File size: 3,193 Bytes
8769036 | 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 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitAssignmentUpdates
namespace OAI
/-! Complete query blocks along a canonical DNF path. A prefix of a path is
enough for the switching encoding; the final Boolean answer is irrelevant. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
structure DNFQueryBlock (n : ℕ) where
term : CubeTerm n
queried : Finset (Fin n)
answer : BooleanCube n
namespace DNFQueryBlock
def support {n : ℕ} : List (DNFQueryBlock n) → Finset (Fin n)
| [] => ∅
| b :: bs => b.queried ∪ support bs
def length {n : ℕ} : List (DNFQueryBlock n) → ℕ
| [] => 0
| b :: bs => b.queried.card + length bs
def image {n : ℕ} : PartialAssignment n → List (DNFQueryBlock n) → PartialAssignment n
| ρ, [] => ρ
| ρ, b :: bs => image (ρ.assign b.queried b.term.value) bs
end DNFQueryBlock
inductive CanonicalDNFWalk {n : ℕ} :
List (CubeTerm n) → PartialAssignment n → List (DNFQueryBlock n) → Prop
| nil (F ρ) : CanonicalDNFWalk F ρ []
| skip {C F ρ bs} (hC : ¬ C.Compatible ρ) (h : CanonicalDNFWalk F ρ bs) :
CanonicalDNFWalk (C :: F) ρ bs
| step {C F ρ bs} (x : BooleanCube n) (hC : C.Compatible ρ)
(hne : (C.live ρ).Nonempty) (h : CanonicalDNFWalk F (ρ.assign (C.live ρ) x) bs) :
CanonicalDNFWalk (C :: F) ρ (⟨C, C.live ρ, x⟩ :: bs)
namespace CanonicalDNFWalk
lemma support_subset_free {n : ℕ} {F : List (CubeTerm n)} {ρ : PartialAssignment n}
{bs : List (DNFQueryBlock n)} (h : CanonicalDNFWalk F ρ bs) :
DNFQueryBlock.support bs ⊆ ρ.free := by
induction h with
| nil => simp [DNFQueryBlock.support]
| skip _ _ ih => exact ih
| @step C F ρ bs x _ _ _ ih =>
intro i hi
rcases mem_union.mp hi with hi | hi
· exact C.live_subset_free ρ hi
· have hh := ih hi
rw [PartialAssignment.free_assign] at hh
exact (mem_sdiff.mp hh).1
lemma head_disjoint_tail {n : ℕ} {C : CubeTerm n} {F : List (CubeTerm n)}
{ρ : PartialAssignment n} {bs : List (DNFQueryBlock n)} (x : BooleanCube n)
(h : CanonicalDNFWalk F (ρ.assign (C.live ρ) x) bs) :
Disjoint (C.live ρ) (DNFQueryBlock.support bs) := by
apply Finset.disjoint_left.mpr
intro i hi htail
have hh := h.support_subset_free htail
rw [PartialAssignment.free_assign] at hh
exact (mem_sdiff.mp hh).2 hi
lemma length_eq_card_support {n : ℕ} {F : List (CubeTerm n)}
{ρ : PartialAssignment n} {bs : List (DNFQueryBlock n)}
(h : CanonicalDNFWalk F ρ bs) :
DNFQueryBlock.length bs = (DNFQueryBlock.support bs).card := by
induction h with
| nil => simp [DNFQueryBlock.length, DNFQueryBlock.support]
| skip _ _ ih => exact ih
| @step C F ρ bs x _ _ h ih =>
simp only [DNFQueryBlock.length, DNFQueryBlock.support]
rw [card_union_of_disjoint (head_disjoint_tail x h), ih]
lemma length_le_dimension {n : ℕ} {F : List (CubeTerm n)}
{ρ : PartialAssignment n} {bs : List (DNFQueryBlock n)}
(h : CanonicalDNFWalk F ρ bs) : DNFQueryBlock.length bs ≤ n := by
rw [h.length_eq_card_support]
exact (card_le_card (subset_univ _)).trans_eq (by simp)
end CanonicalDNFWalk
end TwoPointCorrelations
end OAI
|