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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyWords
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexConditionalHierarchyStage
import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexPrefixSeparationOrientations
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexGeometricStageCoordinates
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionGeometricValues
open MatrixMultiplication.Foundation
open CompletionHierarchyWords ConditionalPrefixWords
open ConditionalHierarchyStage PrefixSeparationPolynomialStage PrefixSeparationOrientations
open StageHierarchyResources
open scoped Classical
variable {A X Y Z : Type*} [Fintype A] (H : ReadableHierarchy A X Y Z)
variable {Position : Type*} [Fintype Position] {counts : A → ℕ} {n t : ℕ}
abbrev size (counts : A → ℕ) (n t : ℕ) := stageRefinementCount counts H.labels n t
theorem actualPool_decode (old : ExactPrefix counts H.labels n t Position)
(code : Fin (size H counts n t)) :
decodePool (actualPool counts H.labels n t id) (size H counts n t)
(actualPool_card counts H.labels n t id) old code =
((conditionalPoolEquivFin old).symm code).val := rfl
theorem decode_eq_iff_eligible (old : ExactPrefix counts H.labels n t Position)
(code : Fin (size H counts n t))
(u : Position → X) (v : Position → Y) (w : Position → Z)
(word : Position → A) (hword : Represents H id old u v w word) :
decodePool (actualPool counts H.labels n t id) (size H counts n t)
(actualPool_card counts H.labels n t id) old code = (fun i => H.labels n (word i)) ↔
eligible H id (appendCode old code) u v w := by
rw [actualPool_decode]
exact (eligible_appendCode_iff_of_represents H old code u v w word hword).symm
theorem raw_XY_normal
(old : ExactPrefix counts H.labels n t Position) (hn : n < H.depth)
(hp : H.pair n = .xy)
(x : (Position → X) × InformedOutput (size H counts n t))
(y : (Position → Y) × InformedOutput (size H counts n t))
(z : (Position → Z) × MissingOutput (size H counts n t))
(word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) :
rawPoolLeading (actualPool counts H.labels n t id) (size H counts n t)
(stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id)
(readXYX H id hp) (readXYY H id hp) old x y z =
if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then
(GeometricStageCoordinates.xy (size H counts n t)).tensor x.2.2 y.2.2 z.2.2
else 0 := by
have hx : readXYX H id hp old x.1 = fun i => H.labels n (word i) := by
rw [hword.1]
exact readXYX_on_original_word H id hp old word hn hword.2.2.2
have hy : readXYY H id hp old y.1 = fun i => H.labels n (word i) := by
rw [hword.2.1]
exact readXYY_on_original_word H id hp old word hn hword.2.2.2
rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readXYX H id hp) (readXYY H id hp) (fun _ _ _ _ => fun i => H.labels n (word i)) old x y z hx hy]
unfold decodedLeading
simp only [decode_eq_iff_eligible H old x.2.1 x.1 y.1 z.1 word hword, GeometricStageCoordinates.xy_tensor_apply]
simp only [ite_and]
theorem raw_XZ_normal
(old : ExactPrefix counts H.labels n t Position) (hn : n < H.depth)
(hp : H.pair n = .xz)
(x : (Position → X) × InformedOutput (size H counts n t))
(y : (Position → Y) × MissingOutput (size H counts n t))
(z : (Position → Z) × InformedOutput (size H counts n t))
(word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) :
rawPoolLeadingXZ (actualPool counts H.labels n t id) (size H counts n t)
(stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id)
(readXZX H id hp) (readXZZ H id hp) old x y z =
if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then
(GeometricStageCoordinates.xz (size H counts n t)).tensor x.2.2 y.2.2 z.2.2
else 0 := by
have hx : readXZX H id hp old x.1 = fun i => H.labels n (word i) := by
rw [hword.1]
exact readXZX_on_original_word H id hp old word hn hword.2.2.2
have hz : readXZZ H id hp old z.1 = fun i => H.labels n (word i) := by
rw [hword.2.2.1]
exact readXZZ_on_original_word H id hp old word hn hword.2.2.2
unfold rawPoolLeadingXZ
rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readXZX H id hp) (readXZZ H id hp) (fun _ _ _ _ => fun i => H.labels n (word i)) old x z y hx hz]
unfold decodedLeading
simp only [decode_eq_iff_eligible H old x.2.1 x.1 y.1 z.1 word hword, GeometricStageCoordinates.xz_tensor_apply]
have hswap :
(x.2.1 = z.2.1 ∧ x.2.1 = y.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 ∧ x.2.2 = z.2.2) ↔
(x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 ∧ x.2.2 = z.2.2) :=
and_left_comm
simp only [hswap, ite_and]
theorem raw_YZ_normal
(old : ExactPrefix counts H.labels n t Position) (hn : n < H.depth)
(hp : H.pair n = .yz)
(x : (Position → X) × MissingOutput (size H counts n t))
(y : (Position → Y) × InformedOutput (size H counts n t))
(z : (Position → Z) × InformedOutput (size H counts n t))
(word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) :
rawPoolLeadingYZ (actualPool counts H.labels n t id) (size H counts n t)
(stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id)
(readYZY H id hp) (readYZZ H id hp) old x y z =
if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then
(GeometricStageCoordinates.yz (size H counts n t)).tensor x.2.2 y.2.2 z.2.2
else 0 := by
have hy : readYZY H id hp old y.1 = fun i => H.labels n (word i) := by
rw [hword.2.1]
exact readYZY_on_original_word H id hp old word hn hword.2.2.2
have hz : readYZZ H id hp old z.1 = fun i => H.labels n (word i) := by
rw [hword.2.2.1]
exact readYZZ_on_original_word H id hp old word hn hword.2.2.2
unfold rawPoolLeadingYZ
rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readYZY H id hp) (readYZZ H id hp) (fun _ _ _ _ => fun i => H.labels n (word i)) old y z x hy hz]
have hguard :
(y.2.1 = z.2.1 ∧ y.2.1 = x.2.1 ∧
decodePool (actualPool counts H.labels n t id) (size H counts n t)
(actualPool_card counts H.labels n t id) old y.2.1 = (fun i => H.labels n (word i)) ∧
y.2.2 = z.2.2) ↔
(x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
eligible H id (appendCode old x.2.1) x.1 y.1 z.1 ∧ y.2.2 = z.2.2) := by
constructor
· rintro ⟨hyz, hyx, hdecode, hpair⟩
refine ⟨hyx.symm, hyx.symm.trans hyz, ?_, hpair⟩
apply (decode_eq_iff_eligible H old x.2.1 x.1 y.1 z.1 word hword).mp
simpa only [hyx] using hdecode
· rintro ⟨hxy, hxz, he, hpair⟩
refine ⟨hxy.symm.trans hxz, hxy.symm, ?_, hpair⟩
have hdecode := (decode_eq_iff_eligible H old x.2.1 x.1 y.1 z.1 word hword).mpr he
simpa only [hxy] using hdecode
unfold decodedLeading
simp only [hguard, GeometricStageCoordinates.yz_tensor_apply]
simp only [ite_and]
end MatrixMultiplication.CompletionGeometricValues
end
end OAI
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