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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 | import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage
import OAI.Computability.PerfectCompleteness.Decoding.ChildBlockProjection
namespace OAI
section
namespace PerfectCompleteness.ChildCallNumbering
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
variable {C : Type*} [Fintype C] {branch : Nat → Nat} {n t : Nat}
abbrev Raw (rows : Nat → Nat) (slots : Slots branch n → Fin t → MixedSupport.Slot) :=
(C → squareSpace (H slots)) × Arrays slots rows
instance rawFintype (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Fintype (Raw (C := C) rows slots) := Fintype.ofFinite _
instance rawNonempty (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Nonempty (Raw (C := C) rows slots) := ⟨(fun _ => 0, fun _ _ => 0)⟩
def equiv (rows : Nat → Nat) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
Raw (C := C) rows slots ≃ₗ[F2]
ChildBlockCardinality.Raw (Fintype.card C) rows slots where
toFun x := (fun k => x.1 ((Fintype.equivFin C).symm k), x.2)
invFun x := (fun c => x.1 (Fintype.equivFin C c), x.2)
left_inv x := by
apply Prod.ext
· funext c
simp only [Equiv.symm_apply_apply]
· rfl
right_inv x := by
apply Prod.ext
· funext k
simp only [Equiv.apply_symm_apply]
· rfl
map_add' x y := rfl
map_smul' c x := rfl
@[simp] theorem equiv_calls (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(x : Raw (C := C) rows slots) (k : Fin (Fintype.card C)) :
(equiv rows slots x).1 k = x.1 ((Fintype.equivFin C).symm k) := rfl
@[simp] theorem equiv_arrays (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(x : Raw (C := C) rows slots) : (equiv rows slots x).2 = x.2 := rfl
@[simp] theorem equiv_symm_calls (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(x : ChildBlockCardinality.Raw (Fintype.card C) rows slots) (c : C) :
((equiv (C := C) rows slots).symm x).1 c = x.1 (Fintype.equivFin C c) := rfl
@[simp] theorem equiv_symm_arrays (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(x : ChildBlockCardinality.Raw (Fintype.card C) rows slots) :
((equiv (C := C) rows slots).symm x).2 = x.2 := rfl
def pullback (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
Raw (C := C) rows projected →ₗ[F2] Raw (C := C) rows slots where
toFun x := (fun c => ChildBlockProjection.squarePullback p (x.1 c),
ChildBlockProjection.arraysPullback rows p x.2)
map_add' x y := by
apply Prod.ext
· funext c
exact map_add (ChildBlockProjection.squarePullback p) (x.1 c) (y.1 c)
· exact map_add (ChildBlockProjection.arraysPullback rows p) x.2 y.2
map_smul' a x := by
apply Prod.ext
· funext c
exact map_smul (ChildBlockProjection.squarePullback p) a (x.1 c)
· exact map_smul (ChildBlockProjection.arraysPullback rows p) a x.2
omit [Fintype C] in
@[simp] theorem pullback_calls (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(x : Raw (C := C) rows projected) (c : C) :
(pullback rows p x).1 c = ChildBlockProjection.squarePullback p (x.1 c) := rfl
omit [Fintype C] in
@[simp] theorem pullback_arrays (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(x : Raw (C := C) rows projected) :
(pullback rows p x).2 = ChildBlockProjection.arraysPullback rows p x.2 := rfl
theorem equiv_pullback (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(x : Raw (C := C) rows projected) :
equiv rows slots (pullback rows p x) =
ChildBlockProjection.rawPullback (Fintype.card C) rows p
(equiv rows projected x) := rfl
omit [Fintype C] in
theorem pullback_cast (rows : Nat → Nat)
{slots target projected : Slots branch n → Fin t → MixedSupport.Slot}
(h : slots = target)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(x : Raw (C := C) rows projected) :
cast (congrArg (Raw (C := C) rows) h) (pullback rows p x) =
pullback rows
(cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot =>
∀ s k, MixedSupport.Projection (ss s k) (projected s k)) h) p) x := by
cases h
rfl
theorem uniform_equiv (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
(FiniteDistribution.uniform (Raw (C := C) rows slots)).pushforward
(equiv (C := C) rows slots) =
FiniteDistribution.uniform (ChildBlockCardinality.Raw (Fintype.card C) rows slots) :=
UniformLinearImage.uniform_pushforward_linearMap
(equiv (C := C) rows slots).toLinearMap (equiv (C := C) rows slots).surjective
theorem card_le (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Fintype.card (Raw (C := C) rows slots) ≤
ChildBlockCardinality.bound branch n t (Fintype.card C) rows := by
calc
Fintype.card (Raw (C := C) rows slots) =
Fintype.card (ChildBlockCardinality.Raw (Fintype.card C) rows slots) :=
Fintype.card_congr (equiv (C := C) rows slots).toEquiv
_ ≤ _ := by
simpa only [Nat.card_eq_fintype_card] using
ChildBlockCardinality.card_le (Fintype.card C) rows slots
end
end PerfectCompleteness.ChildCallNumbering
end
end OAI
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