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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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