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2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import OAI.Computability.PerfectCompleteness.Decoding.CleanDecoderOddLists
import OAI.Computability.PerfectCompleteness.Decoding.CleanEndpointProjection
import OAI.Computability.PerfectCompleteness.Machines.CleanRecordDecoderInput
import OAI.Computability.PerfectCompleteness.Repetition.CleanRecordObservation
namespace OAI
section
namespace PerfectCompleteness.CleanRecordProjection
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces
variable {branch : Nat → Nat} {h t v m : Nat} [NeZero m]
{C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch h) → Slots branch h)
omit [NeZero m] in
private theorem sourceProjection_congr (b b' : Bool)
(e e' : SourceOddLists.Occurrences m t) (hb : b = b') (he : e = e') (k : Fin t) :
HEq (CleanEndpointProjection.sourceProjection clauses b e k)
(CleanEndpointProjection.sourceProjection clauses b' e' k) := by
cases hb
cases he
rfl
omit [NeZero m] in
private theorem sourceProjection_tree (i : Fin (branch h))
(source : SourceChildKernel.Source (m := m) (t := t) designated i)
(b : Bool) (leaf : Slots branch h) (k : Fin t) :
HEq (CleanEndpointProjection.sourceProjection clauses b
(SourceChildKernel.tree designated i source leaf) k)
(SourceChildKernel.projection clauses designated i source b leaf k) := by
cases b <;> rfl
omit [NeZero m] in
theorem insideProjection_heq
(record : CleanRecordObservation.Record (C := C) (t := t) rows clauses designated)
(x : CleanRecordObservation.Sample (C := C) (t := t) rows clauses designated)
(hevent : CleanRecordChildBlocks.event rows clauses designated record x = true)
(s : Slots branch (h + 1)) (k : Fin t) :
HEq
(CleanEndpointProjection.insideProjection clauses designated
(CleanRecordChildBlocks.clean rows clauses designated record)
(CleanRecordChildBlocks.visible rows clauses designated record)
(CleanRecordChildBlocks.projected rows clauses designated record)
(CleanSourceRecord.clean_projected designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record)
(fun i => (x i.val).1) s k)
(SourceChildKernel.parentProjection rows clauses designated
(fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2) s k) := by
have hp := congrFun (CleanSourceRecord.projected_eq_raw designated
(D := SourceChildKernel.D (C := C) (t := t) rows clauses designated)
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated)
record x hevent) s.1
have hflag : CleanSourceRecord.rawProjected designated
(D := SourceChildKernel.D (C := C) (t := t) rows clauses designated)
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
s.1 (x s.1).2.2 =
SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2 := by
cases (x s.1).2.2 <;> rfl
by_cases hs : CleanRecordChildBlocks.clean rows clauses designated record s.1 ∧
s.2 = designated s.1
· simp only [CleanEndpointProjection.insideProjection, dite_eq_left hs]
apply (CutProjectionAssembly.castProjection_heq _ _ _).trans
have he : (x s.1).1 = SourceChildKernel.tree designated s.1
((x s.1).1, (x s.1).2.1) s.2 := by
rw [hs.2]
exact (CleanSourceRecord.assemble_designated designated s.1 _ _).symm
exact (sourceProjection_congr clauses _ _ _ _ (hp.trans hflag) he k).trans
(sourceProjection_tree clauses designated s.1 ((x s.1).1, (x s.1).2.1)
(SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2) s.2 k)
· simp only [CleanEndpointProjection.insideProjection, dite_eq_right hs]
apply (CutProjectionAssembly.castProjection_heq _ _ _).trans
have hv := CleanSourceRecord.visible_eq_reveal designated
(D := SourceChildKernel.D (C := C) (t := t) rows clauses designated)
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated)
(SourceChildKernel.zero (C := C) (t := t) rows clauses designated)
record x hevent
have he : CleanRecordChildBlocks.visible rows clauses designated record s.1 ⟨s.2, hs⟩ =
SourceChildKernel.tree designated s.1 ((x s.1).1, (x s.1).2.1) s.2 :=
congrArg (fun visible => visible s.1 ⟨s.2, hs⟩) hv
exact (sourceProjection_congr clauses _ _ _ _ (hp.trans hflag) he k).trans
(sourceProjection_tree clauses designated s.1 ((x s.1).1, (x s.1).2.1)
(SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2) s.2 k)
private theorem fillProjection_congr {n : Nat} (p : Path branch n (h + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
{left left' right right' : Slots branch (h + 1) → Fin t → MixedSupport.Slot}
(hl : left = left') (hr : right = right')
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(q' : ∀ s k, MixedSupport.Projection (left' s k) (right' s k))
(hq : ∀ s k, HEq (q s k) (q' s k)) (s : Slots branch n) (k : Fin t) :
HEq (CutProjectionAssembly.fillProjection p outside left right q s k)
(CutProjectionAssembly.fillProjection p outside left' right' q' s k) := by
cases hl
cases hr
have hqq : q = q' := funext (fun s => funext (fun k => eq_of_heq (hq s k)))
cases hqq
rfl
omit [NeZero m] in
theorem fillProjection_heq {n : Nat} (p : Path branch n (h + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(record : CleanRecordObservation.Record (C := C) (t := t) rows clauses designated)
(x : CleanRecordObservation.Sample (C := C) (t := t) rows clauses designated)
(hevent : CleanRecordChildBlocks.event rows clauses designated record x = true)
(s : Slots branch n) (k : Fin t) :
HEq
(CutProjectionAssembly.fillProjection p outside _ _
(CleanEndpointProjection.insideProjection clauses designated
(CleanRecordChildBlocks.clean rows clauses designated record)
(CleanRecordChildBlocks.visible rows clauses designated record)
(CleanRecordChildBlocks.projected rows clauses designated record)
(CleanSourceRecord.clean_projected designated
(SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record)
(fun i => (x i.val).1)) s k)
(CutProjectionAssembly.fillProjection p outside _ _
(SourceChildKernel.parentProjection rows clauses designated
(fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2)) s k) :=
fillProjection_congr p outside
(CleanRecordObservation.leftInside_eq rows clauses designated record x hevent)
(CleanRecordObservation.rightInside_eq rows clauses designated record x hevent) _ _
(insideProjection_heq rows clauses designated record x hevent) s k
theorem decoderProjection_heq {root : Nat} (repeats : Nat → Nat)
(upper : HierarchicalArrays.Nodes branch root)
(d : HierarchicalFrozenTables.LowerNodes upper (h + 1))
(cut : OwnInputReference.Cut upper (HierarchicalLeftDecoder.LowerNode upper (h + 1) d))
(outside : Slots branch root → Fin t → MixedSupport.Slot)
(placeholder : Slots branch (h + 1) → Fin t → MixedSupport.Slot)
(record : CleanRecordDecoderInput.Record (t := t) rows repeats upper d cut clauses designated)
(external : CleanRecordDecoderInput.Exterior rows repeats upper d cut outside placeholder)
(x : CleanRecordDecoderInput.Sample (t := t) rows repeats upper d cut clauses designated)
(hevent : CleanRecordChildBlocks.event rows clauses designated record x = true)
(s : Slots branch root) (k : Fin t) :
HEq (CleanDecoderOddLists.fullProjection
(CleanRecordDecoderInput.exposed rows repeats upper d cut outside placeholder
clauses designated record external)
(CleanRecordDecoderInput.sources rows repeats upper d cut clauses designated record x) s k)
(CutProjectionAssembly.fillProjection (CleanRecordDecoderInput.path upper d cut) outside _ _
(SourceChildKernel.parentProjection rows clauses designated
(fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2)) s k) :=
fillProjection_heq rows clauses designated (CleanRecordDecoderInput.path upper d cut)
outside record x hevent s k
end
end PerfectCompleteness.CleanRecordProjection
end
end OAI
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