| import OAI.NumberTheory.TwoPoint.Circuits.CircuitPartialAssignment |
|
|
| namespace OAI |
|
|
| /-! Updates along the disjoint blocks of a canonical DNF path. These lemmas |
| also justify recovering the original restriction after decoding a block. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| namespace PartialAssignment |
|
|
| lemma free_assign {n : ℕ} (ρ : PartialAssignment n) (S : Finset (Fin n)) |
| (x : BooleanCube n) : (ρ.assign S x).free = ρ.free \ S := by |
| ext i |
| simp only [free, mem_filter, mem_univ, true_and, mem_sdiff] |
| by_cases hi : i ∈ S <;> simp [assign, hi] |
|
|
| lemma assign_comm {n : ℕ} (ρ : PartialAssignment n) (S T : Finset (Fin n)) |
| (hST : Disjoint S T) (x y : BooleanCube n) : |
| (ρ.assign S x).assign T y = (ρ.assign T y).assign S x := by |
| funext i |
| by_cases hiS : i ∈ S <;> by_cases hiT : i ∈ T |
| · exact False.elim (Finset.disjoint_left.mp hST hiS hiT) |
| all_goals simp [assign, hiS, hiT] |
|
|
| lemma assign_injective_of_free {n : ℕ} (ρ τ : PartialAssignment n) |
| (S : Finset (Fin n)) (hρ : S ⊆ ρ.free) (hτ : S ⊆ τ.free) |
| (x y : BooleanCube n) (h : ρ.assign S x = τ.assign S y) : ρ = τ := by |
| funext i |
| by_cases hi : i ∈ S |
| · exact (mem_filter.mp (hρ hi)).2.trans (mem_filter.mp (hτ hi)).2.symm |
| · have he := congrFun h i |
| simpa [assign, hi] using he |
|
|
| lemma assign_congr_on {n : ℕ} (ρ : PartialAssignment n) (S : Finset (Fin n)) |
| (x y : BooleanCube n) (h : ∀ i ∈ S, x i = y i) : ρ.assign S x = ρ.assign S y := by |
| funext i |
| by_cases hi : i ∈ S <;> simp [assign, hi, h i] |
|
|
| end PartialAssignment |
|
|
| namespace CubeTerm |
|
|
| lemma live_update_disjoint {n : ℕ} (C : CubeTerm n) (ρ : PartialAssignment n) |
| (x : BooleanCube n) : Disjoint C.support (ρ.assign (C.live ρ) x).free := by |
| apply Finset.disjoint_left.mpr |
| intro i hi hf |
| rw [PartialAssignment.free_assign] at hf |
| have hn := (mem_filter.mp (mem_sdiff.mp hf).1).2 |
| exact (mem_sdiff.mp hf).2 (mem_filter.mpr ⟨hi, hn⟩) |
|
|
| lemma compatible_assign_disjoint {n : ℕ} (C : CubeTerm n) |
| (ρ : PartialAssignment n) (S : Finset (Fin n)) (hS : Disjoint C.support S) |
| (x : BooleanCube n) (hC : C.Compatible ρ) : C.Compatible (ρ.assign S x) := by |
| intro i hi |
| have hn : i ∉ S := fun hs => Finset.disjoint_left.mp hS hi hs |
| simpa [PartialAssignment.assign, hn] using hC i hi |
|
|
| lemma satisfied_mono {n : ℕ} (C : CubeTerm n) {ρ τ : PartialAssignment n} |
| (hC : C.Satisfied ρ) (h : ρ.Extends τ) : C.Satisfied τ := |
| fun i hi => h i (C.value i) (hC i hi) |
|
|
| end CubeTerm |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
