Module CCS := CCSound VSM. Import CCS.
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 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 positive_superposition (c d : clause) l0 : list clause :=
match c, d with
(* positive (right) superposition *)
| PureClause neg (Eqv (Var s0 as s) t :: pos) _ _,
PureClause neg' (Eqv (Var s'0 as 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 t) (Eqv s' v) &&
greater_than_all s0 neg && greater_than_all s'0 neg'
then mkPureClause
(merge pure_atom_cmp neg neg')
(insu_atm (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
(* negative (left) superposition *)
| PureClause (Eqv s' v :: neg') pos' _ _,
PureClause neg (Eqv (Var s0 as s) t :: pos) _ _ =>
if expr_eq s s' && expr_lt t s && expr_lt v s' &&
pure_atom_gt1 (Eqv s t) pos && greater_than_all s0 neg
then mkPureClause
(insu_atm (norm_pure_atom (Eqv t v)) (merge pure_atom_cmp neg neg'))
(merge pure_atom_cmp pos 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.