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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorLaws
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.BaseTensorBudgets
import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessArithmetic
import Mathlib.Analysis.SpecialFunctions.Exp
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualCompletionLaws
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws
universe u
variable {X Y Z : Type u} [Fintype X] [Fintype Y] [Fintype Z]
abbrev Laws (S : FlaggedTensor X Y Z) := ∀ c, FiniteLaw (ColorSlice S c)
def gibbsWeight (a b : ℝ) : ℝ := Real.exp a / (Real.exp a + Real.exp b)
theorem gibbsWeight_nonneg (a b : ℝ) : 0 ≤ gibbsWeight a b :=
div_nonneg (Real.exp_pos a).le (add_pos (Real.exp_pos a) (Real.exp_pos b)).le
theorem gibbsWeight_lt_one (a b : ℝ) : gibbsWeight a b < 1 := by
apply (div_lt_one (add_pos (Real.exp_pos a) (Real.exp_pos b))).mpr
linarith [Real.exp_pos b]
def minusLaws (S : FlaggedTensor X Y Z) (p : Laws S) (m : ℕ) (hm : 0 < m)
(rho : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1) : Laws (complete S .B m)
| .B => conditionalCompletionLaw S .B .B (p .B) (p .B) m hm 0 (by norm_num) (by norm_num)
| .A => conditionalCompletionLaw S .B .A (p .B) (p .A) m hm rho hrho0.le hrho1
| .C => conditionalCompletionLaw S .B .C (p .B) (p .C) m hm rho hrho0.le hrho1
def plusALaws (S : FlaggedTensor X Y Z) (p : Laws S) (m : ℕ) (hm : 0 < m)
(rho hA hC : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1) : Laws (complete S .A m)
| .B => conditionalCompletionLaw S .A .B (p .A) (p .B) m hm (1 - rho)
(sub_nonneg.mpr hrho1.le) (by linarith)
| .A => conditionalCompletionLaw S .A .A (p .A) (p .A) m hm 0 (by norm_num) (by norm_num)
| .C => conditionalCompletionLaw S .A .C (p .A) (p .C) m hm (gibbsWeight hA hC)
(gibbsWeight_nonneg hA hC) (gibbsWeight_lt_one hA hC)
def plusCLaws (S : FlaggedTensor X Y Z) (p : Laws S) (m : ℕ) (hm : 0 < m)
(rho hA hC : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1) : Laws (complete S .C m)
| .B => conditionalCompletionLaw S .C .B (p .C) (p .B) m hm (1 - rho)
(sub_nonneg.mpr hrho1.le) (by linarith)
| .A => conditionalCompletionLaw S .C .A (p .C) (p .A) m hm (gibbsWeight hC hA)
(gibbsWeight_nonneg hC hA) (gibbsWeight_lt_one hC hA)
| .C => conditionalCompletionLaw S .C .C (p .C) (p .C) m hm 0 (by norm_num) (by norm_num)
def baseLeaf : (c : Color) → ColorSlice BaseTwo.flagged c
| .B => ⟨⟨(1, 0, 0), by norm_num [BaseTwo.flagged, BaseTwo.tensor]⟩,
by simp [leafColor, BaseTwo.flagged, BaseTwo.flagB]⟩
| .A => ⟨⟨(0, 0, 1), by norm_num [BaseTwo.flagged, BaseTwo.tensor]⟩,
by simp [leafColor, BaseTwo.flagged, BaseTwo.flagB, BaseTwo.flagA]⟩
| .C => ⟨⟨(0, 1, 0), by norm_num [BaseTwo.flagged, BaseTwo.tensor]⟩,
by simp [leafColor, BaseTwo.flagged, BaseTwo.flagB, BaseTwo.flagA]⟩
def baseLaws : Laws BaseTwo.flagged := fun c => FiniteLaw.pure (baseLeaf c)
def script2 : Script := .step .base .B 2
def script4 : Script := .step script2 .A 2
def script8 : Script := .step script4 .C 2
def script24 : Script := .step script8 .B 3
def script48 : Script := .step script24 .C 2
def script96 : Script := .step script48 .B 2
def laws2 : Laws (Script.tensor BaseTwo.flagged script2) :=
minusLaws BaseTwo.flagged baseLaws 2 (Nat.zero_lt_succ 1) (1 / 2) (by norm_num) (by norm_num)
def laws4 : Laws (Script.tensor BaseTwo.flagged script4) :=
plusALaws (Script.tensor BaseTwo.flagged script2) laws2 2 (Nat.zero_lt_succ 1)
DualWitness.state2.rho DualWitness.state2.hA DualWitness.state2.hC
(by norm_num [DualWitness.state2, DualWitness.initial, DualWitness.minus])
(by norm_num [DualWitness.state2, DualWitness.initial, DualWitness.minus])
def laws8 : Laws (Script.tensor BaseTwo.flagged script8) :=
plusCLaws (Script.tensor BaseTwo.flagged script4) laws4 2 (Nat.zero_lt_succ 1)
DualWitness.state4.rho DualWitness.state4.hA DualWitness.state4.hC
(by norm_num [DualWitness.state4, DualWitness.state2, DualWitness.initial,
DualWitness.minus, DualWitness.plusA])
(by norm_num [DualWitness.state4, DualWitness.state2, DualWitness.initial,
DualWitness.minus, DualWitness.plusA])
def laws24 : Laws (Script.tensor BaseTwo.flagged script24) :=
minusLaws (Script.tensor BaseTwo.flagged script8) laws8 3 (Nat.zero_lt_succ 2)
DualWitness.state8.rho
(by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.1)
(by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.2)
def laws48 : Laws (Script.tensor BaseTwo.flagged script48) :=
plusCLaws (Script.tensor BaseTwo.flagged script24) laws24 2 (Nat.zero_lt_succ 1)
DualWitness.state24.rho DualWitness.state24.hA DualWitness.state24.hC
(by rw [DualWitness.state24_rho]; exact DualWitness.rho24_range.1)
(by rw [DualWitness.state24_rho]; exact DualWitness.rho24_range.2)
def laws96 : Laws (Script.tensor BaseTwo.flagged script96) :=
minusLaws (Script.tensor BaseTwo.flagged script48) laws48 2 (Nat.zero_lt_succ 1)
DualWitness.state48.rho
(by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.1)
(by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.2)
def laws288 : Laws (Script.tensor BaseTwo.flagged Script.dual) :=
plusALaws (Script.tensor BaseTwo.flagged script96) laws96 3 (Nat.zero_lt_succ 2)
DualWitness.state96.rho DualWitness.state96.hA DualWitness.state96.hC
(by rw [DualWitness.state96_rho]; exact DualWitness.rho96_range.1)
(by rw [DualWitness.state96_rho]; exact DualWitness.rho96_range.2)
end MatrixMultiplication.DualCompletionLaws
end
end OAI
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