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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.LinearAlgebra.MatrixMultiplication.Duality.RootIndependence
import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformInstances
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.BaseConditionalReaders
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualProgramIndependence
universe u w
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
open CompletionColorLaws DualCompletionLaws DualInformation TopologicalFlatten DualUniform
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
variable {X Y Z : Type u} {Code : Type w} [Fintype X] [Fintype Y] [Fintype Z]
[Fintype Code] [Inhabited Code] {depth m : ℕ}
theorem minusLaws_independent (S : FlaggedTensor X Y Z) (r : ReadableTensor S)
(p : Laws S) (m : ℕ) (hm : 0 < m) (rho : ℝ)
(hrho0 : 0 < rho) (hrho1 : rho < 1) (h : ProgramLawsIndependent S r p) :
ProgramLawsIndependent (complete S .B m) (ReadableTensor.raise S r .B m)
(minusLaws S p m hm rho hrho0 hrho1) := by
intro c n hn
cases c
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.B .B (p .B) (p .B) 0 (le_refl 0) zero_lt_one (Or.inr rfl) hm
(by intro hh; simp [ownerPair] at hh) (h .B) (h .B) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.B .A (p .B) (p .A) rho hrho0.le hrho1 (Or.inl (by intro he; cases he)) hm
(by intro hh; simp [ownerPair] at hh) (h .B) (h .A) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.B .C (p .B) (p .C) rho hrho0.le hrho1 (Or.inl (by intro he; cases he)) hm
(by intro hh; simp [ownerPair] at hh) (h .B) (h .C) n hn
theorem plusALaws_independent (S : FlaggedTensor X Y Z) (r : ReadableTensor S)
(p : Laws S) (m : ℕ) (hm : 0 < m) (rho hA hC : ℝ)
(hrho0 : 0 < rho) (hrho1 : rho < 1) (h : ProgramLawsIndependent S r p)
(hu : UniformLaws S p) :
ProgramLawsIndependent (complete S .A m) (ReadableTensor.raise S r .A m)
(plusALaws S p m hm rho hA hC hrho0 hrho1) := by
intro c n hn
cases c
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.A .B (p .A) (p .B) (1-rho) (sub_nonneg.mpr hrho1.le) (sub_lt_self 1 hrho0)
(Or.inl (by intro he; cases he)) hm (by intro hh; simp [ownerPair] at hh) (h .A) (h .B) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.A .A (p .A) (p .A) 0 (le_refl 0) zero_lt_one (Or.inr rfl) hm
(by intro hh; simp [ownerPair] at hh) (h .A) (h .A) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.A .C (p .A) (p .C) (gibbsWeight hA hC)
(gibbsWeight_nonneg hA hC) (gibbsWeight_lt_one hA hC) (Or.inl (by intro he; cases he)) hm
(fun _ x => (hu.A x).trans (hu.C x).symm) (h .A) (h .C) n hn
theorem plusCLaws_independent (S : FlaggedTensor X Y Z) (r : ReadableTensor S)
(p : Laws S) (m : ℕ) (hm : 0 < m) (rho hA hC : ℝ)
(hrho0 : 0 < rho) (hrho1 : rho < 1) (h : ProgramLawsIndependent S r p)
(hu : UniformLaws S p) :
ProgramLawsIndependent (complete S .C m) (ReadableTensor.raise S r .C m)
(plusCLaws S p m hm rho hA hC hrho0 hrho1) := by
intro c n hn
cases c
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.C .B (p .C) (p .B) (1-rho) (sub_nonneg.mpr hrho1.le) (sub_lt_self 1 hrho0)
(Or.inl (by intro he; cases he)) hm (by intro hh; simp [ownerPair] at hh) (h .C) (h .B) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.C .A (p .C) (p .A) (gibbsWeight hC hA)
(gibbsWeight_nonneg hC hA) (gibbsWeight_lt_one hC hA) (Or.inl (by intro he; cases he)) hm
(fun _ x => (hu.C x).trans (hu.A x).symm) (h .C) (h .A) n hn
· exact conditionalCompletionLaw_program_factorization S r.program r.context_eq r.view_eq
.C .C (p .C) (p .C) 0 (le_refl 0) zero_lt_one (Or.inr rfl) hm
(by intro hh; simp [ownerPair] at hh) (h .C) (h .C) n hn
abbrev dualReadable (s : Script) :=
ReadableTensor.script BaseTwo.flagged BaseConditionalReaders.baseTwoReadable s
theorem base_independent : ProgramLawsIndependent BaseTwo.flagged
BaseConditionalReaders.baseTwoReadable baseLaws := by
intro c n
exact Fin.elim0 n
theorem laws2_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script2)
(dualReadable script2) laws2 :=
minusLaws_independent _ _ _ _ _ _ _ _ base_independent
theorem laws4_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script4)
(dualReadable script4) laws4 :=
plusALaws_independent _ _ _ _ _ _ _ _ _ _ laws2_independent laws2_uniform
theorem laws8_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script8)
(dualReadable script8) laws8 :=
plusCLaws_independent _ _ _ _ _ _ _ _ _ _ laws4_independent laws4_uniform
theorem laws24_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script24)
(dualReadable script24) laws24 :=
minusLaws_independent _ _ _ _ _ _ _ _ laws8_independent
theorem laws48_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script48)
(dualReadable script48) laws48 :=
plusCLaws_independent _ _ _ _ _ _ _ _ _ _ laws24_independent laws24_uniform
theorem laws96_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged script96)
(dualReadable script96) laws96 :=
minusLaws_independent _ _ _ _ _ _ _ _ laws48_independent
theorem laws288_independent : ProgramLawsIndependent (Script.tensor BaseTwo.flagged Script.dual)
(dualReadable Script.dual) laws288 :=
plusALaws_independent _ _ _ _ _ _ _ _ _ _ laws96_independent laws96_uniform
end MatrixMultiplication.DualProgramIndependence
end
end OAI
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