File size: 2,953 Bytes
cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Partitions
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualHiddenRates
open MatrixMultiplication.Foundation RecursiveCompletion CompletionPartitions
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
variable {X Y Z : Type*} [Fintype X]
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
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
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]
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
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
end MatrixMultiplication.DualHiddenRates
end
end OAI
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