File size: 2,570 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 | 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
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