| 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 |
| |
