| import OAI.Combinatorics.GotsmanLinial.Model |
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| namespace OAI |
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| open scoped BigOperators |
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| namespace LeanBlast.GotsmanLinial |
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| @[simp] theorem flip_apply_same {n : ℕ} (i : Fin n) (x : Cube n) : |
| flip i x i = !(x i) := by |
| simp [flip] |
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| theorem flip_apply_ne {n : ℕ} (i : Fin n) (x : Cube n) (j : Fin n) (h : j ≠ i) : |
| flip i x j = x j := by |
| simp [flip, h] |
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| @[simp] theorem flip_flip {n : ℕ} (i : Fin n) (x : Cube n) : |
| flip i (flip i x) = x := by |
| funext j |
| by_cases h : j = i |
| · subst j |
| simp [flip] |
| · simp [flip, h] |
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| theorem flip_ne_self {n : ℕ} (i : Fin n) (x : Cube n) : flip i x ≠ x := by |
| intro h |
| have hi := congrFun h i |
| cases hxi : x i <;> simp [flip, hxi] at hi |
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| theorem flip_injective_direction {n : ℕ} (x : Cube n) : |
| Function.Injective (fun i : Fin n => flip i x) := by |
| intro i j h |
| by_contra hij |
| have hi := congrFun h i |
| cases hxi : x i <;> simp [flip, hij, hxi] at hi |
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| theorem flip_involutive {n : ℕ} (i : Fin n) : Function.Involutive (flip i) := |
| flip_flip i |
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| theorem flip_bijective {n : ℕ} (i : Fin n) : Function.Bijective (flip i) := |
| (flip_involutive i).bijective |
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| @[simp] theorem cubeCoord_flip_same {n : ℕ} (x : Cube n) (i : Fin n) : |
| cubeCoord (flip i x) i = -cubeCoord x i := by |
| cases hxi : x i <;> simp [cubeCoord, flip, hxi] |
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| theorem cubeCoord_flip_ne {n : ℕ} (x : Cube n) (i j : Fin n) (h : j ≠ i) : |
| cubeCoord (flip i x) j = cubeCoord x j := by |
| simp [cubeCoord, flip, h] |
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| theorem cubeCoord_cases {n : ℕ} (x : Cube n) (i : Fin n) : |
| cubeCoord x i = 1 ∨ cubeCoord x i = -1 := by |
| cases hxi : x i <;> simp [cubeCoord, hxi] |
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| @[simp] theorem cubeCoord_sq {n : ℕ} (x : Cube n) (i : Fin n) : |
| (cubeCoord x i) ^ 2 = 1 := by |
| cases hxi : x i <;> simp [cubeCoord, hxi] |
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| @[simp] theorem thresholdSign_zero : thresholdSign 0 = 1 := by |
| simp [thresholdSign] |
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| theorem thresholdSign_eq_one_iff (t : ℝ) : thresholdSign t = 1 ↔ 0 ≤ t := by |
| by_cases h : 0 ≤ t |
| · simp [thresholdSign, h] |
| · simp [thresholdSign, h, show (-1 : ℝ) ≠ 1 by norm_num] |
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| theorem thresholdSign_eq_neg_one_iff (t : ℝ) : thresholdSign t = -1 ↔ t < 0 := by |
| by_cases h : 0 ≤ t |
| · simp [thresholdSign, h, not_lt.mpr h, show (1 : ℝ) ≠ -1 by norm_num] |
| · simp [thresholdSign, h, lt_of_not_ge h] |
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| theorem thresholdSign_cases (t : ℝ) : thresholdSign t = 1 ∨ thresholdSign t = -1 := by |
| unfold thresholdSign |
| split <;> simp |
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| @[simp] theorem thresholdSign_sq (t : ℝ) : thresholdSign t ^ 2 = 1 := by |
| rcases thresholdSign_cases t with h | h <;> simp [h] |
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| @[simp] theorem card_cube (n : ℕ) : Fintype.card (Cube n) = 2 ^ n := by |
| simp [Cube] |
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| / |
| noncomputable def cubeEquivSignVectors (n : ℕ) : |
| Cube n ≃ {x : Fin n → ℝ // ∀ i, x i = 1 ∨ x i = -1} := by |
| classical |
| exact |
| { toFun := fun x => ⟨cubeCoord x, cubeCoord_cases x⟩ |
| invFun := fun x i => decide (x.1 i = 1) |
| left_inv := by |
| intro x |
| funext i |
| cases hxi : x i <;> simp [cubeCoord, hxi, show (-1 : ℝ) ≠ 1 by norm_num] |
| right_inv := by |
| intro x |
| apply Subtype.ext |
| funext i |
| rcases x.2 i with hxi | hxi <;> |
| simp [cubeCoord, hxi, show (-1 : ℝ) ≠ 1 by norm_num] } |
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| theorem averageSensitivity_eq_count {n : ℕ} (f : Cube n → ℝ) : |
| averageSensitivity f = (sensitiveEdgeCount f : ℝ) / (2 : ℝ) ^ n := by |
| simp [averageSensitivity, sensitiveEdgeCount, Finset.sum_div] |
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| theorem averageSensitivity_eq_uniform {n : ℕ} (f : Cube n → ℝ) : |
| averageSensitivity f = |
| ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (Fintype.card (Cube n) : ℝ) := by |
| simp [averageSensitivity] |
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| end LeanBlast.GotsmanLinial |
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| end OAI |
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