File size: 7,636 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.LinearAlgebra.MatrixMultiplication.Duality.CompletionLaws
import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformUpdates
import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformMixtures
import OAI.LinearAlgebra.MatrixMultiplication.Completion.Partitions
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
universe uCoord
namespace MatrixMultiplication.DualUniform
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws
open CompletionLaws DualCompletionLaws DualUniformConditioning DualUniformUpdates
open DualUniformMixtures CompletionPartitions
open scoped BigOperators
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
variable {X Y Z : Type uCoord} [Fintype X] [Fintype Y] [Fintype Z]
structure UniformLaws (S : FlaggedTensor X Y Z) (p : Laws S) : Prop where
B : UniformCoordinate (p .B) (fun a => a.val.val.1) S.flagB
A : UniformCoordinate (p .A) (fun a => a.val.val.1) (fun x => ¬ S.flagB x)
C : UniformCoordinate (p .C) (fun a => a.val.val.1) (fun x => ¬ S.flagB x)
theorem UniformLaws.color (S : FlaggedTensor X Y Z) (p : Laws S)
(h : UniformLaws S p) (c : Color) :
UniformCoordinate (p c) (fun a => a.val.val.1)
(fun x => if c = .B then S.flagB x else ¬ S.flagB x) := by
cases c
· simpa using h.B
· simpa using h.A
· simpa using h.C
theorem UniformLaws.entropy_B (S : FlaggedTensor X Y Z) (p : Laws S)
(h : UniformLaws S p) :
finiteEntropy ((p .B).map (fun a => a.val.val.1)).mass = Real.log (activeCount S) :=
uniform_coordinate_entropy (p .B) _ _ h.B
theorem UniformLaws.entropy_A (S : FlaggedTensor X Y Z) (p : Laws S)
(h : UniformLaws S p) :
finiteEntropy ((p .A).map (fun a => a.val.val.1)).mass = Real.log (inactiveCount S) :=
uniform_coordinate_entropy (p .A) _ _ h.A
theorem UniformLaws.entropy_C (S : FlaggedTensor X Y Z) (p : Laws S)
(h : UniformLaws S p) :
finiteEntropy ((p .C).map (fun a => a.val.val.1)).mass = Real.log (inactiveCount S) :=
uniform_coordinate_entropy (p .C) _ _ h.C
theorem conditionalCompletionLaw_uniform_common
(S : FlaggedTensor X Y Z) (center output : Color)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(m : ℕ) (hm : 0 < m) (α : ℝ) (hα : 0 ≤ α) (hα' : α < 1)
(P : X → Prop)
(hc : UniformCoordinate pc (fun a => a.val.val.1) P)
(ho : UniformCoordinate po (fun a => a.val.val.1) P) :
UniformCoordinate
(conditionalCompletionLaw S center output pc po m hm α hα hα')
(fun a => a.val.val.1) (fun x => ∀ i, P (x i)) :=
conditionalCompletionLaw_uniform_independent S center output pc po m hm α hα hα' P
(colorMix_uniform_common S center output pc po α hα hα'.le P hc ho)
(colorMix_independent_uniform S center output pc po α hα hα'.le P hc ho)
private theorem inactive_uniform_update
(S : FlaggedTensor X Y Z) (center output : Color) (hcenter : center ≠ .B)
(pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output))
(m : ℕ) (hm : 0 < m) (α : ℝ) (hα : 0 ≤ α) (hα' : α < 1)
(hc : UniformCoordinate pc (fun a => a.val.val.1) (fun x => ¬ S.flagB x))
(ho : UniformCoordinate po (fun a => a.val.val.1) (fun x => ¬ S.flagB x)) :
UniformCoordinate
(conditionalCompletionLaw S center output pc po m hm α hα hα')
(fun a => a.val.val.1) (fun x => ¬ (complete S center m).flagB x) := by
apply uniform_coordinate_pred_congr _ _ (fun x => ∀ i, ¬ S.flagB (x i))
· intro x
change (∀ i, ¬ S.flagB (x i)) ↔ ¬ raisedFlag center .B (fun i => S.flagB (x i))
simp [raisedFlag, hcenter]
· exact conditionalCompletionLaw_uniform_common S center output pc po m hm α hα hα'
(fun x => ¬ S.flagB x) hc ho
theorem minusLaws_uniform (S : FlaggedTensor X Y Z) (p : Laws S)
(m : ℕ) (hm : 0 < m) (rho : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1)
(h : UniformLaws S p) (hrho : rho = activeFraction S) :
UniformLaws (complete S .B m) (minusLaws S p m hm rho hrho0 hrho1) := by
constructor
· exact conditionalCompletionLaw_uniform_common S .B .B (p .B) (p .B) m hm 0
(by norm_num) (by norm_num) S.flagB h.B h.B
· have hu := conditionalCompletionLaw_uniform_incident S .B .A (Or.inl rfl)
(p .B) (p .A) m hm rho hrho0.le hrho1
(colorMix_uniform_complement S .B .A (p .B) (p .A) rho hrho0.le hrho1.le
S.flagB h.B h.A hrho)
simpa only [minusLaws, reduceCtorEq, ite_false] using hu
· have hu := conditionalCompletionLaw_uniform_incident S .B .C (Or.inl rfl)
(p .B) (p .C) m hm rho hrho0.le hrho1
(colorMix_uniform_complement S .B .C (p .B) (p .C) rho hrho0.le hrho1.le
S.flagB h.B h.C hrho)
simpa only [minusLaws, reduceCtorEq, ite_false] using hu
theorem plusALaws_uniform (S : FlaggedTensor X Y Z) (p : Laws S)
(m : ℕ) (hm : 0 < m) (rho hA hC : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1)
(h : UniformLaws S p) (hrho : rho = activeFraction S) :
UniformLaws (complete S .A m) (plusALaws S p m hm rho hA hC hrho0 hrho1) := by
constructor
· have hu := conditionalCompletionLaw_uniform_incident S .A .B (Or.inr rfl)
(p .A) (p .B) m hm (1-rho) (sub_nonneg.mpr hrho1.le) (by linarith)
(colorMix_uniform_complement_reverse S .A .B (p .A) (p .B) rho
(sub_nonneg.mpr hrho1.le) (by linarith) S.flagB h.A h.B hrho)
simpa only [plusALaws, ite_eq_left rfl, ite_true] using hu
· exact inactive_uniform_update S .A .A (by intro h; cases h) (p .A) (p .A) m hm 0
(by norm_num) (by norm_num) h.A h.A
· exact inactive_uniform_update S .A .C (by intro h; cases h) (p .A) (p .C) m hm
(gibbsWeight hA hC) (gibbsWeight_nonneg hA hC) (gibbsWeight_lt_one hA hC) h.A h.C
theorem plusCLaws_uniform (S : FlaggedTensor X Y Z) (p : Laws S)
(m : ℕ) (hm : 0 < m) (rho hA hC : ℝ) (hrho0 : 0 < rho) (hrho1 : rho < 1)
(h : UniformLaws S p) (hrho : rho = activeFraction S) :
UniformLaws (complete S .C m) (plusCLaws S p m hm rho hA hC hrho0 hrho1) := by
constructor
· have hu := conditionalCompletionLaw_uniform_incident S .C .B (Or.inr rfl)
(p .C) (p .B) m hm (1-rho) (sub_nonneg.mpr hrho1.le) (by linarith)
(colorMix_uniform_complement_reverse S .C .B (p .C) (p .B) rho
(sub_nonneg.mpr hrho1.le) (by linarith) S.flagB h.C h.B hrho)
simpa only [plusCLaws, ite_eq_left rfl, ite_true] using hu
· exact inactive_uniform_update S .C .A (by intro h; cases h) (p .C) (p .A) m hm
(gibbsWeight hC hA) (gibbsWeight_nonneg hC hA) (gibbsWeight_lt_one hC hA) h.C h.A
· exact inactive_uniform_update S .C .C (by intro h; cases h) (p .C) (p .C) m hm 0
(by norm_num) (by norm_num) h.C h.C
private theorem pure_map_mass {A U : Type*} [Fintype A] [Fintype U]
(a : A) (f : A → U) (x : U) :
((FiniteLaw.pure a).map f).mass x = if f a = x then 1 else 0 := by
rw [FiniteLaw.map_mass, Finset.sum_eq_single a]
· simp [FiniteLaw.pure]
· intro b _ hb
simp [FiniteLaw.pure, hb]
· simp
theorem baseLaws_uniform : UniformLaws BaseTwo.flagged baseLaws := by
have hc : Fintype.card {x // BaseTwo.flagged.flagB x} = 1 := by
exact @Fintype.card_subtype_eq (Fin 2) 1
(inferInstanceAs (Fintype {x // BaseTwo.flagged.flagB x}))
constructor <;> intro x <;> rw [baseLaws, pure_map_mass] <;>
fin_cases x <;>
norm_num [baseLeaf, BaseTwo.flagged, BaseTwo.flagB, Fintype.card_subtype_compl]
all_goals rw [hc]
end MatrixMultiplication.DualUniform
end
end OAI
|