% =================================================================
:- pred constr(bool).
:- mode constr(in).
:- ignore constr/1.
% =================================================================
% Program

% rev(L,R): 
% list R is the reversal of list L
:- pred rev(list(int),list(int)).
:- mode rev(in,out).

rev([],[]).
rev([H|T],R) :-
  rev(T,S),
  snoc(S,H,R).

% snoc(L,X,S): 
% list S is obtained by adding element X at the end of list L
:- pred snoc(list(int),int,list(int)).
:- mode snoc(in,in,out).

snoc([],X,[X]).
snoc([X|Xs],Y,[X|Zs]) :-
  snoc(Xs,Y,Zs).

   
% =================================================================
% Catamorphisms
% ------ hd(L,IsDef,Hd): IsDef is true iff L is a nonempty list and
% if IsDef=true then Hd is the head of L
:- pred hd(list(int),bool,int).   
:- mode hd(in,out,out).
:- cata hd/3-1.

hd([],IsDef,Hd) :- constr((~IsDef) & (Hd=0)).   % 0 can be any int.
hd([H|T],IsDef,Hd) :- constr(IsDef & (Hd=H)). 

% ------ isASorted(L,Res):  
% Res= true iff L is weakly-ascending.
:- pred isASorted(list(int),bool).   
:- mode isASorted(in,out). 
:- cata isASorted/2-1.

isASorted([],Res) :- constr(Res).
isASorted([H|T],Res) :-
    hd(T,IsDefT,HdT), 
    isASorted(T,ResT),
    constr(Res = (IsDefT => (H=<HdT & ResT))).
    
% ------ isDSorted(L,Res): 
% Res= true iff L is weakly-descending.
:- pred isDSorted(list(int),bool).   
:- mode isDSorted(in,out). 
:- cata isDSorted/2-1.

isDSorted([],Res) :- constr(Res).
isDSorted([H|T],Res) :-
    hd(T,IsDefT,HdT), 
    isDSorted(T,ResT),
    constr(Res = (IsDefT => (H>=HdT & ResT))).
%
%% ------ listMin(L,IsDef,Min):  IsDef=true iff L is a nonempty list and
%%    if IsDef then Min is the minimum element on the list L
%:- pred listMin(list(int),bool,int).  
%:- mode listMin(in,out,out).  
%:- cata listMin/3-1.
%
%listMin([],IsDef,Min) :- constr( (~IsDef) & Min=0 ).    % 0 can be any int.
%listMin([H|T],IsDef,Min) :-  
%     listMin(T,IsDefT,MinT),
%     constr( (IsDef & (Min = ite(IsDefT, ite(H<MinT, H ,MinT), H ) ) ) ).   

% ------ leq_all(N,L,B):    B=true  iff forall element X of L,  N=<X. 
:- pred leq_all(int,list(int),bool).
:- mode leq_all(in,in,out).
:- cata leq_all/3-2.

leq_all(N,[],B) :-
 constr( B ).
leq_all(N,[X|Xs],B) :-
 constr( B = (N=<X & B1) ),
 leq_all(N,Xs,B1).

% =================================================================
% ------ Verification of contract on rev:                    
:- pred ff1.
ff1 :-
  constr( ~(BL=BR) ), 
  isASorted(L,BL),
  rev(L,R),
  isDSorted(R,BR).
%
%% ------ Verification of contract on snoc:          
%:- pred ff2.
%ff2 :-                                                     
% constr( ~(BE => (BA=BC)) ),
% isDSorted(A,BA), 
% leq_all(X,A,BE), 
% snoc(A,X,C), 
% isDSorted(C,BC).
% =================================================================
:- query ff1/0.
%:- query ff2/0.
% =================================================================

%  Catamorphic abstraction

:- cata_abs list(int) ==> isASorted(L,BL1), isDSorted(L,BL2),
                          leq_all(X,L,LA).  

% =================================================================