%------------------------------------------------------------------------------
% File     : SWV381+2 : TPTP v8.2.0. Released v3.3.0.
% Domain   : Software Verification
% Problem  : Priority queue checker: lemma_min_elem_smallest
% Version  : [dNP05] axioms : Augmented & Reduced > Redundant.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v3.3.0
% Syntax   : Number of formulae    :   69 (  23 unt;   0 def)
%            Number of atoms       :  151 (  40 equ)
%            Maximal formula atoms :    5 (   2 avg)
%            Number of connectives :  102 (  20   ~;   6   |;  26   &)
%                                         (  18 <=>;  32  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   5 avg)
%            Maximal term depth    :    5 (   1 avg)
%            Number of predicates  :   21 (  19 usr;   1 prp; 0-3 aty)
%            Number of functors    :   26 (  26 usr;   4 con; 0-3 aty)
%            Number of variables   :  193 ( 188   !;   5   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Version by Geoff Sutcliffe, with more distant lemmas.
%------------------------------------------------------------------------------
%----Include the axioms about priority queues and checked priority queues
include('Axioms/SWV007+0.ax').
include('Axioms/SWV007+1.ax').
include('Axioms/SWV007+2.ax').
include('Axioms/SWV007+3.ax').
include('Axioms/SWV007+4.ax').
%------------------------------------------------------------------------------
%----lemma_contains_s_I (cpq_l005.p, cpq_l006.p)
fof(l17_li56,lemma,
    ! [U,V,W,X] :
      ( contains_cpq(triple(U,V,W),X)
    <=> contains_pq(i(triple(U,V,W)),X) ) ).

%----lemma_not_min_elem_not_check_induction (cpq_l020.p)
fof(l19_l20,lemma,
    ! [U,V,W] :
      ( ( ~ check_cpq(triple(U,V,W))
        | ~ ok(triple(U,V,W)) )
     => ! [X,Y,Z] :
          ( succ_cpq(triple(U,V,W),triple(X,Y,Z))
         => ( ~ ok(triple(X,Y,Z))
            | ~ check_cpq(triple(X,Y,Z)) ) ) ) ).

%----lemma_contains_pair (cpq_l046.p, cpq_l047.p)
fof(l45_li4647,lemma,
    ! [U,V] :
      ( contains_slb(U,V)
     => ? [W] : pair_in_list(U,V,W) ) ).

%----lemma_contains_update_01 (cpq_l048.p)
fof(l45_l48,lemma,
    ! [U,V,W,X] :
      ( ( pair_in_list(U,V,W)
        & strictly_less_than(V,X)
        & strictly_less_than(W,X) )
     => pair_in_list(update_slb(U,X),V,X) ) ).

%----lemma_contains_update_02 (cpq_l049.p)
fof(l45_l49,lemma,
    ! [U,V,W,X] :
      ( ( pair_in_list(U,V,W)
        & strictly_less_than(V,X)
        & less_than(X,W) )
     => ? [Y] :
          ( pair_in_list(update_slb(U,X),V,Y)
          & less_than(X,Y) ) ) ).

%----lemma_check_characterization (cpq_l041.p, cpq_l042.p)
fof(l43_li4142,lemma,
    ! [U,V,W] :
      ( check_cpq(triple(U,V,W))
    <=> ! [X,Y] :
          ( pair_in_list(V,X,Y)
         => less_than(Y,X) ) ) ).

%----lemma_min_elem_smallest (conjecture)
fof(l17_co,conjecture,
    ! [U,V,W] :
      ( phi(findmin_cpq_eff(triple(U,V,W)))
     => issmallestelement_pq(i(triple(U,V,W)),findmin_cpq_res(triple(U,V,W))) ) ).

%------------------------------------------------------------------------------