Section ListFacts.
Definition sat φ l := Forall (h10uc_sem φ) l.
(* needs 2 fresh variables *)
(* x is 0 if 1 + x + x * x = y and 1 + x + y * y = z and 1 + y + x * x = z *)
Definition express_0 (x: nat) (f: nat -> nat) := [(x, x, f 0); (x, f 0, f 1); (f 0, x, f 1)].
Definition express_0f := [1; 2].
(* needs 2 fresh variables *)
(* x is 1 if 1 + y + y * y = x and 1 + y + x * x = z and 1 + x + y * y = z *)
Definition express_1 (x: nat) (f: nat -> nat) := [(f 0, f 0, x); (f 0, x, f 1); (x, f 0, f 1)].
Definition express_1f := [0; 2].

Lemma express_1E x f φ: sat φ (express_1 x f) -> φ x = 1.