End SP_SOUND.
Module SP_Sound (VSM : VERISTAR_MODEL) : SP_SOUND with Module VSM := VSM.
Module VSM := VSM.
Implicit Arguments list_denote [A T].
Local Notation "'listd'" := list_denote.
Implicit Arguments andp [A].
Local Notation "'inter'" := (@andp _). (*implicit arguments acting weird*)
(* the following five rules (barring reflexivity resolution, which is performed during the simplification phase) are collected in rule "sp" *)
Definition positive_superposition (c d : clause) l0 : list clause :=
match c , d with
| PureClause nil (Eqv s t :: pos) _ _ ,
PureClause nil (Eqv s' v :: pos') _ _ =>
if expr_eq s s' && expr_lt t s && expr_lt v s' &&
pure_atom_gt1 (Eqv s t) pos && pure_atom_gt1 (Eqv s' v) pos' &&
pure_atom_gt (Eqv s' v) (Eqv s t)
then mkPureClause nil (insert_uniq pure_atom_cmp (norm_pure_atom (Eqv t v))
(merge pure_atom_cmp pos pos')) :: l0
else l0
| _ , _ => l0
end.
Definition negative_superposition (c d : clause) l0 : list clause :=
match c , d with
| PureClause nil (Eqv s t :: pos) _ _,
PureClause (Eqv s' v :: neg') pos' _ _ =>
if expr_eq s s' && expr_lt t s && expr_lt v s'
then mkPureClause (insert_uniq pure_atom_cmp (norm_pure_atom (Eqv t v)) neg')
(merge pure_atom_cmp pos pos') :: l0
else l0
| _ , _ => l0
end.
Definition reflexivity_resolution (d : clause) l0 : list clause :=
match d with
| PureClause (Eqv u v :: neg) pos _ _ =>
if expr_eq u v then mkPureClause neg pos :: l0 else l0
| _ => l0
end.
Definition ordered_factoring (c : clause) l0 : list clause :=
match c with
| PureClause nil (Eqv s t :: Eqv u v :: pos) _ _ =>
if pure_atom_eq (Eqv s t) (Eqv u v)
then mkPureClause nil
(insert_uniq pure_atom_cmp (norm_pure_atom (Eqv s t)) pos)
:: l0
else l0
| _ => l0
end.
Definition equality_factoring (c : clause) l0 : list clause :=
match c with
| PureClause nil (Eqv u v :: Eqv s t :: pos) _ _ =>
if expr_eq s u && pure_atom_gt (Eqv u v) (Eqv s t)
then mkPureClause [norm_pure_atom (Eqv v t)]
(insert_uniq pure_atom_cmp (norm_pure_atom (Eqv u t)) pos) :: l0
else l0
| _ => l0
end.

Lemma positive_superposition_sound c d l0 s :
clause_denote c s -> clause_denote d s -> listd clause_denote inter TT l0 s ->
listd clause_denote inter TT (positive_superposition c d l0) s.