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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Basic.Real.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Tactic.NormNum
import OAI.Computability.PerfectCompleteness.Algebra.BilinearGramCompressionLemmas
namespace OAI
namespace PerfectCompleteness.TensorSliceDimension
noncomputable section
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
theorem mask_card (n : Nat) : Fintype.card (Fin n → Bool) = 2 ^ n := by
simp only [Fintype.card_fun, Fintype.card_bool, Fintype.card_fin]
theorem mask_card_le {n N : Nat} (h : n ≤ N) :
Fintype.card (Fin n → Bool) ≤ 2 ^ N := by
rw [mask_card]
exact Nat.pow_le_pow_right (by decide : 0 < 2) h
theorem reciprocal_mask_card_ge {n N : Nat} (h : n ≤ N) :
1 / (2 ^ N : ℝ) ≤ 1 / (Fintype.card (Fin n → Bool) : ℝ) := by
simpa only [mask_card, Nat.cast_pow, Nat.cast_ofNat] using
(one_div_pow_le_one_div_pow_of_le (by norm_num : (1 : ℝ) ≤ 2) h)
section GeneralEquationSpace
variable {Q W : Type*} [AddCommGroup Q] [Module F2 Q]
[AddCommGroup W] [Module F2 W]
[FiniteDimensional F2 Q] [FiniteDimensional F2 W]
theorem linearMap_finrank_le {r ℓ : Nat}
(hQ : Module.finrank F2 Q ≤ r) (hW : Module.finrank F2 W ≤ ℓ) :
Module.finrank F2 (Q →ₗ[F2] W) ≤ r * ℓ := by
rw [Module.finrank_linearMap]
exact Nat.mul_le_mul hQ hW
theorem reciprocal_linearMap_masks_ge {r ℓ : Nat}
(hQ : Module.finrank F2 Q ≤ r) (hW : Module.finrank F2 W ≤ ℓ) :
1 / (2 ^ (r * ℓ) : ℝ) ≤
1 / (Fintype.card (Fin (Module.finrank F2 (Q →ₗ[F2] W)) → Bool) : ℝ) :=
reciprocal_mask_card_ge (linearMap_finrank_le hQ hW)
end GeneralEquationSpace
variable {H : Type*} [AddCommGroup H] [Module F2 H] [FiniteDimensional F2 H]
{ℓ : Nat} (W : Submodule F2 (Fin ℓ → F2))
(Q : Submodule F2 (Module.Dual F2 H))
theorem row_finrank_le : Module.finrank F2 W ≤ ℓ := by
simpa only [Module.finrank_fin_fun] using (Submodule.finrank_le W)
theorem equation_finrank_le {r : Nat} (hQ : Module.finrank F2 Q ≤ r) :
Module.finrank F2 (Q →ₗ[F2] W) ≤ r * ℓ :=
linearMap_finrank_le (Q := Q) (W := W) hQ (row_finrank_le W)
theorem equation_mask_card :
Fintype.card (Fin (Module.finrank F2 (Q →ₗ[F2] W)) → Bool) =
2 ^ (Module.finrank F2 Q * Module.finrank F2 W) := by
let : Module.Free F2 Q := Module.Free.of_divisionRing F2 Q
let : Module.Free F2 W := Module.Free.of_divisionRing F2 W
rw [mask_card, Module.finrank_linearMap F2 F2 Q W]
theorem equation_mask_card_le {r : Nat} (hQ : Module.finrank F2 Q ≤ r) :
Fintype.card (Fin (Module.finrank F2 (Q →ₗ[F2] W)) → Bool) ≤ 2 ^ (r * ℓ) :=
mask_card_le (equation_finrank_le W Q hQ)
theorem reciprocal_equation_mask_card_ge {r : Nat} (hQ : Module.finrank F2 Q ≤ r) :
1 / (2 ^ (r * ℓ) : ℝ) ≤
1 / (Fintype.card (Fin (Module.finrank F2 (Q →ₗ[F2] W)) → Bool) : ℝ) :=
reciprocal_mask_card_ge (equation_finrank_le W Q hQ)
theorem restriction_finrank_le {r : Nat} (hQ : Module.finrank F2 Q ≤ r) :
Module.finrank F2 (LinearMap.range (TensorRestriction.restrictionMap (W := W) Q)) ≤
r * ℓ := by
rw [TensorRestriction.finrank_range, Nat.mul_comm]
exact Nat.mul_le_mul hQ (row_finrank_le W)
end
end PerfectCompleteness.TensorSliceDimension
end OAI
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