openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /UniformInstances.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.Uniform | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualUniform | |
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws | |
| open CompletionLaws DualCompletionLaws CompletionPartitions | |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq | |
| attribute [local instance 11000] instDecidableEqFin | |
| private theorem coord_nonempty (s : Script) : Nonempty (Script.Coord (Fin 2) s) := by | |
| induction s with | |
| | base => exact inferInstanceAs (Nonempty (Fin 2)) | |
| | step prior center m ih => | |
| let _ := ih | |
| exact inferInstanceAs (Nonempty (Fin m → Script.Coord (Fin 2) prior)) | |
| attribute [local instance] coord_nonempty | |
| theorem base_activeFraction : activeFraction BaseTwo.flagged = (1/2 : ℝ) := by | |
| have hc : activeCount BaseTwo.flagged = 1 := | |
| @Fintype.card_subtype_eq (Fin 2) 1 | |
| (inferInstanceAs (Fintype {x // BaseTwo.flagged.flagB x})) | |
| rw [activeFraction, hc] | |
| norm_num | |
| theorem rho2 : activeFraction (Script.tensor BaseTwo.flagged script2) = | |
| DualWitness.state2.rho := by | |
| change activeFraction (complete BaseTwo.flagged .B 2) = _ | |
| rw [activeFraction_minus, base_activeFraction] | |
| rfl | |
| theorem rho4 : activeFraction (Script.tensor BaseTwo.flagged script4) = | |
| DualWitness.state4.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script2) .A 2) = _ | |
| rw [activeFraction_plus _ .A (by intro h; cases h), rho2] | |
| rfl | |
| theorem rho8 : activeFraction (Script.tensor BaseTwo.flagged script8) = | |
| DualWitness.state8.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script4) .C 2) = _ | |
| rw [activeFraction_plus _ .C (by intro h; cases h), rho4] | |
| rfl | |
| theorem rho24 : activeFraction (Script.tensor BaseTwo.flagged script24) = | |
| DualWitness.state24.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script8) .B 3) = _ | |
| rw [activeFraction_minus, rho8] | |
| rfl | |
| theorem rho48 : activeFraction (Script.tensor BaseTwo.flagged script48) = | |
| DualWitness.state48.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script24) .C 2) = _ | |
| rw [activeFraction_plus _ .C (by intro h; cases h), rho24] | |
| rfl | |
| theorem rho96 : activeFraction (Script.tensor BaseTwo.flagged script96) = | |
| DualWitness.state96.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script48) .B 2) = _ | |
| rw [activeFraction_minus, rho48] | |
| rfl | |
| theorem rho288 : activeFraction (Script.tensor BaseTwo.flagged Script.dual) = | |
| DualWitness.state288.rho := by | |
| change activeFraction (complete (Script.tensor BaseTwo.flagged script96) .A 3) = _ | |
| rw [activeFraction_plus _ .A (by intro h; cases h), rho96] | |
| rfl | |
| theorem laws2_uniform : UniformLaws (Script.tensor BaseTwo.flagged script2) laws2 := | |
| minusLaws_uniform BaseTwo.flagged baseLaws 2 (Nat.zero_lt_succ _) (1/2) (by norm_num) | |
| (by norm_num) baseLaws_uniform base_activeFraction.symm | |
| theorem laws4_uniform : UniformLaws (Script.tensor BaseTwo.flagged script4) laws4 := | |
| plusALaws_uniform _ laws2 2 (Nat.zero_lt_succ _) | |
| 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]) | |
| laws2_uniform rho2.symm | |
| theorem laws8_uniform : UniformLaws (Script.tensor BaseTwo.flagged script8) laws8 := | |
| plusCLaws_uniform _ laws4 2 (Nat.zero_lt_succ _) | |
| 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]) laws4_uniform rho4.symm | |
| theorem laws24_uniform : UniformLaws (Script.tensor BaseTwo.flagged script24) laws24 := | |
| minusLaws_uniform _ laws8 3 (Nat.zero_lt_succ _) DualWitness.state8.rho | |
| (by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.1) | |
| (by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.2) | |
| laws8_uniform rho8.symm | |
| theorem laws48_uniform : UniformLaws (Script.tensor BaseTwo.flagged script48) laws48 := | |
| plusCLaws_uniform _ laws24 2 (Nat.zero_lt_succ _) | |
| 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) | |
| laws24_uniform rho24.symm | |
| theorem laws96_uniform : UniformLaws (Script.tensor BaseTwo.flagged script96) laws96 := | |
| minusLaws_uniform _ laws48 2 (Nat.zero_lt_succ _) DualWitness.state48.rho | |
| (by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.1) | |
| (by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.2) | |
| laws48_uniform rho48.symm | |
| theorem laws288_uniform : UniformLaws (Script.tensor BaseTwo.flagged Script.dual) laws288 := | |
| plusALaws_uniform _ laws96 3 (Nat.zero_lt_succ _) | |
| 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) | |
| laws96_uniform rho96.symm | |
| end MatrixMultiplication.DualUniform | |
| end | |
| end OAI | |