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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection
import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly
import OAI.Computability.PerfectCompleteness.Foundations.CutSlotOutside
namespace OAI
section
namespace PerfectCompleteness.CutProjectionGeometry
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n t : Nat}
theorem sourceProjection_eq_cast
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(hs : slots = projected)
(hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k)))
(x : Domain slots) :
sourceProjection q x = cast (congrArg Domain hs) x := by
cases hs
have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by
funext s k
exact eq_of_heq (hk s k)
subst q
rfl
theorem HPullback_eq_cast
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(hs : slots = projected)
(hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k)))
(f : H projected) :
HPullback q f = cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot => (H ss : Type)) hs.symm) f := by
cases hs
have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by
funext s k
exact eq_of_heq (hk s k)
subst q
apply Subtype.ext
rfl
theorem squarePullback_eq_cast
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(hs : slots = projected)
(hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k)))
(f : squareSpace (H projected)) :
ChildBlockProjection.squarePullback q f = cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot =>
(squareSpace (H ss) : Type)) hs.symm) f := by
cases hs
have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by
funext s k
exact eq_of_heq (hk s k)
subst q
apply Subtype.ext
rfl
theorem arraysPullback_eq_cast (rows : Nat → Nat)
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(hs : slots = projected)
(hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k)))
(arrays : Arrays projected rows) :
ChildBlockProjection.arraysPullback rows q arrays = cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot => Arrays ss rows) hs.symm) arrays := by
cases hs
have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by
funext s k
exact eq_of_heq (hk s k)
subst q
funext node row
apply Subtype.ext
rfl
theorem rawPullback_eq_cast (calls : Nat) (rows : Nat → Nat)
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(hs : slots = projected)
(hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k)))
(block : ChildBlockCardinality.Raw calls rows projected) :
ChildBlockProjection.rawPullback calls rows q block = cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot =>
ChildBlockCardinality.Raw calls rows ss) hs.symm) block := by
cases hs
have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by
funext s k
exact eq_of_heq (hk s k)
subst q
apply Prod.ext
· funext call
apply Subtype.ext
rfl
· funext node row
apply Subtype.ext
rfl
theorem childPullback_heq {C : Type*} [Fintype C] (rows : Nat → Nat)
{left right left' right' : Slots branch (n + 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)) (i : Fin (branch n))
(block : CutChildGrouping.Child (C := C) right rows i)
(block' : CutChildGrouping.Child (C := C) right' rows i)
(hb : HEq block block') :
HEq (ChildAssemblyProjection.childPullback rows q i block)
(ChildAssemblyProjection.childPullback rows q' i block') := by
cases hL
cases hR
have hqq' : q = q' := by
funext s k
exact eq_of_heq (hq s k)
cases hqq'
have hbb' : block = block' := eq_of_heq hb
cases hbb'
rfl
private theorem keep_heq {a b : MixedSupport.Slot} (h : a = b) :
HEq (MixedSupport.Projection.keep a) (MixedSupport.Projection.keep b) := by
cases h
rfl
theorem fillProjection_outside {m : Nat} (p : Path branch n m)
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(left right : Slots branch m → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(s : Slots branch n) (hout : s ∉ Set.range p.slotEmbedding) (k : Fin t) :
HEq (CutProjectionAssembly.fillProjection p outside left right q s k)
(MixedSupport.Projection.keep (CutSlotAssembly.fill p outside left s k)) := by
revert outside left right q s hout k
induction p with
| refl _ =>
intro outside left right q s hout k
exact False.elim (hout ⟨s, rfl⟩)
| step i p ih =>
intro outside left right q s hout k
obtain ⟨j, s⟩ := s
by_cases hji : j = i
· subst j
have htail : s ∉ Set.range p.slotEmbedding := by
rintro ⟨a, ha⟩
exact hout ⟨a, congrArg (fun b => (i, b)) ha⟩
have hfill : CutSlotAssembly.fill (.step i p) outside left (i, s) =
CutSlotAssembly.fill p (fun b => outside (i, b)) left s :=
ite_eq_left (show i = i from rfl)
simp only [CutProjectionAssembly.fillProjection]
exact (CutProjectionAssembly.castProjection_heq _ _ _).trans
((ih (fun b => outside (i, b)) left right q s htail k).trans
(keep_heq (congrFun hfill k).symm))
· have hfill : CutSlotAssembly.fill (.step i p) outside left (j, s) = outside (j, s) :=
ite_eq_right hji
simp only [CutProjectionAssembly.fillProjection, dite_eq_right hji]
exact (CutProjectionAssembly.castProjection_heq _ _ _).trans
(keep_heq (congrFun hfill k).symm)
end
end PerfectCompleteness.CutProjectionGeometry
end
end OAI
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