.cons.intro.intro.intro α : Type u_1 l l' l₁✝ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n b : α l₂ : List α hl : a :: l₀ <+ l₂ ∨ a = b ∧ l₀ <+ l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₁₂ : l₁ <+ b :: l₂ h₀ : (a :: l₀).length ≤ l₁.length + 1 h₂ : l₁.length + 1 ≤ (b :: l₂).length ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1	case cons.succ.cons.intro.intro.intro.inl α : Type u_1 l l' l₁✝ : List α a✝ b✝ : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n b : α l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₁₂ : l₁ <+ b :: l₂ h₀ : (a :: l₀).length ≤ l₁.length + 1 h₂ : l₁.length + 1 ≤ (b :: l₂).length hl : a :: l₀ <+ l₂ ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ b :: l₂ ∧ l₁_1.length = l₁.length + 1 case cons.succ.cons.intro.intro.intro.inr.intro α : Type u_1 l l' l₁✝ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ l₁ : List α h₀₁ : l₀ <+ l₁ h₀ : (a :: l₀).length ≤ l₁.length + 1 hl : l₀ <+ l₂ h₁₂ : l₁ <+ a :: l₂ h₂ : l₁.length + 1 ≤ (a :: l₂).length ⊢ ∃ l₁_1, a :: l₀ <+ l₁_1 ∧ l₁_1 <+ a :: l₂ ∧ l₁_1.length = l₁.length + 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	exact (exists_sublist_length_eq h₂).imp (fun l₁ h ↦ ⟨nil_sublist _, h⟩)	case nil α : Type u_1 l l' l₁ : List α a b : α m : ℕ l₂ : List α n : ℕ hl : [] <+ l₂ h₀ : [].length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, [] <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	cases h₀	case cons.zero α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n l₂ : List α hl : a :: l₀ <+ l₂ h₀ : (a :: l₀).length ≤ 0 h₂ : 0 ≤ l₂.length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = 0	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Sublist.exists_intermediate	[71, 1]	[84, 23]	cases h₂	case cons.succ.nil α : Type u_1 l l' l₁ : List α a✝ b : α m : ℕ a : α l₀ : List α ih : ∀ {l₂ : List α} {n : ℕ}, l₀ <+ l₂ → l₀.length ≤ n → n ≤ l₂.length → ∃ l₁, l₀ <+ l₁ ∧ l₁ <+ l₂ ∧ l₁.length = n n✝ : ℕ h₀ : (a :: l₀).length ≤ n✝ + 1 hl : a :: l₀ <+ [] h₂ : n✝ + 1 ≤ [].length ⊢ ∃ l₁, a :: l₀ <+ l₁ ∧ l₁ <+ [] ∧ l₁.length = n✝ + 1	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Subperm.exists_intermediate	[86, 1]	[90, 45]	obtain ⟨l₀', hl₀, hl'⟩ := hl	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hl : l₀ <+~ l₂ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length ⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n	case intro.intro α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length l₀' : List α hl₀ : l₀' ~ l₀ hl' : l₀' <+ l₂ ⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Subperm.exists_intermediate	[86, 1]	[90, 45]	obtain ⟨l₁, h₀₁, h₁₂, hn⟩ := hl'.exists_intermediate (hl₀.length_eq.trans_le h₀) h₂	case intro.intro α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length l₀' : List α hl₀ : l₀' ~ l₀ hl' : l₀' <+ l₂ ⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n	case intro.intro.intro.intro.intro α : Type u_1 l l' l₀ l₁✝ l₂ : List α a b : α m n : ℕ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length l₀' : List α hl₀ : l₀' ~ l₀ hl' : l₀' <+ l₂ l₁ : List α h₀₁ : l₀' <+ l₁ h₁₂ : l₁ <+ l₂ hn : l₁.length = n ⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.Subperm.exists_intermediate	[86, 1]	[90, 45]	exact ⟨l₁, ⟨_, hl₀, h₀₁⟩, h₁₂.subperm, hn⟩	case intro.intro.intro.intro.intro α : Type u_1 l l' l₀ l₁✝ l₂ : List α a b : α m n : ℕ h₀ : l₀.length ≤ n h₂ : n ≤ l₂.length l₀' : List α hl₀ : l₀' ~ l₀ hl' : l₀' <+ l₂ l₁ : List α h₀₁ : l₀' <+ l₁ h₁₂ : l₁ <+ l₂ hn : l₁.length = n ⊢ ∃ l₁, l₀ <+~ l₁ ∧ l₁ <+~ l₂ ∧ l₁.length = n	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/Data/List/DropRight.lean	List.exists_subperm_length_eq	[92, 1]	[93, 65]	simpa using nil_subperm.exists_intermediate (Nat.zero_le _) hn	α : Type u_1 l l' l₀ l₁ l₂ : List α a b : α m n : ℕ hn : n ≤ l.length ⊢ ∃ l', l' <+~ l ∧ l'.length = n	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean	ZMod.pow_card_sub_one	[6, 1]	[9, 75]	split_ifs with hx	p : ℕ inst✝ : Fact p.Prime x : ZMod p ⊢ x ^ (p - 1) = if x ≠ 0 then 1 else 0	case pos p : ℕ inst✝ : Fact p.Prime x : ZMod p hx : x ≠ 0 ⊢ x ^ (p - 1) = 1 case neg p : ℕ inst✝ : Fact p.Prime x : ZMod p hx : ¬x ≠ 0 ⊢ x ^ (p - 1) = 0
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean	ZMod.pow_card_sub_one	[6, 1]	[9, 75]	exact pow_card_sub_one_eq_one hx	case pos p : ℕ inst✝ : Fact p.Prime x : ZMod p hx : x ≠ 0 ⊢ x ^ (p - 1) = 1	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean	ZMod.pow_card_sub_one	[6, 1]	[9, 75]	simp [of_not_not hx, (Fact.out : p.Prime).one_lt, tsub_eq_zero_iff_le]	case neg p : ℕ inst✝ : Fact p.Prime x : ZMod p hx : ¬x ≠ 0 ⊢ x ^ (p - 1) = 0	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	refine' ⟨fun h => ⟨h.1.superset, fun s hs => _⟩, fun h => ⟨h.1.antisymm' fun x hx => _, fun s₁ hs₁ => _⟩⟩	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ K₁.Subdivides K₂ ↔ K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂	case refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂ case refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx : x ∈ K₁.space ⊢ x ∈ K₂.space case refine'_3 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ s₁ : Finset E hs₁ : s₁ ∈ K₁ ⊢ ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨t, ht, hst⟩ := h.2 hs	case refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂	case refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨u, hut, hu, hsu⟩ := simplex_combiInteriors_split_interiors_nonempty (K₁.nonempty hs) (K₂.indep ht) hst	case refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂	case refine'_1.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t u : Finset E hut : u ⊆ t hu : u.Nonempty hsu : combiInterior ℝ s ⊆ combiInterior ℝ u ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	exact ⟨u, K₂.down_closed' ht hut hu, hsu⟩	case refine'_1.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t u : Finset E hut : u ⊆ t hu : u.Nonempty hsu : combiInterior ℝ s ⊆ combiInterior ℝ u ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨s₁, hs₁, hx⟩ := mem_space_iff.1 hx	case refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx : x ∈ K₁.space ⊢ x ∈ K₂.space	case refine'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ ⊢ x ∈ K₂.space
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨s₂, hs₂, hs₁s₂⟩ := h.2 _ hs₁	case refine'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ ⊢ x ∈ K₂.space	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ s₂ : Finset E hs₂ : s₂ ∈ K₂ hs₁s₂ : combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ ⊢ x ∈ K₂.space
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	rw [mem_space_iff]	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ s₂ : Finset E hs₂ : s₂ ∈ K₂ hs₁s₂ : combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ ⊢ x ∈ K₂.space	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ s₂ : Finset E hs₂ : s₂ ∈ K₂ hs₁s₂ : combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ ⊢ ∃ s ∈ K₂.faces, x ∈ (convexHull ℝ) ↑s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	exact ⟨s₂, hs₂, convexHull_subset_convexHull_of_combiInterior_subset_combiInterior (K₁.indep hs₁) (K₂.indep hs₂) hs₁s₂ hx⟩	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ K₁.faces hx : x ∈ (convexHull ℝ) ↑s₁ s₂ : Finset E hs₂ : s₂ ∈ K₂ hs₁s₂ : combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ ⊢ ∃ s ∈ K₂.faces, x ∈ (convexHull ℝ) ↑s	no goals

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