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namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionLabels.TopologicalFlatten
open MatrixMultiplication.Foundation RecursiveCompletion
universe u v w z
theorem labelRecordOf_eq_iff {A : Type u} {Label : ℕ → Type v}
(labels : ∀ n, A → Label n) (n : ℕ) (a b : A) :
labelRecordOf labels n a = labelRecordOf labels n b ↔
∀ k, k < n → labels k a = labels k b := by
constructor
· intro h k hk
exact label_eq_of_record_eq labels h hk
· intro h
revert h
induction n with
| zero => intro h; rfl
| succ n ih =>
intro h
apply Prod.ext
· exact ih (fun k hk => h k (Nat.lt_succ_of_lt hk))
· exact h n (Nat.lt_succ_self n)
namespace Program
variable {A : Type u} {Coord : Color → Type v}
{Context OtherContext : Type z} {Code : Type w} {d e : ℕ}
theorem append_record_eq_iff
(p : Program A Coord Context Code d) (q : Program A Coord OtherContext Code e)
(sameView : ∀ side a, p.view side a = q.view side a)
(derive : Context → (Fin d → Code) → OtherContext)
(derive_correct : ∀ a, derive (p.context a) (fun j => p.labels j.val a) = q.context a)
(k : ℕ) (a b : A) :
labelRecordOf (p.append q sameView derive derive_correct).labels (d + k) a =
labelRecordOf (p.append q sameView derive derive_correct).labels (d + k) b ↔
labelRecordOf p.labels d a = labelRecordOf p.labels d b ∧
labelRecordOf q.labels k a = labelRecordOf q.labels k b := by
simp only [labelRecordOf_eq_iff]
constructor
· intro h
constructor
· intro n hn
have heq := h n (by omega)
simpa only [append_labels_left p q sameView derive derive_correct n hn] using heq
· intro n hn
have heq := h (d + n) (by omega)
simpa only [append_labels_right] using heq
· rintro ⟨hp, hq⟩ n hn
by_cases hd : n < d
· simpa only [append_labels_left p q sameView derive derive_correct n hd] using hp n hd
· have hdn : d + (n - d) = n := Nat.add_sub_of_le (Nat.le_of_not_gt hd)
have heq := hq (n - d) (by omega)
have ha := append_labels_right p q sameView derive derive_correct (n - d) a
have hb := append_labels_right p q sameView derive derive_correct (n - d) b
rw [hdn] at ha hb
exact ha.trans (heq.trans hb.symm)
theorem recode_record_eq_iff {NewCode : Type*}
(p : Program A Coord Context Code d) (encode : Code → NewCode)
(decode : NewCode → Code) (roundtrip : Function.LeftInverse decode encode)
(k : ℕ) (a b : A) :
labelRecordOf (p.recode encode decode roundtrip).labels k a =
labelRecordOf (p.recode encode decode roundtrip).labels k b ↔
labelRecordOf p.labels k a = labelRecordOf p.labels k b := by
simp only [labelRecordOf_eq_iff]
constructor
· intro h n hn
exact roundtrip.injective (h n hn)
· intro h n hn
exact congrArg encode (h n hn)
theorem pow_record_eq_iff [Inhabited Code]
(p : Program A Coord Context Code d) (m : ℕ) (i : Fin m)
(k : ℕ) (hk : k ≤ d) (a b : Fin m → A) :
labelRecordOf (p.pow m).labels (i.val * d + k) a =
labelRecordOf (p.pow m).labels (i.val * d + k) b ↔
(∀ j : Fin m, j.val < i.val →
labelRecordOf p.labels d (a j) = labelRecordOf p.labels d (b j)) ∧
labelRecordOf p.labels k (a i) = labelRecordOf p.labels k (b i) := by
simp only [labelRecordOf_eq_iff]
constructor
· intro h
constructor
· intro j hj n hn
have hindex : j.val * d + n < i.val * d + k := by
calc
j.val * d + n < j.val * d + d := Nat.add_lt_add_left hn _
_ = (j.val + 1) * d := by rw [Nat.add_mul, Nat.one_mul]
_ ≤ i.val * d := Nat.mul_le_mul_right d (Nat.succ_le_of_lt hj)
_ ≤ i.val * d + k := Nat.le_add_right _ _
have heq := h (j.val * d + n) hindex
simpa only [pow_labels p m a j n hn, pow_labels p m b j n hn] using heq
· intro n hn
have hnd : n < d := lt_of_lt_of_le hn hk
have heq := h (i.val * d + n) (Nat.add_lt_add_left hn _)
simpa only [pow_labels p m a i n hnd, pow_labels p m b i n hnd] using heq
· rintro ⟨hp, hc⟩ n hn
have hd : 0 < d := by
by_contra h
have hz : d = 0 := by omega
simp [hz] at hk hn
omega
have hq : n / d ≤ i.val := by
have hbound : n < (i.val + 1) * d := by
rw [Nat.add_mul, Nat.one_mul]
exact lt_of_lt_of_le hn (Nat.add_le_add_left hk _)
exact Nat.le_of_lt_succ ((Nat.div_lt_iff_lt_mul hd).2 hbound)
let j : Fin m := ⟨n / d, lt_of_le_of_lt hq i.isLt⟩
have hrem : n % d < d := Nat.mod_lt n hd
have hsplit : j.val * d + n % d = n := by
simpa only [j, Nat.mul_comm] using Nat.div_add_mod n d
have heq : p.labels (n % d) (a j) = p.labels (n % d) (b j) := by
by_cases hj : j.val < i.val
· exact hp j hj (n % d) hrem
· have hji : j = i := Fin.ext (by dsimp [j] at hj ⊢; omega)
have hcur : n % d < k := by
rw [hji] at hsplit
omega
simpa only [hji] using hc (n % d) hcur
rw [← hsplit, pow_labels p m a j (n % d) hrem, pow_labels p m b j (n % d) hrem]
exact heq
end Program
end MatrixMultiplication.CompletionLabels.TopologicalFlatten
namespace MatrixMultiplication.CompletionLabels
open MatrixMultiplication.Foundation RecursiveCompletion TopologicalFlatten
attribute [local instance] Classical.propDecidable Classical.decEq
universe uCoord
variable {X Y Z : Type uCoord}
theorem completionLocalProgram_label_root {Code : Type*} [Inhabited Code] {depth : ℕ}
(S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(p : Program (Leaf S) (Coordinate X Y Z) Color Code depth)
(context_eq : ∀ a, p.context a = leafColor S a)
(view_eq : ∀ side a, p.view side a = coordinate a.val side)
(n : ℕ) (hn : n < 3) (a : Leaf (complete S center m)) :
(completionLocalProgram S center m p context_eq view_eq).labels n a =
encodePattern (if leafColor (complete S center m) a = rootColor n then
some (pattern S center m a) else none) := by
change (if n < 3 then encodePattern ((patternProgram S center m).labels n a)
else (inheritedProgram S center m p view_eq).labels (n - 3) a) = _
rw [ite_eq_left hn]
rfl
theorem completionLocalProgram_pair_root {Code : Type*} [Inhabited Code] {depth : ℕ}
(S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(p : Program (Leaf S) (Coordinate X Y Z) Color Code depth)
(context_eq : ∀ a, p.context a = leafColor S a)
(view_eq : ∀ side a, p.view side a = coordinate a.val side)
(n : ℕ) (hn : n < 3) :
(completionLocalProgram S center m p context_eq view_eq).pair n =
ownerPair center (rootColor n) := by
change (if n < 3 then ownerPair center (rootColor n) else p.pair ((n - 3) % depth)) = _
rw [ite_eq_left hn]
theorem completionLocalProgram_label_slot {Code : Type*} [Inhabited Code] {depth : ℕ}
(S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(p : Program (Leaf S) (Coordinate X Y Z) Color Code depth)
(context_eq : ∀ a, p.context a = leafColor S a)
(view_eq : ∀ side a, p.view side a = coordinate a.val side)
(i : Fin m) (k : ℕ) (hk : k < depth) (a : Leaf (complete S center m)) :
(completionLocalProgram S center m p context_eq view_eq).labels (3 + (i.val * depth + k)) a =
encodeOld (p.labels k (slot S center m a i)) := by
change (if 3 + (i.val * depth + k) < 3 then
encodePattern ((patternProgram S center m).labels (3 + (i.val * depth + k)) a)
else (inheritedProgram S center m p view_eq).labels
(3 + (i.val * depth + k) - 3) a) = _
have hsub : 3 + (i.val * depth + k) - 3 = i.val * depth + k := by omega
rw [ite_eq_right (by omega), hsub]
change ((p.recode encodeOld decodeOld oldCode_roundtrip).pow m).labels
(i.val * depth + k) (slot S center m a) = _
rw [Program.pow_labels _ m _ i k hk]
rfl
theorem completionLocalProgram_pair_slot {Code : Type*} [Inhabited Code] {depth : ℕ}
(S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(p : Program (Leaf S) (Coordinate X Y Z) Color Code depth)
(context_eq : ∀ a, p.context a = leafColor S a)
(view_eq : ∀ side a, p.view side a = coordinate a.val side)
(i : Fin m) (k : ℕ) (hk : k < depth) :
(completionLocalProgram S center m p context_eq view_eq).pair (3 + (i.val * depth + k)) =
p.pair k := by
change (if 3 + (i.val * depth + k) < 3 then _
else p.pair ((3 + (i.val * depth + k) - 3) % depth)) = _
rw [ite_eq_right (by omega)]
have hsub : 3 + (i.val * depth + k) - 3 = i.val * depth + k := by omega
rw [hsub, Nat.mul_add_mod_self_right, Nat.mod_eq_of_lt hk]
theorem patternProgram_record_eq_iff (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(a b : Leaf (complete S center m)) :
labelRecordOf (patternProgram S center m).labels 3 a =
labelRecordOf (patternProgram S center m).labels 3 b ↔
leafColor (complete S center m) a = leafColor (complete S center m) b ∧
pattern S center m a = pattern S center m b := by
constructor
· intro h
have heq := label_eq_of_record_eq (patternProgram S center m).labels h
(rootIndex (leafColor (complete S center m) a)).isLt
change (if leafColor (complete S center m) a =
rootColor (rootIndex (leafColor (complete S center m) a)).val
then some (pattern S center m a) else none) =
(if leafColor (complete S center m) b =
rootColor (rootIndex (leafColor (complete S center m) a)).val
then some (pattern S center m b) else none) at heq
rw [rootColor_rootIndex, ite_eq_left rfl] at heq
by_cases hb : leafColor (complete S center m) b = leafColor (complete S center m) a
· rw [ite_eq_left hb] at heq
exact ⟨hb.symm, Option.some.inj heq⟩
· rw [ite_eq_right hb] at heq
cases heq
· rintro ⟨hc, hp⟩
apply (labelRecordOf_eq_iff _ _ _ _).mpr
intro n hn
change (if leafColor (complete S center m) a = rootColor n then
some (pattern S center m a) else none) =
(if leafColor (complete S center m) b = rootColor n then
some (pattern S center m b) else none)
rw [hc, hp]
theorem completionProgram_record_eq_iff {Code : Type*} [Inhabited Code] {depth : ℕ}
(S : FlaggedTensor X Y Z) (center : Color) (m : ℕ)
(p : Program (Leaf S) (Coordinate X Y Z) Color Code depth)
(context_eq : ∀ a, p.context a = leafColor S a)
(view_eq : ∀ side a, p.view side a = coordinate a.val side)
(i : Fin m) (k : ℕ) (hk : k ≤ depth)
(a b : Leaf (complete S center m)) :
labelRecordOf (completionLocalProgram S center m p context_eq view_eq).labels
(3 + (i.val * depth + k)) a =
labelRecordOf (completionLocalProgram S center m p context_eq view_eq).labels
(3 + (i.val * depth + k)) b ↔
(leafColor (complete S center m) a = leafColor (complete S center m) b ∧
pattern S center m a = pattern S center m b) ∧
((∀ j : Fin m, j.val < i.val →
labelRecordOf p.labels depth (slot S center m a j) =
labelRecordOf p.labels depth (slot S center m b j)) ∧
labelRecordOf p.labels k (slot S center m a i) =
labelRecordOf p.labels k (slot S center m b i)) := by
change labelRecordOf (completionProgram S center m p context_eq view_eq).labels
(3 + (i.val * depth + k)) a =
labelRecordOf (completionProgram S center m p context_eq view_eq).labels
(3 + (i.val * depth + k)) b ↔ _
trans
(labelRecordOf ((patternProgram S center m).recode (encodePattern (Code := Code))
decodePatternCode patternCode_roundtrip).labels 3 a =
labelRecordOf ((patternProgram S center m).recode (encodePattern (Code := Code))
decodePatternCode patternCode_roundtrip).labels 3 b ∧
labelRecordOf (inheritedProgram S center m p view_eq).labels (i.val * depth + k) a =
labelRecordOf (inheritedProgram S center m p view_eq).labels (i.val * depth + k) b)
· exact Program.append_record_eq_iff _ _ _ _ _ (i.val * depth + k) a b
rw [Program.recode_record_eq_iff, patternProgram_record_eq_iff]
have htail :
(labelRecordOf (inheritedProgram S center m p view_eq).labels (i.val * depth + k) a =
labelRecordOf (inheritedProgram S center m p view_eq).labels (i.val * depth + k) b) ↔
(labelRecordOf ((p.recode (encodeOld (m := m)) decodeOld oldCode_roundtrip).pow m).labels
(i.val * depth + k) (slot S center m a) =
labelRecordOf ((p.recode (encodeOld (m := m)) decodeOld oldCode_roundtrip).pow m).labels
(i.val * depth + k) (slot S center m b)) := by
simp only [labelRecordOf_eq_iff]
rfl
rw [htail, Program.pow_record_eq_iff _ m i k hk]
simp only [Program.recode_record_eq_iff]
end MatrixMultiplication.CompletionLabels
end
end OAI
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