| 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 |
|
|
