| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Partitions |
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| namespace OAI |
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| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
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| noncomputable section |
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| namespace MatrixMultiplication.DualHiddenRates |
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| open MatrixMultiplication.Foundation RecursiveCompletion CompletionPartitions |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
| attribute [local instance 11000] instDecidableEqFin |
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| variable {X Y Z : Type*} [Fintype X] |
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| theorem inactiveCount_minus_real (S : FlaggedTensor X Y Z) (m : ℕ) : |
| (inactiveCount (complete S .B m) : ℝ) = |
| ((activeCount S : ℝ) + (inactiveCount S : ℝ)) ^ m - |
| (activeCount S : ℝ) ^ m := by |
| have h := partition_count (complete S .B m) |
| rw [activeCount_minus, Fintype.card_fun, Fintype.card_fin, |
| ← partition_count S] at h |
| have hreal : (activeCount S : ℝ) ^ m + |
| (inactiveCount (complete S .B m) : ℝ) = |
| ((activeCount S : ℝ) + (inactiveCount S : ℝ)) ^ m := by |
| simpa only [Nat.cast_add, Nat.cast_pow] using |
| congrArg (fun n : ℕ => (n : ℝ)) h |
| linarith |
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| theorem activeCount_plus_real (S : FlaggedTensor X Y Z) (center : Color) |
| (hc : center ≠ .B) (m : ℕ) : |
| (activeCount (complete S center m) : ℝ) = |
| ((inactiveCount S : ℝ) + (activeCount S : ℝ)) ^ m - |
| (inactiveCount S : ℝ) ^ m := by |
| have h := partition_count (complete S center m) |
| rw [inactiveCount_plus S center hc m, Fintype.card_fun, |
| Fintype.card_fin, ← partition_count S] at h |
| have hreal : (activeCount (complete S center m) : ℝ) + |
| (inactiveCount S : ℝ) ^ m = |
| ((activeCount S : ℝ) + (inactiveCount S : ℝ)) ^ m := by |
| simpa only [Nat.cast_add, Nat.cast_pow] using |
| congrArg (fun n : ℕ => (n : ℝ)) h |
| rw [add_comm (activeCount S : ℝ) (inactiveCount S : ℝ)] at hreal |
| linarith |
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| theorem activeFraction_ratio (S : FlaggedTensor X Y Z) : |
| activeFraction S = (activeCount S : ℝ) / |
| ((activeCount S : ℝ) + (inactiveCount S : ℝ)) := by |
| rw [activeFraction, ← Nat.cast_add, partition_count] |
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| theorem inactiveFraction_ratio [Nonempty X] (S : FlaggedTensor X Y Z) : |
| 1 - activeFraction S = (inactiveCount S : ℝ) / |
| ((inactiveCount S : ℝ) + (activeCount S : ℝ)) := by |
| rw [add_comm (inactiveCount S : ℝ) (activeCount S : ℝ), |
| ← Nat.cast_add, partition_count] |
| have h := fraction_sum S |
| unfold inactiveFraction at h |
| linarith |
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| theorem counts_pos [Nonempty X] (S : FlaggedTensor X Y Z) |
| (h0 : 0 < activeFraction S) (h1 : activeFraction S < 1) : |
| 0 < (activeCount S : ℝ) ∧ 0 < (inactiveCount S : ℝ) := by |
| have hcard : (0 : ℝ) < Fintype.card X := |
| Nat.cast_pos.mpr Fintype.card_pos |
| have hi : 0 < inactiveFraction S := by |
| linarith [fraction_sum S] |
| constructor |
| · exact (div_pos_iff_of_pos_right hcard).mp h0 |
| · exact (div_pos_iff_of_pos_right hcard).mp hi |
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| end MatrixMultiplication.DualHiddenRates |
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| end |
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| end OAI |
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