openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /ScriptIndependence.lean
| 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 | |