Require Import List.


Inductive h10c : Set :=
  | h10c_one : h10c
  | h10c_plus : h10c
  | h10c_mult : h10c.

Definition h10c_sem c φ :=
  match c with
    | h10c_one x φ x = 1
    | h10c_plus x y z φ x + φ y = φ z
    | h10c_mult x y z φ x * φ y = φ z
  end.

Definition H10C_SAT (cs: list h10c) := (φ: ), c, In c cs h10c_sem c φ.


Inductive h10sqc : Set :=
  | h10sqc_one : h10sqc
  | h10sqc_plus : h10sqc
  | h10sqc_sq : h10sqc.

Definition h10sqc_sem φ c :=
  match c with
    | h10sqc_one x φ x = 1
    | h10sqc_plus x y z φ x + φ y = φ z
    | h10sqc_sq x y φ x * φ x = φ y
  end.

Definition H10SQC_SAT (cs: list h10sqc) := (φ: ), c, In c cs h10sqc_sem φ c.

Definition h10uc := ( * * )%type.

Definition h10uc_sem φ (c : h10uc) :=
  match c with
    | (x, y, z) 1 + φ x + φ y * φ y = φ z
  end.

Definition H10UC_SAT (cs: list h10uc) := (φ: ), c, In c cs h10uc_sem φ c.