## TPTP Axioms File: SWV007+3.ax

```%------------------------------------------------------------------------------
% File     : SWV007+3 : TPTP v7.5.0. Released v3.3.0.
% Domain   : Software Verification
% Axioms   : Priority queue checker: checked priority queues
% Version  : [dNP05] axioms.
% English  : This axiom set fully describes checked priority queues. Checked
%            priority queues are defined as triples of a "priority queue
%            pretender", a system of lower bounds and Boolean value that keep
%            track of errors.

% Refs     : [Pis06] Piskac (2006), Email to Geoff Sutcliffe
%          : [dNP05] de Nivelle & Piskac (2005), Verification of an Off-Lin
% Source   : [Pis06]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   7 unit)
%            Number of atoms       :   48 (  17 equality)
%            Maximal formula depth :    8 (   5 average)
%            Number of connectives :   32 (   7 ~  ;   0  |;   7  &)
%                                         (   4 <=>;  14 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    9 (   1 propositional; 0-2 arity)
%            Number of functors    :   18 (   3 constant; 0-3 arity)
%            Number of variables   :   70 (   4 singleton;  70 !;   0 ?)
%            Maximal term depth    :    4 (   2 average)
% SPC      :

% Comments : Requires SWV007+0 SWV007+2
%------------------------------------------------------------------------------
fof(ax31,axiom,(
! [U] : succ_cpq(U,U) )).

fof(ax32,axiom,(
! [U,V,W] :
( succ_cpq(U,V)
=> succ_cpq(U,insert_cpq(V,W)) ) )).

fof(ax33,axiom,(
! [U,V,W] :
( succ_cpq(U,V)
=> succ_cpq(U,remove_cpq(V,W)) ) )).

fof(ax34,axiom,(
! [U,V] :
( succ_cpq(U,V)
=> succ_cpq(U,findmin_cpq_eff(V)) ) )).

fof(ax35,axiom,(
! [U,V] :
( succ_cpq(U,V)
=> succ_cpq(U,removemin_cpq_eff(V)) ) )).

fof(ax36,axiom,(
! [U,V] : check_cpq(triple(U,create_slb,V)) )).

fof(ax37,axiom,(
! [U,V,W,X,Y] :
( less_than(Y,X)
=> ( check_cpq(triple(U,insert_slb(V,pair(X,Y)),W))
<=> check_cpq(triple(U,V,W)) ) ) )).

fof(ax38,axiom,(
! [U,V,W,X,Y] :
( strictly_less_than(X,Y)
=> ( check_cpq(triple(U,insert_slb(V,pair(X,Y)),W))
<=> \$false ) ) )).

fof(ax39,axiom,(
! [U,V,W,X] :
( contains_cpq(triple(U,V,W),X)
<=> contains_slb(V,X) ) )).

fof(ax40,axiom,(
! [U,V] :
<=> \$false ) )).

fof(ax41,axiom,(
! [U,V,W] :
( ~ ok(triple(U,V,W))
=> W = bad ) )).

fof(ax42,axiom,(
! [U,V,W,X] : insert_cpq(triple(U,V,W),X) = triple(insert_pqp(U,X),insert_slb(V,pair(X,bottom)),W) )).

fof(ax43,axiom,(
! [U,V,W,X] :
( ~ contains_slb(V,X)
=> remove_cpq(triple(U,V,W),X) = triple(U,V,bad) ) )).

fof(ax44,axiom,(
! [U,V,W,X] :
( ( contains_slb(V,X)
& less_than(lookup_slb(V,X),X) )
=> remove_cpq(triple(U,V,W),X) = triple(remove_pqp(U,X),remove_slb(V,X),W) ) )).

fof(ax45,axiom,(
! [U,V,W,X] :
( ( contains_slb(V,X)
& strictly_less_than(X,lookup_slb(V,X)) )
=> remove_cpq(triple(U,V,W),X) = triple(remove_pqp(U,X),remove_slb(V,X),bad) ) )).

fof(ax46,axiom,(
! [U,V] : findmin_cpq_eff(triple(U,create_slb,V)) = triple(U,create_slb,bad) )).

fof(ax47,axiom,(
! [U,V,W,X] :
( ( V != create_slb
& ~ contains_slb(V,findmin_pqp_res(U)) )
=> findmin_cpq_eff(triple(U,V,W)) = triple(U,update_slb(V,findmin_pqp_res(U)),bad) ) )).

fof(ax48,axiom,(
! [U,V,W,X] :
( ( V != create_slb
& contains_slb(V,findmin_pqp_res(U))
& strictly_less_than(findmin_pqp_res(U),lookup_slb(V,findmin_pqp_res(U))) )
=> findmin_cpq_eff(triple(U,V,W)) = triple(U,update_slb(V,findmin_pqp_res(U)),bad) ) )).

fof(ax49,axiom,(
! [U,V,W,X] :
( ( V != create_slb
& contains_slb(V,findmin_pqp_res(U))
& less_than(lookup_slb(V,findmin_pqp_res(U)),findmin_pqp_res(U)) )
=> findmin_cpq_eff(triple(U,V,W)) = triple(U,update_slb(V,findmin_pqp_res(U)),W) ) )).

fof(ax50,axiom,(
! [U,V] : findmin_cpq_res(triple(U,create_slb,V)) = bottom )).

fof(ax51,axiom,(
! [U,V,W,X] :
( V != create_slb
=> findmin_cpq_res(triple(U,V,W)) = findmin_pqp_res(U) ) )).

fof(ax52,axiom,(
! [U] : removemin_cpq_eff(U) = remove_cpq(findmin_cpq_eff(U),findmin_cpq_res(U)) )).

fof(ax53,axiom,(
! [U] : removemin_cpq_res(U) = findmin_cpq_res(U) )).

%------------------------------------------------------------------------------
```