1		% (c) 2004-2024 Lehrstuhl fuer Softwaretechnik und Programmiersprachen,
2		% Heinrich Heine Universitaet Duesseldorf
3		% This software is licenced under EPL 1.0 (http://www.eclipse.org/org/documents/epl-v10.html)
4
5		:- module(bsets_clp,
6		[empty_sequence/1,
7		is_sequence/2, is_sequence_wf/3, not_is_sequence/2, not_is_sequence_wf/3,
8		not_is_non_empty_sequence_wf/3,
9		injective_sequence_wf/3,
10		not_injective_sequence/3,
11		not_non_empty_injective_sequence/3,
12		injective_non_empty_sequence/3,
13		finite_non_empty_sequence/3,
14		test_finite_non_empty_sequence/4,
15		permutation_sequence_wf/3,
16		not_permutation_sequence/3,
17		size_of_sequence/3,
18		prepend_sequence/4, append_sequence/4, prefix_sequence_wf/4,
19		suffix_sequence/4, concat_sequence/4,
20		disjoint_union_wf/4,
21		concatentation_of_sequences/3,
22		tail_sequence/4, first_sequence/4, front_sequence/4, last_sequence/4,
23		rev_sequence/3,
24
25
26		% maplet/3,
27		% relation/1,
28		relation_over/3, relation_over_wf/4,
29		not_relation_over/4,
30		domain_wf/3,
31
32		range_wf/3,
33		identity_relation_over_wf/3, in_identity/3, not_in_identity/3,
34		invert_relation_wf/3,
35		tuple_of/3,
36		in_composition_wf/4, not_in_composition_wf/4, rel_composition_wf/5,
37		direct_product_wf/4,
38		parallel_product_wf/4, in_parallel_product_wf/4, not_in_parallel_product_wf/4,
39		rel_iterate_wf/5,
40		event_b_identity_for_type/3,
41
42		not_partial_function/4,
43		partial_function/3, partial_function_wf/4, partial_function_test_wf/5,
44
45		total_function/3, total_function_wf/4, total_function_test_wf/5,
46
47		% enumerate_total_bijection/3,
48		total_bijection/3, total_bijection_wf/4,
49
50		not_total_function/4,
51		not_total_bijection/4,
52
53
54		range_restriction_wf/4, range_subtraction_wf/4,
55		in_range_restriction_wf/4, not_in_range_restriction_wf/4,
56		in_range_subtraction_wf/4, not_in_range_subtraction_wf/4,
57		domain_restriction_wf/4, domain_subtraction_wf/4,
58		in_domain_restriction_wf/4, not_in_domain_restriction_wf/4,
59		in_domain_subtraction_wf/4, not_in_domain_subtraction_wf/4,
60		override_relation/4,
61		in_override_relation_wf/4, not_in_override_relation_wf/4,
62		image_wf/4, image_for_closure1_wf/4,
63		special_operator_for_image/3, image_for_special_operator/5, apply_fun_for_special_operator/6,
64
65		in_domain_wf/3, not_in_domain_wf/3,
66		apply_to/4, apply_to/5, apply_to/6,
67		override/5,
68
69		%sum_over_range/2, mul_over_range/2,
70
71		disjoint_union_generalized_wf/3,
72
73		partial_surjection/3, not_partial_surjection_wf/4,
74		partial_surjection_test_wf/5,
75
76		total_relation_wf/4,
77		not_total_relation_wf/4,
78
79		surjection_relation_wf/4, total_surjection_relation_wf/4,
80		not_surjection_relation_wf/4, not_total_surjection_relation_wf/4,
81
82		total_surjection/3, total_surjection_wf/4,
83		not_total_surjection_wf/4,
84
85		partial_injection/3, partial_injection_wf/4,
86		not_partial_injection/4,
87
88		total_injection/3, total_injection_wf/4,
89		not_total_injection/4,
90
91		partial_bijection/3, partial_bijection_wf/4,
92		not_partial_bijection/4,
93
94		relational_trans_closure_wf/3, %relational_reflexive_closure/2,
95		in_closure1_wf/3, not_in_closure1_wf/3
96		]).
97
98
99		:- use_module(library(terms)).
100		:- use_module(self_check).
101
102		:- use_module(debug).
103		:- use_module(tools).
104
105		:- use_module(module_information,[module_info/2]).
106		:- module_info(group,kernel).
107		:- module_info(description,'This module provides more advanced operations for the basic datatypes of ProB (mainly for relations, functions, sequences).').
108
109		:- use_module(tools_printing).
110
111		:- use_module(delay).
112
113		:- use_module(typechecker).
114		:- use_module(error_manager).
115
116		:- use_module(kernel_objects).
117		:- use_module(kernel_records).
118		:- use_module(kernel_tools).
119
120		:- use_module(kernel_waitflags).
121		:- use_module(kernel_equality,[equality_objects_wf/4]).
122
123		:- use_module(custom_explicit_sets).
124		:- use_module(avl_tools,[avl_fetch_pair/3]).
125		:- use_module(bool_pred,[negate/2]).
126		:- use_module(closures,[is_symbolic_closure/1]).
127		:- use_module(bsyntaxtree, [conjunct_predicates/2,
128		mark_bexpr_as_symbolic/2,
129		create_texpr/4,
130		safe_create_texpr/3,
131		get_texpr_type/2
132		]).
133
134		/* --------- */
135		/* SEQUENCES */
136		/* ------- - */
137
138		:- assert_must_succeed((bsets_clp:empty_sequence([]))).
139		:- assert_must_fail((bsets_clp:empty_sequence([int(1)]))).
140	?	empty_sequence(X) :- empty_set(X). % TO DO: add WF
141
142		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_empty_sequence([(int(2),int(33)),(int(1),int(22))]))).
143		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_empty_sequence([(int(1),int(33))]))).
144		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_empty_sequence([]))).
145
146		not_empty_sequence(X) :- var(X),!,
147		X = [(int(1),_)|_].
148		not_empty_sequence(X) :- is_custom_explicit_set_nonvar(X),!,
149		is_non_empty_explicit_set(X).
150		not_empty_sequence([(int(_),_)|_]). % clousure, avl_set dealt with clause above
151
152		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_empty_sequence_wf([(int(1),int(33))],WF),WF)).
153		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_empty_sequence_wf([(int(1),pred_true),(int(2),pred_false)],WF),WF)).
154		not_empty_sequence_wf(X,_WF) :- nonvar(X),!, not_empty_sequence(X).
155		not_empty_sequence_wf(X,WF) :-
156		(preferences:preference(use_smt_mode,true) -> not_empty_sequence(X)
157		; get_enumeration_starting_wait_flag(not_empty_sequence_wf,WF,LWF),
158		not_empty_sequence_lwf(X,LWF)).
159
160		:- block not_empty_sequence_lwf(-,-).
161		not_empty_sequence_lwf(S,_) :- nonvar(S),!,not_empty_sequence(S).
162		not_empty_sequence_lwf([(int(1),_)|_],_).
163
164		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:is_sequence([(int(1),int(22))],[int(22)]))).
165		:- assert_must_succeed(bsets_clp:is_sequence(closure(['_zzzz_unit_tests'],[couple(integer,integer)],b(member(b(identifier('_zzzz_unit_tests'),couple(integer,integer),[generated]),b(value([(int(1),int(22))]),set(couple(integer,integer)),[])),pred,[])),[int(22)])).
166
167		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:is_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
168		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)]))).
169		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(1),int(33)),(int(0),int(22))],[int(22),int(33),int(44)]))).
170		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(3),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
171		:- assert_must_succeed((is_sequence(R,global_set('Name')),R = [])).
172		:- assert_must_succeed((is_sequence(R,global_set('Name')),
173		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
174		:- assert_must_succeed((is_sequence(R,global_set('Name')),
175		R = [(int(1),fd(2,'Name'))] )).
176		:- assert_must_succeed((is_sequence(R,global_set('Name')),
177		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
178		:- assert_must_succeed((is_sequence(R,global_set('Name')),
179		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
180		:- assert_must_succeed((is_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
181		global_set('Name')) )).
182		:- assert_must_succeed((is_sequence(R,[int(1),int(2)]),
183		R = [(int(2),int(2)),(int(1),int(1))] )).
184		:- assert_must_fail((is_sequence(R,[int(1),int(2)]),
185		R = [(int(2),int(2)),(int(3),int(1))] )).
186		:- assert_must_fail((is_sequence(R,[int(1),int(2)]),
187		R = [(int(2),int(2)),(int(1),int(3))] )).
188		:- assert_must_fail((is_sequence(R,global_set('Name')),
189		R = [(int(0),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
190		:- assert_must_succeed((is_sequence(X,global_set('Name')),
191		(preferences:get_preference(randomise_enumeration_order,true) -> true
192		; kernel_objects:enumerate_basic_type(X,seq(global('Name')))),
193		X = [(int(1),fd(1,'Name'))])). % can take a long time with RANDOMISE_ENUMERATION_ORDER
194
195		is_sequence(X,Type) :- init_wait_flags(WF,[is_sequence]),
196		is_sequence_wf(X,Type,WF),
197	?	ground_wait_flags(WF).
198
199		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([int(1),int(2),int(3)],WF),WF)).
200		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([int(1)],WF),WF)).
201		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([],WF),WF)).
202		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:is_sequence_domain([int(0)],WF),WF)).
203		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:is_sequence_domain([int(2),int(3)],WF),WF)).
204		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_is_sequence_domain([int(2),int(3)],WF),WF)).
205		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_is_sequence_domain([int(0)],WF),WF)).
206		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_is_sequence_domain([int(1)],WF),WF)).
207		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_is_sequence_domain([],WF),WF)).
208
209		% check if a set is the domain of a sequence, i.e., an interval 1..n with n>=0
210		:- use_module(custom_explicit_sets,[construct_interval_closure/3]).
211		:- use_module(kernel_cardinality_attr,[finite_cardinality_as_int_wf/3]).
212		:- block is_sequence_domain(-,?).
213		is_sequence_domain(Domain,WF) :-
214		finite_cardinality_as_int_wf(Domain,int(Max),WF),
215		construct_interval_closure(1,Max,Interval), equal_object_wf(Domain,Interval,is_sequence_domain,WF).
216		:- block not_is_sequence_domain(-,?).
217		not_is_sequence_domain(Domain,WF) :-
218		finite_cardinality_as_int_wf(Domain,int(Max),WF),
219		construct_interval_closure(1,Max,Interval), not_equal_object_wf(Domain,Interval,WF).
220
221		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_wf([(int(1),pred_true)],
222		[pred_true,pred_false],WF),WF)).
223		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_wf([(int(1),pred_true),(int(2),pred_false),(int(3),pred_true)],
224		[pred_true,pred_false],WF),WF)).
225		:- assert_must_succeed((bsets_clp:is_sequence_wf([(int(X),R)],[pred_true],_WF),X==1, R==pred_true)).
226		:- assert_must_succeed((bsets_clp:is_sequence_wf([(int(X),R),(int(Y),R)],[pred_true],_WF),X=2,
227		(preferences:preference(use_clpfd_solver,true) -> Y==1 ; Y=1), R==pred_true)).
228
229		is_sequence_wf(Seq,Range,WF) :- is_sequence_wf_ex(Seq,Range,WF,_).
230		% is_sequence_wf_ex also returns expansion; if it was done
231		:- block is_sequence_wf_ex(-,?,?,?).
232		is_sequence_wf_ex(FF,Range,WF,FF) :-
233		nonvar(FF), FF = closure(_,_,_),
234		custom_explicit_sets:is_definitely_maximal_set(Range),
235		% we do not need the Range; this means we can match more closures (e.g., lambda)
236		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
237		is_sequence_domain(FFDomain,WF).
238		is_sequence_wf_ex(Seq,Range,WF,Res) :-
239		expand_and_convert_to_avl_set_warn(Seq,AER,is_sequence_wf_ex,'ARG : seq(?)',WF),!,
240		is_avl_sequence(AER),
241		is_avl_relation_over_range(AER,Range,WF),
242		custom_explicit_sets:construct_avl_set(AER,Res).
243		is_sequence_wf_ex(X,Type,WF,EX) :-
244		% try_ensure_seq_numbering(X,1),
245		expand_custom_set_to_list_wf(X,EX,_,is_sequence_wf_ex,WF),
246		is_sequence2(EX,[],Type,0,_MinSize,WF).
247
248		% will make this much faster x:seq(STRING) & card(x)=400 & 401:dom(x) (40 ms rather than > 2 secs)
249		% but this does not work -eval_file /Users/leuschel/git_root/prob_examples/examples/Setlog/prob-ttf/plavis-TransData_SP_21_simplified.prob
250		%:- block try_ensure_seq_numbering(-,?).
251		%try_ensure_seq_numbering([H|T],NextNr) :- var(H),!, print(nr(NextNr)),nl,
252		% H = (int(NextNr),_), N1 is NextNr+1,
253		% try_ensure_seq_numbering(T,N1).
254		%try_ensure_seq_numbering(_,_).
255
256		:- block is_sequence2(-,?,?,?,?,?).
257		is_sequence2([],IndexesSoFar,_Type,Size,MinSize,_WF) :- MinSize = Size,
258		contiguous_set_of_indexes(IndexesSoFar,Size).
259		/* not very good to do the checking at the end; can we move part of the checking earlier ? */
260		is_sequence2([(int(Idx),X)|Tail],IndexesSoFar,Type,Size,MinSize,WF) :-
261		less_than_direct(0,Idx), %is_index_greater_zero(Idx),
262		not_element_of_wf(int(Idx),IndexesSoFar,WF),
263		check_element_of_wf(X,Type,WF), S1 is Size+1,
264		clpfd_interface:clpfd_leq(Idx,MinSize,_Posted),
265		(var(Tail)
266		-> clpfd_interface:clpfd_domain(MinSize,Low,_Up), % TO DO: ensure that final size at least Low
267		(number(Low),Low>S1 -> Tail = [_|_] % TO DO: proper reification; what if MinSize gets constrained later
268		; expand_seq_if_necessary(Idx,S1,Tail)) % the sequence must be longer; force it
269		; true
270		),
271		is_sequence2(Tail,[int(Idx)|IndexesSoFar],Type,S1,MinSize,WF).
272
273		:- block expand_seq_if_necessary(-,?,-).
274		expand_seq_if_necessary(MinSize,S1,Tail) :- % TO DO: proper reification on MinSize above
275		number(MinSize), MinSize>S1, (var(Tail) ; Tail==[]),
276		!,
277		Tail = [_|_].
278		expand_seq_if_necessary(_,_,_).
279
280		:- block contiguous_set_of_indexes(-,?).
281		contiguous_set_of_indexes([],_).
282		contiguous_set_of_indexes([H|T],Size) :- contiguous_set_of_indexes1(T,H,Size).
283
284		:- block contiguous_set_of_indexes1(-,?,?).
285		contiguous_set_of_indexes1([],int(1),_).
286		contiguous_set_of_indexes1([int(H2)|T],int(H1),Size) :- less_than_equal_direct(H1,Size),
287		less_than_equal_direct(H2,Size), less_than_equal_indexes(T,[H1,H2],Size).
288
289
290		less_than_equal_indexes([],All,_) :- clpfd_interface:clpfd_alldifferent(All).
291		less_than_equal_indexes([int(H)|T],All,Size) :- less_than_equal_direct(H,Size),less_than_equal_indexes(T,[H|All],Size).
292
293		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(4),int(7))],[int(7),int(6)],WF),WF)).
294		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(3),int(8))],[int(7),int(6)],WF),WF)).
295		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(7),int(6)],WF),WF)).
296		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(2),int(6)),(int(3),int(7)),(int(4),int(7))],[int(7),int(6)],WF),WF)).
297		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(0),int(7)),(int(2),int(7))],[int(7),int(6)],WF),WF)).
298		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_sequence([(int(1),int(22))],[int(22)]))).
299		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
300		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)]))).
301		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_sequence([(int(3),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
302		:- assert_must_fail((not_is_sequence(R,global_set('Name')),R = [])).
303		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
304		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
305		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
306		R = [(int(1),fd(2,'Name'))] )).
307		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
308		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
309		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
310		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
311		:- assert_must_fail((not_is_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
312		global_set('Name')) )).
313		:- assert_must_fail((not_is_sequence(R,[int(1),int(2)]),
314		R = [(int(2),int(2)),(int(1),int(1))] )).
315		:- assert_must_succeed((not_is_sequence(R,[int(1),int(2)]),
316		R = [(int(2),int(2)),(int(3),int(1))] )).
317		:- assert_must_succeed((not_is_sequence(R,[int(1),int(2)]),
318		R = [(int(2),int(2)),(int(1),int(3))] )).
319
320
321		not_is_sequence(X,Type) :- init_wait_flags(WF,[not_is_sequence]),
322		not_is_sequence_wf(X,Type,WF),
323		ground_wait_flags(WF).
324
325		:- block not_is_sequence_wf(-,?,?).
326		not_is_sequence_wf(FF,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
327		% we do not need the Range; this means we can match more closures (e.g., lambda)
328		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
329		not_is_sequence_domain(FFDomain,WF).
330		not_is_sequence_wf(Seq,Range,WF) :-
331		expand_and_convert_to_avl_set_warn(Seq,AER,not_is_sequence_wf,'ARG /: seq(?)',WF),
332		!,
333		(is_avl_sequence(AER) -> is_not_avl_relation_over_range(AER,Range,WF)
334		; true).
335		not_is_sequence_wf(X,Type,WF) :- expand_custom_set_to_list_wf(X,EX,_Done,not_is_sequence_wf,WF),
336		not_is_sequence2(EX,[],Type,WF).
337
338		:- block not_is_sequence2(-,?,?,?).
339	?	not_is_sequence2([],IndexesSoFar,_,_WF) :- not_contiguous_set_of_indexes(IndexesSoFar).
340		not_is_sequence2([(int(Idx),X)|Tail],IndexesSoFar,Type,WF) :-
341		membership_test_wf(IndexesSoFar,int(Idx),MemRes,WF),
342	?	not_is_sequence3(MemRes,Idx,X,Tail,IndexesSoFar,Type,WF).
343
344		:- block not_is_sequence3(-,?,?,?,?,?,?).
345		not_is_sequence3(pred_true,_Idx,_X,_Tail,_IndexesSoFar,_Type,_WF).
346		not_is_sequence3(pred_false,Idx,_X,_Tail,_IndexesSoFar,_Type,_WF) :- nonvar(Idx),Idx<1,!.
347		not_is_sequence3(pred_false,Idx,X,Tail,IndexesSoFar,Type,WF) :-
348		membership_test_wf(Type,X,MemRes,WF),
349	?	not_is_sequence4(MemRes,Idx,Tail,IndexesSoFar,Type,WF).
350
351		:- block not_is_sequence4(-,?,?,?,?,?).
352		not_is_sequence4(pred_false,_Idx,_Tail,_IndexesSoFar,_Type,_WF).
353		not_is_sequence4(pred_true,Idx,Tail,IndexesSoFar,Type,WF) :-
354	?	not_is_sequence2(Tail,[int(Idx)|IndexesSoFar],Type,WF).
355
356		not_contiguous_set_of_indexes(Indexes) :-
357	?	when(ground(Indexes),(sort(Indexes,Sorted),not_contiguous_set_of_indexes2(Sorted,1))).
358		not_contiguous_set_of_indexes2([int(N)|T],N1) :-
359	?	when(?=(N,N1),
360		((N \= N1) ; (N=N1, N2 is N1+1, not_contiguous_set_of_indexes2(T,N2)))).
361
362
363
364
365
366		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_non_empty_sequence([(int(1),int(22))],[int(22)]))).
367		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_non_empty_sequence([(int(1),int(2))],[int(22)]))).
368		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),R = [])).
369		:- assert_must_fail((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
370		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
371		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
372		R = [(int(2),fd(1,'Name')),(int(4),fd(2,'Name'))] )).
373		:- assert_must_fail((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
374		R = [(int(1),fd(1,'Name')),(int(2),fd(1,'Name'))] )).
375		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,[int(1),int(2)]),
376		R = [(int(1),int(2)),(int(2),int(3))] )).
377
378		% S /: seq1(T)
379		not_is_non_empty_sequence_wf(S,T,_) :- not_is_non_empty_sequence(S,T).
380		:- block not_is_non_empty_sequence(-,?).
381		not_is_non_empty_sequence([],_) :- !.
382		not_is_non_empty_sequence(X,Type) :-
383		empty_sequence(X) ; not_is_sequence(X,Type).
384
385
386
387		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([(int(1),int(22))],[int(22)],WF),WF)).
388		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
389		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
390		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(22)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
391		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([],global_set('Name'),WF),WF)).
392		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
393		kernel_waitflags:ground_det_wait_flag(WF), R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
394		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
395		ground_det_wait_flag(WF), R = [(int(1),fd(2,'Name'))] )).
396		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
397		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
398		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
399		ground_det_wait_flag(WF), R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
400		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_sequence_wf([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
401		global_set('Name'),WF),WF) ).
402		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
403		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(1),int(1))] )).
404		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
405		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(3),int(1))] )).
406		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
407		ground_det_wait_flag(WF), R = [(int(2),int(2)),(int(1),int(3))] )).
408
409
410
411		:- block injective_sequence_wf(-,-,?).
412		injective_sequence_wf(Seq,Type,WF) :- /* corresponds to iseq */
413		nonvar(Seq),
414		%expand_and_convert_to_avl_set_warn(Seq,AER,injective_sequence_wf_aux,'ARG : iseq(?)',WF),
415		Seq=avl_set(AER),
416		!,
417		is_avl_sequence(AER),
418		is_injective_avl_relation(AER,_ExactRange), % Should we check _ExactRange <: Type ??
419		is_avl_relation_over_range(AER,Type,WF).
420		injective_sequence_wf(Seq,Type,WF) :-
421		cardinality_as_int_for_wf(Type,MaxCard),
422		custom_explicit_sets:blocking_nr_iseq(MaxCard,ISeqSize),
423		block_get_wait_flag(ISeqSize,injective_sequence_wf,WF,LWF),
424		injective_sequence_wf_aux(Seq,Type,MaxCard,WF,LWF).
425
426		:- block injective_sequence_wf_aux(-,?,?,?,-).
427		injective_sequence_wf_aux(Seq,Type,_,WF,_) :- /* corresponds to iseq */
428		nonvar(Seq),
429		expand_and_convert_to_avl_set_warn(Seq,AER,injective_sequence_wf_aux,'ARG : iseq(?)',WF),!,
430		%Seq=avl_set(AER),
431		!,
432		is_avl_sequence(AER),
433		is_injective_avl_relation(AER,_ExactRange), % Should we check _ExactRange <: Type ??
434		is_avl_relation_over_range(AER,Type,WF).
435		injective_sequence_wf_aux(Seq,Type,MaxCard,WF,LWF) :-
436		expand_custom_set_to_list_wf(Seq,ESeq,_,injective_sequence_wf,WF),
437		is_sequence_wf(ESeq,Type,WF),
438		injective_sequence2(ESeq,0,[],Type,WF,MaxCard,LWF).
439
440		:- block injective_sequence2(-,?,?,?,?,?,-),injective_sequence2(-,?,?,?,?,-,?).
441		injective_sequence2([],_,_,_Type,_WF,_MaxCard,_LWF).
442		injective_sequence2([(int(Index),X)|Tail],CardSoFar,SoFar,Type,WF,MaxCard,LWF) :-
443		(number(MaxCard) -> CardSoFar< MaxCard, %less_than_equal_direct(Index,MaxCard) % does not enumerate index
444		in_nat_range_wf(int(Index),int(0),int(MaxCard),WF) % ensures the index gets enumerated, see test 1914, x:iseq(50001..50002) & y:1..100005 & SIGMA(yy).(yy:dom(x)|x(yy)) = y & y>50002
445		; true),
446		check_element_of_wf(X,Type,WF),
447		not_element_of_wf(X,SoFar,WF),
448		add_new_element_wf(X,SoFar,SoFar2,WF),
449		C1 is CardSoFar+1,
450		(C1 == MaxCard -> Tail=[] ; true),
451		injective_sequence2(Tail,C1,SoFar2,Type,WF,MaxCard,LWF).
452
453
454		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_injective_sequence([(int(1),int(22))],[int(22)],WF),WF)).
455		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_injective_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
456		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_injective_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
457		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_injective_sequence([(int(2),int(22)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
458		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),_WF),R = [])).
459		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
460		ground_det_wait_flag(WF),
461		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
462		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
463		ground_det_wait_flag(WF),
464		R = [(int(1),fd(2,'Name'))] )).
465		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
466		ground_det_wait_flag(WF),
467		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
468		:- assert_must_fail((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
469		ground_det_wait_flag(WF),
470		R = [(int(2),int(2)),(int(1),int(1))] )).
471		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
472		ground_det_wait_flag(WF),
473		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
474		:- assert_must_succeed((bsets_clp:not_injective_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
475		global_set('Name'),WF),
476		ground_det_wait_flag(WF) )).
477		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
478		ground_det_wait_flag(WF),
479		R = [(int(2),int(2)),(int(3),int(1))] )).
480		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
481		ground_det_wait_flag(WF),
482		R = [(int(2),int(2)),(int(1),int(3))] )).
483		:- block not_injective_sequence(-,?,?), not_injective_sequence(?,-,?).
484		not_injective_sequence(Seq,_,_) :- Seq==[],!,fail.
485		not_injective_sequence(Seq,Type,WF) :- nonvar(Seq),
486		expand_and_convert_to_avl_set_warn(Seq,AER,not_injective_sequence,'ARG /: iseq(?)',WF),!,
487		(\+ is_avl_sequence(AER) -> true
488		; is_injective_avl_relation(AER,ExactRange) -> not_subset_of_wf(ExactRange,Type,WF)
489		; true).
490		not_injective_sequence(Seq,Type,WF) :- /* corresponds to Iseq */
491		%get_middle_wait_flag(not_injective_sequence,WF,LWF),
492		ground_value_check(Seq,SV),
493		not_injective_sequence1(Seq,Type,WF,SV).
494		:- block not_injective_sequence1(?,?,?,-).
495		not_injective_sequence1(Seq,Type,WF,_) :-
496		expand_custom_set_to_list_wf(Seq,ESeq,_,not_injective_sequence1,WF),
497		(not_is_sequence_wf(ESeq,Type,WF)
498		; /* CHOICE POINT !! */
499		(is_sequence_wf(ESeq,Type,WF),not_injective_sequence2(ESeq,[],Type,WF))).
500		:- block not_injective_sequence2(-,?,?,?).
501		not_injective_sequence2([(int(_),X)|Tail],SoFar,Type,WF) :-
502		membership_test_wf(SoFar,X,MemRes,WF),
503		not_injective_sequence3(MemRes,X,Tail,SoFar,Type,WF).
504
505		:- block not_injective_sequence3(-,?,?,?,?,?).
506		not_injective_sequence3(pred_true,_X,_Tail,_SoFar,_Type,_WF).
507		not_injective_sequence3(pred_false,X,Tail,SoFar,Type,WF) :-
508		add_new_element_wf(X,SoFar,SoFar2,WF),
509		not_injective_sequence2(Tail,SoFar2,Type,WF).
510
511		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(1),int(22))],[int(22)],WF),WF)).
512		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
513		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
514		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(33))],[int(44),int(33),int(22)],WF),WF)).
515		:- assert_must_succeed((bsets_clp:not_non_empty_injective_sequence(R,global_set('Name'),WF),
516		ground_det_wait_flag(WF), R = [])).
517		:- assert_must_fail((bsets_clp:not_non_empty_injective_sequence(R,global_set('Name'),WF),
518		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
519		:- assert_must_succeed((bsets_clp:not_non_empty_injective_sequence(R,[int(1),int(2)],WF),
520		ground_det_wait_flag(WF), R = [(int(2),int(2)),(int(1),int(3))] )).
521
522		:- block not_non_empty_injective_sequence(-,?,?).
523		not_non_empty_injective_sequence([],_Type,_WF) :- !.
524		not_non_empty_injective_sequence(X,Type,WF) :-
525		empty_sequence(X) ; not_injective_sequence(X,Type,WF).
526
527
528		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_non_empty_sequence([(int(1),int(22))],[int(22)],WF),WF)).
529		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_non_empty_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
530		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_non_empty_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
531		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_non_empty_sequence([(int(2),int(44)),(int(1),int(44))],[int(22),int(33),int(44)],WF),WF)).
532		:- assert_must_fail((bsets_clp:injective_non_empty_sequence(R,global_set('Name'),WF),
533		ground_det_wait_flag(WF),R = [])).
534		:- assert_must_succeed((bsets_clp:injective_non_empty_sequence(R,global_set('Name'),WF),
535		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
536		:- block injective_non_empty_sequence(-,-,?). /* corresponds to iseq1 */
537		injective_non_empty_sequence(A,Type,WF) :- nonvar(A),A=avl_set(AS), !,
538		injective_sequence_wf(avl_set(AS),Type,WF),is_non_empty_explicit_set_wf(avl_set(AS),WF).
539		injective_non_empty_sequence(Seq,Type,WF) :-
540		((nonvar(Seq),Seq=closure(_,_,_)) -> try_expand_custom_set_wf(Seq,ESeq,injective_non_empty_sequence,WF) ; ESeq=Seq),
541		injective_sequence_wf(ESeq,Type,WF),not_empty_sequence_wf(ESeq,WF).
542
543		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:finite_non_empty_sequence([(int(1),int(22))],[int(22)],WF),WF)).
544		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:finite_non_empty_sequence([(int(1),int(33)),(int(2),int(33))],[int(22),int(33)],WF),WF)).
545		:- assert_must_fail((bsets_clp:finite_non_empty_sequence(R,global_set('Name'),WF),ground_det_wait_flag(WF),ground_det_wait_flag(WF),R = [])).
546		:- assert_must_succeed((bsets_clp:finite_non_empty_sequence(R,global_set('Name'),WF),
547		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
548		:- block finite_non_empty_sequence(-,?,?).
549		finite_non_empty_sequence(Seq,Type,WF) :- /* corresponds to Seq1 */
550		is_sequence_wf_ex(Seq,Type,WF,ESeq),
551		(var(ESeq) -> not_empty_sequence_wf(Seq,WF) ; not_empty_sequence_wf(ESeq,WF)).
552
553
554		:- block test_finite_non_empty_sequence(-,?,-,?).
555		test_finite_non_empty_sequence(Seq,_Type,Res,_WF) :-
556		Seq == [],!, Res=pred_false.
557		test_finite_non_empty_sequence(Seq,Type,Res,WF) :- var(Res),!,
558		ground_value_check(Seq,GrSeq),
559		test_finite_non_empty_sequence2(Res,Seq,Type,GrSeq,WF). % will trigger and enumerate Res below
560		% Note: we cannot rely on Res being enumerated; e.g., in case a WD error occurs
561		test_finite_non_empty_sequence(Seq,Type,Res,WF) :-
562		test_finite_non_empty_sequence2(Res,Seq,Type,_,WF).
563
564		% TODO: improve to incrementally check if something is a sequence
565		:- block test_finite_non_empty_sequence2(-,?,?,-,?).
566		test_finite_non_empty_sequence2(pred_true,Seq,Type,_,WF) :-
567		finite_non_empty_sequence(Seq,Type,WF).
568		test_finite_non_empty_sequence2(pred_false,Seq,Type,_,WF) :-
569		not_is_non_empty_sequence_wf(Seq,Type,WF).
570
571
572
573		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:permutation_sequence_wf([(int(1),int(22))],[int(22)],WF),WF)).
574		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:permutation_sequence_wf([(int(2),int(33)),(int(1),int(22))],[int(22),int(33)],WF),WF)).
575		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:permutation_sequence_wf([(int(2),int(33)),(int(1),int(23))],[int(23),int(33),int(44)],WF),WF)).
576		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:permutation_sequence_wf([(int(2),int(44)),(int(1),int(44))],[int(44)],WF),WF)).
577		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1)],WF),
578		ground_det_wait_flag(WF),R = [(int(1),int(1))] )).
579		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
580		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(2))] )).
581		:- assert_must_succeed((bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
582		ground_det_wait_flag(WF),R = [(int(1),int(2)),(int(2),int(1))] )).
583		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[pred_true /* bool_true */,pred_false /* bool_false */],WF), kernel_waitflags:ground_wait_flags(WF), nonvar(R),
584		R = [(int(1),pred_false /* bool_false */),(int(2),pred_true /* bool_true */)] )).
585		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1)],WF),
586		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(1))] )).
587		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),ground_det_wait_flag(WF),R = [])).
588		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
589		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
590		:- assert_must_succeed((bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
591		ground_det_wait_flag(WF),
592		kernel_objects:equal_object(R,[(int(1),fd(1,'Name')),(int(3),fd(2,'Name')),(int(2),fd(3,'Name'))]) )).
593		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
594		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(3))] )).
595		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
596		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
597		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(4),int(3),int(2),int(1)],WF),
598		ground_det_wait_flag(WF), R=[(int(1),int(1)),(int(2),int(2)),(int(3),int(3)),(int(4),int(4))])).
599
600		:- block permutation_sequence_wf(-,-,?).
601		permutation_sequence_wf(SeqFF,Type,WF) :- nonvar(SeqFF),
602		custom_explicit_sets:dom_range_for_specific_closure(SeqFF,FFDomain,FFRange,function(bijection),WF),!,
603		equal_object_wf(FFRange,Type,permutation_sequence_wf_1,WF),
604		is_sequence_domain(FFDomain,WF).
605		permutation_sequence_wf(Seq,Type,WF) :-
606		expand_and_convert_to_avl_set_warn(Seq,AER,permutation_sequence_wf,'ARG : perm(?)',WF),!,
607		is_avl_sequence(AER),
608		is_injective_avl_relation(AER,Range),
609		kernel_objects:equal_object_wf(Range,Type,permutation_sequence_wf_2,WF).
610		permutation_sequence_wf(Seq,Type,WF) :-
611		try_expand_custom_set_wf(Seq,ESeq,permutation_sequence_wf,WF),
612		cardinality_as_int_wf(Type,int(Card),WF),
613		when(nonvar(Card), (setup_sequence_wf(Card,SkelSeq,perm,WF),
614		CardGround=true,
615		kernel_objects:equal_object_wf(SkelSeq,ESeq,permutation_sequence_wf_3,WF))),
616		%injective_sequence_wf(ESeq,Type,WF,LWF),
617		surjective_iseq_0(SkelSeq,ESeq,Type,WF,Card,CardGround).
618		% quick_all_different_range(ESeq,[],Type,WF).
619
620		:- block surjective_iseq_0(-,-,?,?,?,-).
621		surjective_iseq_0(SkelSeq,_ESeq,Type,WF,_Card,Ground) :-
622		nonvar(Ground),
623		nonvar(SkelSeq),
624		preference(use_clpfd_solver,true), % try and use an optimized version calling global_cardinality in CLPFD module
625	?	get_global_cardinality_list(Type,YType,GCL,_,WF),
626		% this dramatically reduces runtime for NQueens40_perm; maybe we should do this only when necessary, i.e., when surjective_iseq blocks on PreviousRemoveDone
627		% check why it slows down SortByPermutation_v2
628		!,
629		global_cardinality_range(SkelSeq,[],YType,GCL,WF).
630		surjective_iseq_0(_,ESeq,Type,WF,Card,_) :-
631		%quick_propagate_range(ESeq,Type,WF), % ensure that we propagate type information to all elements; p:perm(5..20) & p(10)=21 will fail straightaway (surjective_iseq will block);
632		% but this slows down EulerWay.mch ; maybe because it sets up enumerators ? TO DO: investigate
633		surjective_iseq(ESeq,Type,WF,Card).
634
635		%:- use_module(clpfd_interface,[clpfd_alldifferent/1]).
636		% collect range and then call CLPFD global_cardinality using GCL (Global Cardinality List Ki-Vi)
637		:- use_module(library(clpfd), [global_cardinality/3]).
638		:- block global_cardinality_range(-,?,?,?,?).
639		global_cardinality_range([],Acc,_Type,GCL,WF) :-
640		global_cardinality(Acc,GCL,[consistency(value)]),
641		add_fd_variables_for_labeling(Acc,WF). % this is needed for efficiency for NQueens40_perm !!
642		global_cardinality_range([(_,Y)|T],Acc,Type,GCL,WF) :-
643		get_simple_fd_value(Type,Y,FDYVAL),
644		global_cardinality_range(T,[FDYVAL|Acc],Type,GCL,WF).
645
646
647		:- use_module(library(avl), [avl_domain/2]).
648		:- use_module(b_global_sets,[all_elements_of_type_wf/3,b_integer_set/1]).
649		% try and convert a B set into a list suitable for calling clpfd:global_cardinality
650		% get_global_cardinality_list(avl_set(A) % TO DO: extend to integer_lists
651		get_global_cardinality_list(global_set(G),Type,GCL,list,WF) :- !,
652		all_elements_of_type_wf(G,Values,WF), % can only work for finite sets, not for STRING, NATURAL, REAL, ...
653		(b_integer_set(G) -> Type=integer ; Type=global(G)),
654		findall(X-1,(get_simple_fd_value(Type,VV,X),member(VV,Values)),GCL).
655		get_global_cardinality_list(avl_set(A),Type,GCL,list,_WF) :- !,
656		A = node(TopValue,_True,_,_,_),
657	?	get_simple_fd_value(Type,TopValue,_), % we have CLPFD values
658		avl_domain(A,Values),
659		findall(X-1,(get_simple_fd_value(Type,VV,X),member(VV,Values)),GCL).
660		get_global_cardinality_list(Set,integer,GCL,interval(L1,U1),_WF) :- nonvar(Set),
661		is_interval_closure_or_integerset(Set,L1,U1), number(L1),number(U1),
662		global_cardinality_list_interval(L1,U1,GCL).
663
664		global_cardinality_list_interval(From,To,[]) :- From>To, !.
665		global_cardinality_list_interval(From,To,[From-1|T]) :-
666		F1 is From+1, global_cardinality_list_interval(F1,To,T).
667
668		%try_get_simple_fd_value(Type,V,Val) :- nonvar(V),get_simple_fd_value(Type,V,Val).
669		get_simple_fd_value(integer,int(X),X).
670		get_simple_fd_value(global(T),fd(X,T),X).
671		% try_get_simple_fd_value(pred_false,0). try_get_simple_fd_value(pred_true,1). ??
672		% TO DO: maybe also treat pairs ? but we need complete values; see module clpfd_lists !
673
674		setup_sequence_wf(0,R,_,_) :- !, R=[].
675		setup_sequence_wf(Card,_,PP,WF) :- \+ number(Card), !,
676		add_error_wf(infinite_sequence,'Cannot generate infinite sequence for', PP,unkown,WF). % triggered in test 1979
677		setup_sequence_wf(Card,[(int(1),_)|T] ,_PP,_WF) :- Card>0, C1 is Card-1,
678		setup_sequence(C1,T,2).
679		setup_sequence(0,R,_) :- !, R=[].
680		setup_sequence(Card,[(int(Nr),_)|T], Nr ) :- Card>0, C1 is Card-1,
681		N1 is Nr+1,
682		setup_sequence(C1,T,N1).
683
684		:- block surjective_iseq(?,?,?,-),surjective_iseq(?,-,?,?), surjective_iseq(-,?,?,?).
685		surjective_iseq(avl_set(S),Type,WF,Done) :-
686		expand_custom_set_wf(avl_set(S),ES,surjective_iseq,WF),
687		surjective_iseq(ES,Type,WF,Done).
688		surjective_iseq(closure(P,T,B),Type,WF,Done) :-
689		expand_custom_set_wf(closure(P,T,B),ES,surjective_iseq,WF),
690		surjective_iseq(ES,Type,WF,Done).
691		% no case for global_set: cannot be a relation
692		surjective_iseq([],T,WF,_) :- empty_set_wf(T,WF).
693		surjective_iseq([(int(_Nr),El)|Tail],Type,WF,_PreviousRemoveDone) :-
694		remove_element_wf(El,Type,NType,WF,Done),
695		surjective_iseq(Tail,NType,WF,Done).
696		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_permutation_sequence([(int(1),int(22))],[int(22)],WF),WF)).
697		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33)],WF),WF)).
698		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(33)),(int(1),int(23))],[int(23),int(33),int(44)],WF),WF)).
699		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(44)),(int(1),int(44))],[int(44)],WF),WF)).
700		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1)],WF),
701		ground_det_wait_flag(WF),R = [(int(1),int(1))] )).
702		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
703		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(1),int(1))] )).
704		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
705		ground_det_wait_flag(WF),R = [(int(1),int(2)),(int(2),int(1))] )).
706		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
707		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(3),fd(2,'Name')),(int(2),fd(3,'Name'))] )).
708		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,[int(1)],WF),
709		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(1))] )).
710		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),_WF),R = [])).
711		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
712		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
713		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
714		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(3))] )).
715		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
716		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
717		:- block not_permutation_sequence(-,?,?).
718		not_permutation_sequence(SeqFF,Type,WF) :- nonvar(SeqFF),
719		custom_explicit_sets:dom_range_for_specific_closure(SeqFF,FFDomain,FFRange,function(bijection),WF),!,
720		equality_objects_wf(FFRange,Type,Result,WF),
721		when(nonvar(Result),(Result=pred_false -> true ; not_is_sequence_domain(FFDomain,WF))).
722		not_permutation_sequence(Seq,Type,WF) :-
723		ground_value_check(Seq,SV),
724	?	not_permutation_sequence1(Seq,Type,SV,WF).
725		:- block not_permutation_sequence1(?,-,?,?), not_permutation_sequence1(?,?,-,?).
726		not_permutation_sequence1(avl_set(A),Type,_,WF) :- is_ground_set(Type), !, Seq=avl_set(A),
727		if(not_injective_sequence(Seq,Type,WF),
728		true, % no backtracking required; we could even use regular if with ->
729		not_surj_avl(Seq,Type,WF)
730		).
731		not_permutation_sequence1(avl_set(A),Type,_,WF) :- !, Seq=avl_set(A),
732		(not_injective_sequence(Seq,Type,WF)
733		; injective_sequence_wf(Seq,Type,WF),
734		not_surj_avl(Seq,Type,WF)).
735		not_permutation_sequence1(Seq,Type,_,WF) :-
736		expand_custom_set_to_list_wf(Seq,ESeq,Done,not_permutation_sequence1,WF),
737	?	not_permutation_sequence2(ESeq,Type,WF,Done).
738
739		not_surj_avl(Seq,Type,WF) :- range_wf(Seq,Range,WF),
740		not_equal_object_wf(Range,Type,WF). % TO DO: one could even just check cardinality as Seq is inj
741		%expand_custom_set_to_list_wf(Seq,ESeq,_,not_permutation_sequence1,WF),
742		% not_surjective_seq(ESeq,Type,WF).
743		% check if it is a ground set that cannot be instantiated
744		is_ground_set(V) :- var(V),!,fail.
745		is_ground_set(avl_set(_)).
746		is_ground_set(global_set(_)).
747		is_ground_set([]).
748
749		% here we could have a choice point in WF0
750		:- block not_permutation_sequence2(?,?,?,-).
751		not_permutation_sequence2(Seq,Type,WF,_) :- not_injective_sequence(Seq,Type,WF).
752		not_permutation_sequence2(Seq,Type,WF,_) :-
753		injective_sequence_wf(Seq,Type,WF), not_surjective_seq(Seq,Type,WF).
754
755		:- block not_surjective_seq(-,?,?).
756		not_surjective_seq([],T,WF) :- not_empty_set_wf(T,WF).
757		not_surjective_seq([(int(_),El)|Tail],Type,WF) :-
758		delete_element_wf(El,Type,NType,WF),
759		not_surjective_seq(Tail,NType,WF).
760
761		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(1),int(22))],int(1),_WF))).
762		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22))],int(2),_WF))).
763		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22))],int(3),_WF))).
764		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22)),(int(3),int(33))],int(3),_WF))).
765		:- assert_must_succeed((bsets_clp:size_of_sequence(X,R,_WF),
766		X = [(int(1),int(2)),(int(2),int(1))],
767		R = int(2))).
768		:- assert_must_succeed((preferences:preference(use_clpfd_solver,false) -> true
769		; preferences:preference(use_smt_mode,false) -> true
770		; bsets_clp:size_of_sequence(X,R,_WF), R=int(RI),
771		clpfd_interface:clpfd_geq2(RI,2,_), nonvar(X), X = [(I1,_),(I2,_)|T],
772		I1==int(1), I2==int(2), T=[], RI==2 )).
773		:- assert_must_succeed((bsets_clp:size_of_sequence(X,R,_WF),X = [(int(1),_),(int(2),_)],R = int(2))).
774		:- assert_must_succeed((bsets_clp:size_of_sequence(X,_R,_WF),X =[(int(1),_),(int(2),_)] )).
775		:- assert_must_succeed_any((bsets_clp:size_of_sequence(X,int(2),_WF),nonvar(X),X=[_|Y],nonvar(Y),Y=[_|Z],Z==[])).
776		:- assert_must_succeed((bsets_clp:size_of_sequence([],int(0),_WF))).
777		:- assert_must_succeed((bsets_clp:size_of_sequence([],int(0),_WF))).
778		:- assert_must_succeed((bsets_clp:size_of_sequence([(int(1),int(4))],int(1),_WF))).
779		:- assert_must_succeed((bsets_clp:size_of_sequence([],_,_WF))).
780		:- assert_must_fail((bsets_clp:size_of_sequence(X,int(1),_WF),
781		X = [(int(1),_),(int(2),_)|_])).
782		:- block size_of_sequence(-,-,?).
783	?	size_of_sequence(Seq,int(Res),WF) :- size_of_sequence1(Seq,Res,WF),
784		set_up_sequence_skel(Seq,Res,WF).
785
786		% setup sequence skeleton if we have some CLPFD bounds information about the size
787		% currently still quite limited: only sets up if sequence is a variable; + does the setup only once
788		:- use_module(library(clpfd), [(#<=>)/2]).
789		:- use_module(clpfd_interface,[clpfd_domain/3]).
790		set_up_sequence_skel(Seq,Res,WF) :-
791		var(Seq), % to do: also deal with cases when Seq partially instantiated
792		var(Res),
793		preferences:preference(use_clpfd_solver,true),
794		!,
795		clpfd_interface:clpfd_geq2(Res,0,_), % assert that size must not be negative
796		clpfd_interface:try_post_constraint((Res#>0) #<=> Trigger), % generate reified trigger for when we can instantiate Seq
797		set_up_sequence_skel_aux(Seq,Res,Trigger,WF).
798		set_up_sequence_skel(_,_,_). % TO DO: check if Size interval shrinks
799		:- block set_up_sequence_skel_aux(-,?,-,?).
800		set_up_sequence_skel_aux(Seq,_Res,_Trigger,_WF) :-
801		nonvar(Seq),
802		!. % to do: also deal with cases when Seq partially instantiated
803		set_up_sequence_skel_aux(Seq,Res,_Trigger,_WF) :-
804		(number(Res) ; preferences:preference(use_smt_mode,true)),
805		!,
806		gen_seq_for_res(Res,Seq).
807		set_up_sequence_skel_aux(Seq,Res,_Trigger,WF) :-
808		get_large_finite_wait_flag(set_up_sequence_skel,WF,LWF), % delay, avoid costly unification with partially instantaited list skeleton; TO DO: in future we may use the kernel_cardinality attribute instead
809		when((nonvar(LWF) ; nonvar(Seq) ; nonvar(Res)), (nonvar(Seq) -> true ; gen_seq_for_res(Res,Seq))).
810
811		gen_seq_for_res(Res,Seq) :-
812		clpfd_domain(Res,FDLow,FDUp), % FDLow could also be 0
813		gen_sequence_skeleton(1,FDLow,FDUp,S),
814		Seq=S.
815		gen_sequence_skeleton(N,M,FDUp,S) :- N>M,!,(FDUp==M -> S=[] ; true).
816		gen_sequence_skeleton(N,Max,FDUp,[(int(N),_)|T]) :-
817		N1 is N+1,
818		gen_sequence_skeleton(N1,Max,FDUp,T).
819
820		:- block size_of_sequence1(-,-,?).
821		size_of_sequence1(Seq,ResInt,WF) :-
822		nonvar(Seq),is_custom_explicit_set_nonvar(Seq),
823		size_of_custom_explicit_set(Seq,Size,WF),!,
824	?	equal_object_wf(Size,int(ResInt),size_of_sequence1,WF).
825		/* TO DO: CHECK BELOW: would it not be better to use cardinality ?? */
826		/*
827		size_of_sequence1(Seq,Size,WF) :- !,kernel_cardinality_attr:finite_cardinality_as_int_wf(Seq,int(Size),WF), check_indexes(Seq,Size).
828
829		construct_interval_closure(1,Size,Domain),
830		total_function_wf(FF,Domain,Range,_WF)
831		% we could also call total_function 1..Size --> _RangeType; would setup domain ?
832		:- block check_indexes(-,?).
833		check_indexes([],_) :- !.
834		check_indexes([(int(X),_)|T],Size) :- !,
835		less_than_equal_direct(X,Size), check_indexes(T,Size).
836		check_indexes(_,_).
837		*/
838	?	size_of_sequence1(Seq,Size,_WF) :- Size==0,!, empty_sequence(Seq).
839		size_of_sequence1(Seq,Size,WF) :-
840		expand_custom_set_to_list_wf(Seq,ESeq,_,size_of_sequence1,WF),
841	?	(var(ESeq),nonvar(Size) -> size_of_var_seq(ESeqR,0,Size),
842		ESeqR=ESeq % unify after to do propagation in one go, without triggering coroutines inbetween
843	?	; size_of_seq2(ESeq,0,Size),
844		(var(Size),var(ESeq) -> less_than_equal_direct(0,Size) % propagate that Size is positive
845		; true)
846		).
847		/* small danger of expanding closure while still var !*/
848		:- block size_of_seq2(-,?,-).
849		size_of_seq2([],Size,Size).
850		size_of_seq2([I|Tail],SizeSoFar,Res) :-
851		S2 is SizeSoFar + 1,
852	?	check_index(I,Res), % don't instantiate I yet; allow other kernel_predicates to freely instantiate it
853		less_than_equal_direct(S2,Res),
854		%(ground(Res) -> safe_less_than_equal(size_of_seq2,S2,Res) ; true),
855	?	size_of_seq2(Tail,S2,Res).
856		size_of_var_seq([],Size,Size).
857		size_of_var_seq([(int(S2),_)|Tail],SizeSoFar,Res) :-
858		S2 is SizeSoFar + 1,safe_less_than_equal(size_of_var_seq,S2,Res),
859	?	(var(Tail) -> size_of_var_seq(Tail,S2,Res) ; size_of_seq2(Tail,S2,Res)).
860
861
862		:- block check_index(-,?).
863	?	check_index((I,_),Res) :- check_index1(I,Res).
864		:- block check_index1(-,?).
865	?	check_index1(int(Idx),Res) :- less_than_equal_direct(Idx,Res).
866
867		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(1),int(22))],[(int(2),int(22)),(int(1),int(33))],WF),WF)).
868		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[],[(int(1),int(33))],WF),WF)).
869		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(2),int(44)),(int(1),int(22))],[(int(1),int(33)),(int(3),int(44)),(int(2),int(22))],WF),WF)).
870		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(1),int(22))],[(int(1),int(22)),(int(2),int(33))],WF),WF)).
871		:- assert_must_succeed((bsets_clp:prepend_sequence(int(7),[],[(int(1),int(7))],_WF))).
872		:- assert_must_succeed((bsets_clp:prepend_sequence(int(7),X,R,_WF),
873		X = [(int(2),int(4)),(int(1),int(3))],
874		kernel_objects:equal_object(R,[(int(1),int(7)),(int(2),int(3)),(int(3),int(4))]))).
875		% code for insert_front operator: El -> Seq
876		:- block prepend_sequence(?,-,-,?).
877		prepend_sequence(El,Seq,Res,_WF) :- Seq==[],!,
878		equal_object_optimized([(int(1),El)],Res,prepend_sequence).
879		prepend_sequence(El,Seq,Res,WF) :- nonvar(Seq),is_custom_explicit_set(Seq,prepend_sequence),
880		prepend_custom_explicit_set(Seq,El,ERes),!,
881		equal_sequence(Res,ERes,WF).
882		prepend_sequence(El,Seq,Res,WF) :- nonvar(Res),is_custom_explicit_set(Res,prepend_sequence),
883		tail_sequence_custom_explicit_set(Res,First,Tail,unknown,WF),!,
884		equal_object_wf(El,First,prepend_sequence,WF),
885		equal_sequence(Seq,Tail,WF).
886		prepend_sequence(El,Seq,Res,WF) :-
887		equal_cons_wf(Res,(int(1),El),ShiftSeq,WF),
888		shift_seq_indexes(Seq,1,ShiftSeq,WF).
889
890		:- block shift_seq_indexes(-,-,?,?),shift_seq_indexes(-,?,-,?).
891		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :-
892		Offset == 0,!, equal_sequence(Seq,ShiftedSeq,WF).
893		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :- nonvar(Seq),!,
894		expand_custom_set_to_list_wf(Seq,ESeq,_,shift_seq_indexes,WF),
895		shift_seq_indexes2(ESeq,Offset,ShiftedSeq,WF,Done),
896		(Done == done
897		-> true
898		; % also propagate in the other way: TO DO: make a more efficient fine-grained two-ways propagation; maybe using CHR
899		NegOffset is -Offset,
900		expand_custom_set_to_list_wf(ShiftedSeq,ESeq1,_,shift_seq_indexes,WF),
901		shift_seq_indexes2(ESeq1,NegOffset,ESeq,WF,_)).
902		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :- NegOffset is -Offset,
903		% compute in the other direction; TO DO: make a more efficient fine-grained two-ways propagation; maybe using CHR
904		expand_custom_set_to_list_wf(ShiftedSeq,ESeq,_,shift_seq_indexes,WF),
905		shift_seq_indexes2(ESeq,NegOffset,Seq,WF,Done),
906		(Done == done
907		-> true
908		; % also propagate in the original way:
909		expand_custom_set_to_list_wf(Seq,ESeq1,_,shift_seq_indexes,WF),
910		shift_seq_indexes2(ESeq1,Offset,ESeq,WF,_)).
911
912		:- block shift_seq_indexes2(-,?,?,?,?).
913	?	shift_seq_indexes2([],_,R,WF,Done) :- !, Done = done, empty_set_wf(R,WF).
914		shift_seq_indexes2([Pair|Tail],Offset,Res,WF,Done) :- !,
915		Pair = (int(N),El),
916	?	equal_cons_wf(Res,(int(NewN),El),ShiftTail,WF),
917		int_plus(int(N),int(Offset),int(NewN)),
918		shift_seq_indexes2(Tail,Offset,ShiftTail,WF,Done).
919		shift_seq_indexes2(Seq,Offset,Res,WF,Done) :-
920		add_internal_error('Unexpected set argument: ',shift_seq_indexes2(Seq,Offset,Res,WF,Done)), fail.
921
922		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([(int(1),int(22))],int(33),[(int(2),int(33)),(int(1),int(22))],WF),WF)).
923		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([],int(33),[(int(1),int(33))],WF),WF)).
924		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([(int(2),int(44)),(int(1),int(22))],int(33),[(int(1),int(22)),(int(3),int(33)),(int(2),int(44))],WF),WF)).
925		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:append_sequence([(int(1),int(22))],int(33),[(int(1),int(33)),(int(2),int(22))],WF),WF)).
926		:- assert_must_succeed((bsets_clp:append_sequence([],int(7),[(int(1),int(7))],_WF))).
927		:- assert_must_succeed((bsets_clp:append_sequence(X,int(7),R,_WF),
928		X = [(int(2),int(4)),(int(1),int(3))],
929		kernel_objects:equal_object(R,[(int(1),int(3)),(int(2),int(4)),(int(3),int(7))]))).
930
931		% code for the insert_tail operator Seq<-El
932		:- block append_sequence(-,?,-,?).
933		append_sequence(Seq,El,Res,_WF) :- Seq==[],!,
934		equal_object_optimized([(int(1),El)],Res,append_sequence).
935		append_sequence(Seq,El,Res,WF) :-
936		nonvar(Seq),is_custom_explicit_set_nonvar(Seq),
937		append_custom_explicit_set(Seq,El,ERes,WF),!,
938		equal_sequence(Res,ERes,WF).
939		append_sequence(Seq,El,Res,WF) :-
940		nonvar(Res),is_custom_explicit_set_nonvar(Res),
941		% we know result: deconstruct into last El and front Seq
942		front_sequence_custom_explicit_set(Res,Last,Front), !,
943		equal_object_wf(El,Last,append_sequence,WF),
944		equal_sequence(Seq,Front,WF).
945		append_sequence(Seq,El,Res,WF) :-
946		(var(Seq) -> size_of_sequence(Res,INewSize,WF), INewSize=int(NewSize) ; true),
947		equal_cons_wf(Res,(int(NewSize),El),ResT,WF),
948		append_sequence2(Seq,ResT,NewSize,WF).
949
950		:- block append_sequence2(-,?,-,?).
951		append_sequence2(Seq,ResT,_NewSize,WF) :- var(Seq),!,
952		equal_sequence(Seq,ResT,WF).
953		append_sequence2(Seq,ResT,NewSize,WF) :-
954		try_expand_custom_set_wf(Seq,ESeq,append_sequence2,WF),
955		equal_sequence(ESeq,ResT,WF),
956		size_of_sequence(ESeq,Size,WF),
957		int_plus(Size,int(1),int(NewSize)).
958
959		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:prefix_sequence([(int(1),int(22))],int(1),[(int(1),int(22))]))).
960		:- assert_must_succeed(exhaustive_kernel_succeed_check(bsets_clp:prefix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(2),[(int(1),int(11)),(int(2),int(22))]))).
961		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:prefix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(3),[(int(1),int(11)),(int(2),int(22))]))).
962		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(1),X),X = [(int(1),int(1))])).
963		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(0),[]),X = [(int(1),int(1))])).
964		:- assert_must_abort_wf((bsets_clp:prefix_sequence_wf(X,int(-1),_R,WF),X = [(int(1),int(1))]),WF).
965		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(2),Y),
966		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
967		kernel_objects:equal_object(Y,[(int(1),int(1)),(int(2),int(3))]) )).
968		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(1),Y),
969		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
970		kernel_objects:equal_object(Y,[(int(1),int(1))]) )).
971		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(3),Y),
972		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
973		kernel_objects:equal_object(Y,X) )).
974
975		prefix_sequence(Seq,N,R) :- init_wait_flags(WF,[prefix_sequence]),
976	?	prefix_sequence_wf(Seq,N,R,WF),
977	?	ground_wait_flags(WF).
978
979		% Prefix of a sequence (s /|\ n)
980		prefix_sequence_wf(Seq,int(Num),Res,WF) :-
981	?	prefix_sequence1(Seq,Num,Res,WF),
982	?	propagate_size(Res,Num,WF).
983
984		% the size of the result of (s /|\ n) is the number n
985		:- block propagate_size(-,-,?).
986		propagate_size(Res,Num,WF) :-
987		var(Res),!,
988		(Num<0 -> preferences:preference(disprover_mode,false) % don't do anything; we may want to generate WD error
989	?	; Num < 4 -> size_of_sequence(Res,int(Num),WF)
990		; Prio is 1+Num // 100,
991		get_wait_flag(Prio,propagate_size,WF,LWF), % avoid setting up very large skeletons too early
992		block_size_of_sequence(LWF,Res,int(Num),WF)
993		).
994		propagate_size(_,Num,_) :- number(Num), !. % no need to propagate
995		propagate_size(_,_Num,_) :- \+ preferences:preference(find_abort_values,false),
996		!. % do not propagate as we could prevent detection of WD errors below
997		propagate_size([],Num,_WF) :- !,
998		Num=0. % Note: this could prevent a wd-error being detected
999		propagate_size(avl_set(A),Num,WF) :- var(Num),
1000		% with partially instantated sets we get slowdowns (SimpleCSGGrammar2_SlowCLPFD)
1001		% TO DO: treat list skeletons
1002		!,
1003	?	size_of_sequence(avl_set(A),int(Num),WF). % Note: this could prevent a wd-error being detected
1004		propagate_size(_,_,_). % should we also propagate the other way around ?
1005
1006		:- block block_size_of_sequence(-,?,?,?).
1007		block_size_of_sequence(_,Seq,Size,WF) :- size_of_sequence(Seq,Size,WF).
1008
1009		:- block prefix_sequence1(-,?,?,?), prefix_sequence1(?,-,?,?).
1010		prefix_sequence1(_Seq,Num,Res,WF) :- Num==0,!, empty_set_wf(Res,WF).
1011		prefix_sequence1(_Seq,Num,_Res,WF) :- Num<0,!, % according to version 1.8.8 of Atelier-B manual Num must be in 0..size(_Seq)
1012		add_wd_error('negative index in prefix_sequence (/|\\)! ', Num,WF).
1013		prefix_sequence1(Seq,Num,Res,WF) :-
1014		is_custom_explicit_set(Seq,prefix),
1015		prefix_of_custom_explicit_set(Seq,Num,ERes,WF),!, % TO DO: check Num <= size(Seq)
1016		equal_object_wf(Res,ERes,prefix_sequence1,WF).
1017		prefix_sequence1(Seq,Num,Res,WF) :-
1018		expand_custom_set_to_list_wf(Seq,ESeq,_,prefix_sequence1,WF),
1019		unify_same_index_elements(Res,ESeq,WF),
1020		unify_same_index_elements(Seq,Res,WF),
1021	?	prefix_seq(ESeq,Num,0,Res,WF).
1022		:- block prefix_seq(-,?,?,?,?).
1023		prefix_seq([],Max,Sze,Res,WF) :-
1024		(less_than_direct(Sze,Max)
1025		-> add_wd_error('index larger than size of sequence in prefix_sequence (/|\\)! ', (Max,Sze),WF)
1026		; true),
1027		empty_set_wf(Res,WF).
1028		%(less_than(int(_Sze),int(_Max))
1029		% -> (print_message('Index bigger than sequence size in prefix_sequence (/|\\) !'),
1030		% print_message(Max))
1031		% /* in the AtelierB book this is allowed, in Wordsworth + AMN on web it is not ?? */
1032		% ; true).
1033		prefix_seq([(int(N),El)|Tail],Max,SizeSoFar,Res,WF) :-
1034	?	prefix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF).
1035		:- block prefix_seq2(-,?,?,?,?,?,?).
1036		prefix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF) :- % SizeSoFar is always ground
1037	?	(less_than_equal_direct(N,Max), equal_cons_wf(Res,(int(N),El),PTail,WF)
1038		;
1039		less_than_direct(Max,N), equal_object_wf(Res,PTail,prefix_seq2,WF)
1040		),
1041		( SizeSoFar<N -> NewSizeSoFar=N ; NewSizeSoFar = SizeSoFar ),
1042	?	prefix_seq(Tail,Max,NewSizeSoFar,PTail,WF).
1043
1044		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:suffix_sequence([(int(1),int(22))],int(0),[(int(1),int(22))],WF),ground_det_wait_flag(WF))).
1045		:- assert_must_succeed(exhaustive_kernel_succeed_check(bsets_clp:suffix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(1),[(int(1),int(22)),(int(2),int(33))],_WF))).
1046		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:suffix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(2),[(int(1),int(22)),(int(2),int(33))],_WF))).
1047		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(0),X,_WF),X = [(int(1),int(1))])).
1048		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(1),[],_WF),X = [(int(1),int(1))])).
1049		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(2),Y,_WF),
1050		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1051		kernel_objects:equal_object(Y,[(int(1),int(4))]) )).
1052		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(1),Y,_WF),
1053		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1054		kernel_objects:equal_object(Y,[(int(1),int(3)),(int(2),int(4))]) )).
1055		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(2),Y,_WF),
1056		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1057		kernel_objects:equal_object(Y,[(int(1),int(4))]) )).
1058		:- assert_must_abort_wf(bsets_clp:suffix_sequence([(int(1),int(11)),(int(2),int(22))],int(-1),_R,WF),WF).
1059		:- assert_must_abort_wf(bsets_clp:suffix_sequence([(int(1),int(11)),(int(2),int(22))],int(3),_R,WF),WF).
1060
1061		% kernel_waitflags:assert_must_abort2_wf(bsets_clp:suffix_sequence([int(11),int(22)],int(-1),_R,WF),WF)
1062
1063		% suffix of a sequence (s \|/ n); also called restrict at tail (Atelier B) or Drop
1064		:- block suffix_sequence(-,?,?,?).
1065		suffix_sequence(Seq,int(Num),Res,WF) :-
1066	?	suffix_sequence1(Seq,Num,Res,WF).
1067		:- block suffix_sequence1(?,-,?,?).
1068		suffix_sequence1(Seq,Num,Res,WF) :- Num==0, !, equal_object_wf(Res,Seq,suffix_sequence1_1,WF).
1069		suffix_sequence1(_Seq,Num,_Res,WF) :- Num<0, !, add_wd_error('negative index in suffix_sequence (\\|/)! ', Num,WF).
1070		suffix_sequence1(Seq,Num,Res,WF) :- is_custom_explicit_set(Seq,suffix),
1071		suffix_of_custom_explicit_set(Seq,Num,ERes,WF),!,
1072		equal_object_wf(Res,ERes,suffix_sequence1_2,WF).
1073		suffix_sequence1(Seq,Num,Res,WF) :-
1074	?	expand_custom_set_to_list_wf(Seq,ESeq,_,suffix_sequence,WF), suffix_seq(ESeq,Num,0,Res,WF).
1075		:- block suffix_seq(-,?,?,?,?).
1076		suffix_seq([],Max,Sze,Res,WF) :-
1077		(less_than_direct(Sze,Max)
1078		-> add_wd_error('index larger than size of sequence in suffix_sequence (\\|/)! ', '>'(Max,Sze),WF)
1079		; true), empty_set_wf(Res,WF).
1080		suffix_seq([(int(N),El)|Tail],Max,SizeSoFar,Res,WF) :-
1081	?	suffix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF).
1082		:- block suffix_seq2(-,?,?,?,?,?,?).
1083		suffix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF) :-
1084		(less_than_equal_direct(N,Max), equal_object_wf(Res,PTail,suffix_seq2,WF)
1085		;
1086		less_than_direct(Max,N),int_minus(int(N),int(Max),int(NN)),
1087		equal_cons_wf(Res,(int(NN),El),PTail,WF)
1088		),
1089		(N>SizeSoFar -> (NewSizeSoFar=N)
1090		; (NewSizeSoFar = SizeSoFar)),
1091	?	suffix_seq(Tail,Max,NewSizeSoFar,PTail,WF).
1092
1093
1094
1095		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:concat_sequence([],[(int(1),int(33))],[(int(1),int(33))],WF),WF)).
1096		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:concat_sequence([(int(1),int(22)),(int(2),int(33))],[(int(1),int(33)),(int(2),int(44))],[(int(2),int(33)),(int(3),int(33)),(int(1),int(22)),(int(4),int(44))],WF),WF)). % not wfdet because of pending label_el_nr from clpfd_lists
1097		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:concat_sequence([(int(1),int(22))],[(int(1),int(33))],[(int(2),int(33)),(int(1),int(22))],WF),WF)).
1098		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:concat_sequence([(int(1),int(22))],[(int(1),int(33))],[(int(2),int(22)),(int(1),int(33))],WF),WF)).
1099		:- assert_must_succeed((bsets_clp:concat_sequence([],X,Y,_WF),
1100		X = [(int(1),int(1))], Y==X)).
1101		:- assert_must_succeed((bsets_clp:concat_sequence(X,[],Y,_WF), X = [(int(1),int(1))], Y==X)).
1102		:- assert_must_succeed((bsets_clp:concat_sequence([(int(1),int(1))],[],Y,_WF), Y==[(int(1),int(1))])).
1103		:- assert_must_succeed((bsets_clp:concat_sequence(X,X,Y,_WF),
1104		X = [(int(1),int(1))], kernel_objects:equal_object(Y,[(int(1),int(1)),(int(2),int(1))]))).
1105		:- assert_must_succeed((bsets_clp:concat_sequence(X,X,Y,_WF),
1106		X = [(int(2),int(88)),(int(1),int(77))],
1107		kernel_objects:equal_object(Y,[(int(1),int(77)),(int(2),int(88)),(int(3),int(77)),(int(4),int(88))]))).
1108
1109		:- block /* concat_sequence(-,-,?,?), */
1110		concat_sequence(?,-,-,?), concat_sequence(-,?,-,?).
1111		concat_sequence(S1,S2,Res,WF) :- Res==[],!, empty_set_wf(S1,WF), empty_set_wf(S2,WF).
1112		concat_sequence(S1,S2,Res,WF) :-
1113		(var(S1),var(S2) -> get_wait_flag(2,concat,WF,LWF) % we have at least two solutions; TODO maybe use cardinality as wait_flag?
1114		; LWF=1),
1115	?	concat_sequence2(LWF,S1,S2,Res,WF).
1116
1117		:- block concat_sequence2(-,?,-,?,?), concat_sequence2(-,-,?,?,?).
1118	?	concat_sequence2(_,S1,S2,Res,WF) :- S1==[],!,equal_sequence(S2,Res,WF).
1119		concat_sequence2(_,S1,S2,Res,WF) :- S2==[],!,equal_sequence(S1,Res,WF).
1120		concat_sequence2(LWF,S1,S2,Res,WF) :-
1121		try_expand_and_convert_to_avl_with_check(S1,AS1,concat1),
1122		try_expand_and_convert_to_avl_with_check(S2,AS2,concat2),
1123	?	concat_sequence3(LWF,AS1,AS2,Res,WF).
1124
1125		concat_sequence3(_,S1,S2,Res,WF) :- nonvar(S1),is_custom_explicit_set(S1,concat_sequence),
1126		concat_custom_explicit_set(S1,S2,ERes,WF),!,
1127		equal_sequence(Res,ERes,WF).
1128		concat_sequence3(_LWF,S1,S2,Res,WF) :-
1129		%try_expand_custom_set_wf(S1,ES1,concat,WF),
1130		size_of_sequence(S1,int(Size1),WF),
1131		(number(Size1) -> true
1132		; size_of_sequence(S2,Size2,WF),
1133		size_of_sequence(Res,SizeRes,WF),
1134	?	int_minus(SizeRes,Size2,int(Size1)),
1135	?	in_nat_range_wf(int(Size1),int(0),SizeRes,WF)
1136		% is this required: ?? ,in_nat_range_wf(Size2,int(0),SizeRes,WF)
1137		),
1138	?	concat_sequence_aux(Size1,S1,S2,Res,WF).
1139
1140
1141		:- assert_must_succeed( (bsets_clp:equal_sequence([(int(1),A)|T1],[(int(1),int(22))|T2],_WF),
1142		A==int(22),T2=[],T1==[] )) .
1143		:- assert_must_succeed( (bsets_clp:equal_sequence([(int(1),A)|T],avl_set(node((int(2),string(a)),true,0,node((int(1),string(c)),true,0,empty,empty),node((int(3),string(b)),true,0,empty,empty))),_WF),
1144		check_eqeq(A,string(c)),
1145		kernel_objects:equal_object(T,[(int(2),B)|T2]), check_eqeq(B,string(a)),
1146		kernel_objects:equal_object(T2,[(int(3),C)]), check_eqeq(C,string(b))) ).
1147		% equal_object optimized for sequences; can infer that pairs with same index have same value
1148		% TO DO: complete and make more efficient
1149		%equal_sequence(X,Y,_WF) :- nonvar(X),nonvar(Y),
1150		% is_custom_explicit_set(X,eval_sequence), is_custom_explicit_set(Y,eval_sequence),!,
1151		% equal_explicit_sets(X,Y).
1152		equal_sequence(X,Y,WF) :- nonvar(X),nonvar(Y),
1153		get_seq_head(X,XI,XEl,XT), get_seq_head(Y,YI,YEl,YT), XI==YI,!,
1154		% THIS CURRENTLY ONLY CHECKS FRONTMOST indexes
1155		equal_object_wf(XEl,YEl,equal_sequence_1,WF),
1156		equal_sequence(XT,YT,WF).
1157		equal_sequence(X,Y,WF) :-
1158		/* (is_custom_explicit_set(Y) -> monitor_equal_sequence_againts_custom_set(X,Y,WF)
1159		; is_custom_explicit_set(X) -> monitor_equal_sequence_againts_custom_set(Y,X,WF)
1160		; true), does not seem to buy anything; equal_object already powerful enough */
1161	?	equal_object_wf(X,Y,equal_sequence_2,WF).
1162
1163		% enforces the constraint that there is only one possible elemenent per index:
1164		%:- block monitor_equal_sequence_againts_custom_set(-,?,?).
1165		%monitor_equal_sequence_againts_custom_set([],_,_) :- !.
1166		%monitor_equal_sequence_againts_custom_set([El|T],CS,WF) :- !,
1167		% element_of_custom_set_wf(El,CS,WF),
1168		% monitor_equal_sequence_againts_custom_set(T,CS,WF).
1169		%monitor_equal_sequence_againts_custom_set(_,_,_).
1170
1171		get_seq_head([(Idx,El)|Tail],Idx,El,Tail).
1172		%get_seq_head(avl_set(AVL),Idx,El,Tail) :- does not seem to buy anything; equal_object already powerful enough
1173		% custom_explicit_sets:avl_min_pair(AVL,Idx,El),
1174		% custom_explicit_sets:direct_remove_element_from_avl(AVL,(Idx,El),Tail). % TO DO: only compute if indexes ==
1175
1176
1177		:- block concat_sequence_aux(-,?,?,?,?).
1178		concat_sequence_aux(Size1,_S1,_S2,Res,WF) :- nonvar(Res),Res=avl_set(_),
1179		size_of_custom_explicit_set(Res,int(RSize),WF), number(RSize),
1180		Size1 > RSize,!, % S1 is longer than Res; no solution (prevent WD error below)
1181		fail.
1182		concat_sequence_aux(Size1,S1,S2,Res,WF) :- nonvar(Res),Res=avl_set(_),
1183		% split Result into prefix and suffix
1184		prefix_of_custom_explicit_set(Res,Size1,Prefix,WF), % we could call versions which do not check WD
1185		suffix_of_custom_explicit_set(Res,Size1,Postfix,WF),
1186		!,
1187		equal_sequence(S1,Prefix,WF), equal_sequence(S2,Postfix,WF).
1188		concat_sequence_aux(Size1,S1,S2,Res,WF) :-
1189		shift_seq_indexes(S2,Size1,NewS2,WF),
1190		% We can do something stronger than disjoint union: we know that the indexes are disjoint!
1191		% Hence: if (int(X),Y) : Res & (int(X),Z) : S1 => Y=Z
1192		% Hence: if (int(X),Y) : Res & (int(X),Z) : S2 => Y=Z
1193		unify_same_index_elements(S1,Res,WF),
1194		unify_same_index_elements(Res,S1,WF),
1195		unify_same_index_elements(NewS2,Res,WF),
1196		unify_same_index_elements(Res,NewS2,WF),
1197	?	disjoint_union_wf(S1,NewS2,Res,WF).
1198
1199		% Check if (int(X),Y) pairs in Seq2 have a matching (int(X),Z) in Seq1 and then unify(Y,Z)
1200		:- block unify_same_index_elements(-,?,?).
1201		unify_same_index_elements(avl_set(A),Seq,WF) :- !,
1202		unify_same_index_elements_aux(Seq,A,WF).
1203		unify_same_index_elements(_,_,_). % TO DO: maybe also support other partially instantiated lists
1204
1205		:- block unify_same_index_elements_aux(-,?,?).
1206		unify_same_index_elements_aux([],_,_) :- !.
1207		unify_same_index_elements_aux([(int(Idx),El)|T],A,WF) :- !,
1208		try_find_index_element(Idx,El,A,WF),
1209		unify_same_index_elements_aux(T,A,WF).
1210		unify_same_index_elements_aux(_,_,_).
1211
1212		:- block try_find_index_element(-,?,?,?).
1213		try_find_index_element(Idx,El,AVL,WF) :-
1214	?	avl_fetch_pair(int(Idx),AVL,AvlEl),
1215		!,
1216		% We have found an entry with the same index: El and AvlEl must be identical:
1217		equal_object_wf(El,AvlEl,try_find_index_element,WF).
1218		try_find_index_element(_Idx,_El,_AVL,_WF). % :- print(not_found(_Idx,_AVL)),nl.
1219
1220		% TO DO: add waitflags + use within partition_wf
1221		% computes union of two sets which are guaranteed to be disjoint: means that if two of three sets known the other one can be determined
1222
1223		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([int(3)],[int(2),int(1)],[int(1),int(3),int(2)],WF),WF)).
1224		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([],[int(2),int(1)],[int(1),int(2)],WF),WF)).
1225		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([int(1),int(2)],[],[int(2),int(1)],WF),WF)).
1226		:- assert_must_succeed((bsets_clp:disjoint_union_wf([int(1)],[int(2)],Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]))).
1227		:- assert_must_succeed((bsets_clp:disjoint_union_wf(A,B,[int(1)],_WF),B=[H],H==int(1),A==[])).
1228
1229		% a union where we know that Set1 and Set2 are disjoint
1230		% this means we can propagate elements of Set1/2 more easily to result
1231		disjoint_union_wf(Set1,Set2,Res,WF) :-
1232		(var(Res)
1233		-> disjoint_union_wf0(Set1,Set2,DRes,DRes,WF),
1234		equal_object_optimized(Res,DRes) % try and convert result to AVL
1235	?	; disjoint_union_wf0(Set1,Set2,Res,Res,WF)).
1236
1237		% disjoint_union_wf0(Set1,Set2,UnionOfSet1Set2, SuperSet, WF)
1238		:- block disjoint_union_wf0(-,-,-,?,?).
1239		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Set1==[],!,equal_object_wf(Set2,Res,disjoint_union_wf0_1,WF).
1240		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Set2==[],!,equal_object_wf(Set1,Res,disjoint_union_wf0_2,WF).
1241		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Res==[],!,empty_set_wf(Set1,WF), empty_set_wf(Set2,WF).
1242		disjoint_union_wf0(Set1,Set2,Res,FullRes,WF) :-
1243		((nonvar(Set1);nonvar(Set2)) -> true ; get_cardinality_powset_wait_flag(Res,disjoint_union_wf0,WF,_Card,CWF)),
1244	?	disjoint_union0(Set1,Set2,Res,FullRes,WF,CWF).
1245
1246		:- block disjoint_union0(-,-,?,?,?,-), disjoint_union0(-,?,-,-,?,?), disjoint_union0(?,-,-,-,?,?).
1247		disjoint_union0(Set1,Set2,Res,_,WF,_) :- Set1==[],!,equal_object_wf(Set2,Res,disjoint_union0_1,WF).
1248		disjoint_union0(Set1,Set2,Res,_,WF,_) :- Set2==[],!,equal_object_wf(Set1,Res,disjoint_union0_2,WF).
1249		disjoint_union0(S1,S2,Res,_F,WF,_CWF) :-
1250		ground_value(Res),
1251		( ground_value(S1) -> !,
1252		check_subset_of_wf(S1,Res,WF), % TO DO: check if we can merge the check_subset and difference set in one predicate
1253		difference_set_wf(Res,S1,S2,WF)
1254		; ground_value(S2) -> !,
1255		check_subset_of_wf(S2,Res,WF),
1256	?	difference_set_wf(Res,S2,S1,WF)
1257		; var(S1),var(S2) -> !, % CWF nonvar
1258		% see test 1408; allows to generate subsets of Res and avoid enumeration warnings
1259		check_subset_of_wf(S1,Res,WF),
1260		%check_subset_of(S1,Res), % without waitflag: will generate ground version
1261		difference_set_wf(Res,S1,S2,WF)
1262		).
1263		disjoint_union0(Set1,Set2,Res,_,WF,_) :- nonvar(Set1),
1264		is_custom_explicit_set_nonvar(Set1),
1265		union_of_explicit_set(Set1,Set2,Union), !, % TODO: if it fails: copy/propagate values to result?
1266	?	equal_object_wf(Union,Res,disjoint_union0_3,WF).
1267		disjoint_union0(Set1,Set2,Res,Full,WF,_) :-
1268		expand_custom_set_to_list_no_dups_wf(Set1,ESet1,_,disjoint_union0_1,WF),
1269		expand_custom_set_to_list_no_dups_wf(Set2,ESet2,_,disjoint_union0_2,WF),
1270	?	disj_union1(ESet1,ESet2,Res,Full,WF).
1271
1272		:- block disj_union1(-,-,?,?,?).
1273		disj_union1(X,Y,Res,FullRes,WF) :-
1274	?	var(X) -> disj_union2(Y,X,Res,FullRes,WF) ; disj_union2(X,Y,Res,FullRes,WF).
1275
1276		disj_union2([],Y,Res,_,_WF) :- equal_object_optimized(Y,Res,disj_union2).
1277		disj_union2([X|TX],Y,Res,FullRes,WF) :-
1278	?	remove_element_wf(X,Res,TR,WF), % was: equal_cons_wf(Res,X,TR,WF) but error was that it could force X to be a certain value
1279		ground_value_check(X,XV),
1280		(nonvar(XV) -> equal_cons_wf(Res,X,TR,WF)
1281		; check_element_of_wf(X,FullRes,WF), % ensure that we set up proper constraints for X; e.g., for x \/ y = 1..10 & x /\ y = {}
1282		when(nonvar(XV), equal_cons_wf(Res,X,TR,WF))
1283		), % ensure that we instantiate Res if TR known; otherwise we may get pending co-routines, e.g. test 506, SyracuseGrammar
1284		disj_union3(TX,Y,TR,FullRes,WF).
1285
1286		:- block disj_union3(-,-,-,?,?).
1287		disj_union3(X,Y,Res,_,WF) :- Res==[],!,empty_set_wf(X,WF),empty_set_wf(Y,WF).
1288		disj_union3(X,Y,Res,FullRes,WF) :- disj_union1(X,Y,Res,FullRes,WF).
1289
1290
1291		:- block disjoint_union_generalized_wf(-,?,?).
1292		%disjoint_union_generalized_wf([Set1|T],Res,_WF) :- T==[],!, % just one set; probably not covered at the moment (ast_cleanup simplifies partition with single set
1293		% equal_object(Set1,Res).
1294		disjoint_union_generalized_wf(ListOfSets,Res,WF) :-
1295		%expand_custom_set_to_list_wf(SetsOfSets,ESetsOfSets,_,disjoint_union_generalized_wf,WF), % this is a list of sets
1296		disjoint_union_generalized2(ListOfSets,[],Res,WF).
1297		:- block disjoint_union_generalized2(-,?,?,?).
1298		disjoint_union_generalized2([],S,Res,WF) :- !, equal_object_optimized_wf(S,Res,disjoint_union_generalized2,WF).
1299		disjoint_union_generalized2([H|T],UnionSoFar,Res,WF) :- !,
1300		disjoint_union_wf0(H,UnionSoFar,UnionSoFar2,Res,WF),
1301		%% print_message(called_disjoint_union(H,UnionSoFar,UnionSoFar2)), %%
1302		disjoint_union_generalized2(T,UnionSoFar2,Res,WF).
1303		disjoint_union_generalized2(L,S,Res,WF) :-
1304		add_internal_error('Not a list: ',disjoint_union_generalized2(L,S,Res,WF)),fail.
1305		% TO DO: if there are more than two sets: it may be interesting to set up constraint that
1306		% each set is a subset of the full set;
1307		% (would avoid enumeration warning in, e.g., x \/ y \/ z = 1..10 & x /\ y = {} & x /\ z = {} & y /\ z = {} & card(x)=card(y)+2 )
1308
1309		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:concatentation_of_sequences([(int(1),[]),(int(3),[(int(1),int(22)),(int(2),int(33))]),(int(2),[(int(1),int(11))])],
1310		[(int(1),int(11)),(int(2),int(22)),(int(3),int(33))],_WF))).
1311		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:concatentation_of_sequences([(int(1),[]),(int(2),[(int(1),int(33))])],[(int(1),int(33))],_WF))).
1312		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:concatentation_of_sequences([(int(1),[]),(int(2),[(int(1),int(55))])],Res,WF),
1313		kernel_waitflags:ground_wait_flags(WF),
1314		kernel_objects:equal_object(Res,[(int(1),int(55))]) )).
1315		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:concatentation_of_sequences([(int(1),[(int(1),int(22))]),(int(2),[(int(1),int(55))])],Res,WF),
1316		kernel_waitflags:ground_wait_flags(WF),
1317		kernel_objects:equal_object(Res,[(int(1),int(22)),(int(2),int(55))]) )).
1318		:- block concatentation_of_sequences(-,?,?).
1319		concatentation_of_sequences(SeqOfSeq,Res,WF) :-
1320		try_expand_and_convert_to_avl_with_check(SeqOfSeq,ES,conc),
1321	?	concs2(ES,Res,WF).
1322
1323		concs2(SeqOfSeq,Res,WF) :- is_custom_explicit_set(SeqOfSeq,conc),
1324		conc_custom_explicit_set(SeqOfSeq,CRes),!,
1325		equal_object_wf(CRes,Res,concs2,WF).
1326		concs2(SeqOfSeq,Res,WF) :- empty_set_wf(SeqOfSeq,WF),empty_set_wf(Res,WF).
1327		concs2(SeqOfSeq,Res,WF) :- not_empty_set_wf(SeqOfSeq,WF),
1328		front_sequence(SeqOfSeq,Front,WF),
1329	?	concatentation_of_sequences(Front,FrontRes,WF),
1330	?	last_sequence(SeqOfSeq,Last,WF),
1331	?	concat_sequence(FrontRes,Last,Res,WF).
1332
1333		:- assert_must_abort_wf(bsets_clp:tail_sequence([],_R,unknown,WF),WF).
1334		:- assert_must_abort_wf(bsets_clp:tail_sequence([],[],unknown,WF),WF).
1335		:- assert_must_succeed(exhaustive_kernel_succeed_check(
1336		bsets_clp:tail_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(5))],unknown,_WF)) ).
1337		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:tail_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],
1338		[(int(1),int(5)),(int(2),int(6))],unknown,_WF)) ).
1339		:- assert_must_succeed((bsets_clp:tail_sequence(X,R,unknown,_),
1340		X = [(int(1),int(6)),(int(2),int(5))],
1341		kernel_objects:equal_object(R,[(int(1),int(5))]) )).
1342		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(1),int(5))],unknown,_),
1343		X = [(int(1),int(6)),(int(2),int(5))] )).
1344		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(1),int(5)),(int(2),int(7))],unknown,_),
1345		X = [(int(1),int(6)),(int(2),int(5)),(int(3),int(7))] )).
1346		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(2),int(7)),(int(1),int(5))],unknown,_),
1347		X = [(int(1),int(6)),(int(2),int(5)),(int(3),int(7))] )).
1348		:- block tail_sequence(-,?,?,?).
1349		tail_sequence(S1,Res,Span,WF) :- is_custom_explicit_set(S1,tail_sequence),
1350		tail_sequence_custom_explicit_set(S1,_,TRes,Span,WF),!,
1351		equal_object_wf(TRes,Res,tail_sequence,WF).
1352		tail_sequence(S1,Res,Span,WF) :- expand_custom_set_to_list_wf(S1,ES1,_,tail_sequence,WF),
1353		tail2(ES1,Res,Span,WF).
1354
1355		tail2([],_,Span,WF) :-
1356		add_wd_error_span('tail applied to empty sequence!',[],Span,WF).
1357		tail2([H|T],Res,_Span,WF) :- domain_subtraction_wf([int(1)],[H|T],IntRes,WF),
1358		shift_seq_indexes(IntRes,-1,Res,WF).
1359
1360
1361		:- assert_must_abort_wf(bsets_clp:first_sequence([],_R,unknown,WF),WF).
1362		:- assert_must_abort_wf(bsets_clp:first_sequence([],int(1),unknown,WF),WF).
1363		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:first_sequence([(int(1),int(4)),(int(2),int(5))],int(4),unknown,_WF)) ).
1364		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:first_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],int(4),unknown,_WF)) ).
1365		:- assert_must_succeed((bsets_clp:first_sequence(X,R,unknown,_WF),
1366		X = [(int(1),int(2)),(int(2),int(1))],
1367		R = int(2))).
1368
1369		:- block first_sequence(-,?,?,?).
1370		first_sequence([],_,Span,WF) :- !,add_wd_error_span('first applied to empty sequence!',[],Span,WF).
1371		first_sequence(Seq,Res,Span,WF) :- apply_to(Seq,int(1),Res,Span,WF).
1372
1373
1374
1375		:- assert_must_abort_wf(bsets_clp:front_sequence([],_R,WF),WF).
1376		:- assert_must_abort_wf(bsets_clp:front_sequence([],[],WF),WF).
1377		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:front_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(4))],_WF)) ).
1378		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:front_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],[(int(1),int(4)),(int(2),int(5))],_WF)) ).
1379		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:front_sequence(X,R,WF),
1380		X = [(int(1),int(2)),(int(2),int(55))],kernel_waitflags:ground_wait_flags(WF),
1381		kernel_objects:equal_object(R,[(int(1),int(2))]))).
1382		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:front_sequence(X,R,WF),
1383		X = [(int(3),int(33))|R], kernel_waitflags:ground_wait_flags(WF),
1384		kernel_objects:equal_object(R,[(int(1),int(2)),(int(2),int(55))]) )).
1385
1386	?	front_sequence(Seq,Res,WF) :- front_sequence(Seq,Res,unknown,WF).
1387		:- block front_sequence(-,?,?,?).
1388		front_sequence(S1,Res,_Span,WF) :-
1389		is_custom_explicit_set(S1,front_sequence),
1390		front_sequence_custom_explicit_set(S1,_,FRes),!,
1391		equal_object_wf(FRes,Res,front_sequence,WF).
1392		front_sequence(Seq,Res,Span,WF) :- expand_custom_set_to_list_wf(Seq,ESeq,_,front_sequence,WF),
1393	?	front2(ESeq,Res,Span,WF).
1394		front2([],_,Span,WF) :- add_wd_error_span('front applied to empty sequence!',[],Span,WF).
1395		front2([H|T],Res,_Span,WF) :- size_of_sequence([H|T],int(Size),WF),
1396	?	(number(Size) -> true ; size_of_sequence(Res,SizeRes,WF), int_plus(SizeRes,int(1),int(Size))),
1397	?	when(ground(Size), domain_subtraction_wf([int(Size)],[H|T],Res,WF)).
1398
1399
1400		:- assert_must_abort_wf(bsets_clp:last_sequence([],_R,WF),WF).
1401		:- assert_must_abort_wf(bsets_clp:last_sequence([],int(1),WF),WF).
1402		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:last_sequence([(int(1),int(4)),(int(2),int(5))],int(5),_WF)) ).
1403		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:last_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],int(6),_WF)) ).
1404		:- assert_must_succeed((bsets_clp:last_sequence(X,R,_WF),
1405		X = [(int(1),int(2)),(int(2),int(55))],R = int(55))).
1406		:- assert_must_succeed((bsets_clp:last_sequence(X,R,_WF), X = [(int(1),int(55))], R = int(55))).
1407		:- assert_must_succeed((bsets_clp:last_sequence([(int(1),[(int(1),int(22))]),(int(2),[(int(1),int(55))])],R,_WF), R == [(int(1),int(55))])).
1408
1409	?	last_sequence(Seq,Res,WF) :- last_sequence(Seq,Res,unknown,WF).
1410		:- block last_sequence(-,?,?,?).
1411		last_sequence(Seq,Res,_Span,WF) :-
1412		is_custom_explicit_set(Seq,last_sequence),
1413		last_sequence_explicit_set(Seq,Last), !,
1414		equal_object_wf(Last,Res,last_sequence,WF).
1415		last_sequence([],_,Span,WF) :- !,add_wd_error_span('last applied to empty sequence!',[],Span,WF).
1416		last_sequence(Seq,Res,Span,WF) :-
1417		size_of_sequence(Seq,int(Size),WF),
1418	?	last_sequence_aux(Size,Seq,Res,Span,WF).
1419		:- block last_sequence_aux(-,?,?,?,?).
1420		last_sequence_aux(Size,Seq,Res,Span,WF) :-
1421	?	apply_to(Seq,int(Size),Res,Span,WF).
1422
1423
1424		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(5)),(int(2),int(4))],WF),WF )).
1425		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4))],[(int(1),int(4))],WF),WF )).
1426		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([],[],WF),WF )).
1427		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],[(int(1),int(6)),(int(3),int(4)),(int(2),int(5))],WF),WF )).
1428		:- assert_must_succeed((bsets_clp:rev_sequence([],[],_WF))).
1429		:- assert_must_succeed((bsets_clp:rev_sequence(X,R,_WF),
1430		X = [(int(1),int(2)),(int(2),int(1))],
1431		kernel_objects:equal_object(R,[(int(2),int(2)),(int(1),int(1))]) )).
1432		:- assert_must_succeed((bsets_clp:rev_sequence(X,R,_WF),
1433		X = [(int(1),int(23)),(int(2),int(1)),(int(3),int(55))],
1434		kernel_objects:equal_object(R,[(int(3),int(23)),(int(2),int(1)),(int(1),int(55))]) )).
1435		:- assert_must_succeed((bsets_clp:rev_sequence(R,X,_WF),
1436		X = [(int(1),int(23)),(int(2),int(1)),(int(3),int(55))],
1437		kernel_objects:equal_object(R,[(int(3),int(23)),(int(2),int(1)),(int(1),int(55))]) )).
1438		:- assert_must_succeed((bsets_clp:rev_sequence(X,_R,_WF),
1439		X = [(int(2),int(1)),(int(1),int(23)),(int(3),int(55))] )).
1440		:- assert_must_succeed((bsets_clp:rev_sequence(_R,X,_WF),
1441		X = [(int(3),int(55)),(int(1),int(23)),(int(2),int(1))] )).
1442
1443		/* reverse of sequence */
1444		:- block rev_sequence(-,-,?).
1445		rev_sequence(S1,Res,WF) :-
1446	?	(nonvar(S1) -> rev_sequence2(S1,Res,WF)
1447		; rev_sequence2(Res,S1,WF)).
1448
1449		rev_sequence2(S1,Res,WF) :- reverse_custom_explicit_set(S1,RS1),!,
1450		equal_object_wf(Res,RS1,WF).
1451		rev_sequence2(S1,Res,WF) :-
1452		expand_custom_set_to_list_wf(S1,ES1,_,rev_sequence2,WF),
1453		size_of_sequence(ES1,int(Size1),WF),
1454		% TO DO: we could also try and get the size from the result Res
1455	?	rev_sequence3(ES1,Size1,Res,WF).
1456
1457		:- block rev_sequence3(?,-,-,?).
1458		rev_sequence3(E,_Size,Res,WF) :- nonvar(Res), reverse_custom_explicit_set(Res,RevRes),!,
1459		equal_object_wf(E,RevRes,WF).
1460		rev_sequence3(E,Size,Res,WF) :- var(Size), !,
1461		% try to obtain size from result as well
1462	?	size_of_sequence(Res,int(Size),WF), rev_sequence3b(E,Size,Res,WF).
1463		rev_sequence3(E,S,R,WF) :- rev_sequence4(E,S,R,WF).
1464
1465		:- block rev_sequence3b(?,-,?,?).
1466		rev_sequence3b(E,S,R,WF) :- rev_sequence4(E,S,R,WF).
1467
1468		:- block rev_sequence4(-,?,?,?).
1469		rev_sequence4([],_,Res,WF) :- empty_set_wf(Res,WF).
1470		rev_sequence4([(int(N),El)|Tail],Size1,Res,WF) :-
1471		equal_cons_wf(Res,(NewN,El),RTail,WF),
1472		% compute NewN = Size - (N-1)
1473		int_minus(int(N),int(1),N1),
1474		int_minus(int(Size1),N1,NewN),
1475		(ground(NewN) -> true ; in_nat_range(NewN,int(0),int(Size1))),
1476		rev_sequence4(Tail,Size1,RTail,WF).
1477
1478
1479		/* --------- */
1480		/* RELATIONS */
1481		/* --------- */
1482
1483		%maplet(X,Y,(X,Y)).
1484
1485		% relation([]).
1486		% relation([(_X,_Y)|T]) :- relation(T).
1487
1488		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:relation_over_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
1489		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:relation_over([],[int(1),int(2)],[int(2)]) )).
1490		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(int(1),int(2))],[int(1),int(2)],[int(2)]) )).
1491		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([([int(1)],[int(2)])],[[int(1)],[],[int(2)]],[[int(2)]]) )).
1492		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(pred_true /* bool_true */,pred_false /* bool_false */)],[pred_false /* bool_false */,pred_true /* bool_true */],[pred_false /* bool_false */]) )).
1493		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((pred_true /* bool_true */,int(2)),fd(1,'Name'))],[(pred_false /* bool_false */,int(1)),(pred_true /* bool_true */,int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1494		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((rec([field(a,fd(1,'Name'))]),int(2)),fd(1,'Name'))],[(rec([field(a,fd(1,'Name'))]),int(1)),(rec([field(a,fd(1,'Name'))]),int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1495		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((rec([field(a,fd(2,'Name')),field(b,fd(1,'Name'))]),int(2)),fd(1,'Name'))],[(rec([field(a,fd(1,'Name')),field(b,fd(1,'Name'))]),int(1)),(rec([field(a,fd(1,'Name')),field(b,fd(2,'Name'))]),int(2)),(rec([field(a,fd(2,'Name')),field(b,fd(1,'Name'))]),int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1496		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((pred_true /* bool_true */,int(2)),string('STRING1'))],[(pred_false /* bool_false */,int(1)),(pred_true /* bool_true */,int(2))],[string('STRING2'),string('STRING1')]) )).
1497		:- assert_must_succeed(exhaustive_kernel_succeed_check( /* multiple solutions !!*/ bsets_clp:relation_over([(int(1),int(2)),(int(2),int(2))],[int(1),int(2)],[int(2)]) )).
1498		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(int(1),int(2)),(int(1),int(3))],[int(1),int(2)],[int(3),int(2)]) )).
1499		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:relation_over([(int(1),int(2)),(int(2),int(1))],[int(1),int(2)],[int(2)]) )).
1500		:- assert_must_fail(( bsets_clp:relation_over([(int(1),int(1))],[int(1),int(2)],[int(2)]) )).
1501		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(3)]),
1502		X==[(int(1),int(3))] )).
1503		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(3)]),
1504		X==[(int(1),int(3)),(int(2),int(3))] )).
1505		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(4),int(5)]),
1506		X==[(int(2),int(4)),(int(2),int(5))] )).
1507
1508		relation_over(R,Dom,Ran) :- init_wait_flags(WF,[relation_over]),
1509	?	relation_over_wf(R,Dom,Ran,WF),
1510	?	ground_wait_flags(WF).
1511
1512		:- block relation_over_wf(-,-,-,?).
1513		relation_over_wf(R,Dom,Ran,WF) :-
1514		kernel_equality:get_cardinality_relation_over_wait_flag(Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels),
1515	?	relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels).
1516
1517		:- block relation_over1(-,?,?,?,-,?,?).
1518		relation_over1(FF,Domain,Range,WF,_WFR,_MaxCard,_MaxNrOfRels) :-
1519		nonvar(FF),
1520		custom_explicit_sets:is_definitely_maximal_set(Range),
1521		% we do not need the Range; this means we can match more closures (e.g., lambda)
1522		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,_,WF),!,
1523		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF).
1524		relation_over1(FF,Domain,Range,WF,_WFR,_MaxCard,_MaxNrOfRels) :- nonvar(FF),
1525		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,_,WF),!,
1526		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
1527		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
1528		relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels) :- var(R),!,
1529		expand_custom_set_to_list_wf(R,ER,_,relation_over1,WF),
1530	?	relation_over2(ER,[],Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels,none).
1531		relation_over1(R,Domain,Range,WF,_WFR,_MaxCard,_) :-
1532		expand_and_convert_to_avl_set_catch(R,AER,relation_over1,'ARG : ? <-> ?',ResultStatus,WF),!,
1533		(ResultStatus=avl_set
1534	?	-> is_avl_relation_over_domain(AER,Domain,WF),
1535		is_avl_relation_over_range(AER,Range,WF)
1536		; (debug_mode(on) -> add_message_wf(relation_over,'SYMBOLIC <-> check: ',R,unknown,WF) ; true),
1537		symbolic_domain_subset_check(R,Domain,WF),
1538		symbolic_range_subset_check(R,Range,WF)
1539		).
1540		relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels) :-
1541		expand_custom_set_to_list_wf(R,ER,_,relation_over1,WF),
1542	?	relation_over2(ER,[],Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels,none).
1543
1544		% check the domain of a symbolic closure value FF whose domain is FFDomain and expected domain is Domain:
1545		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF) :-
1546		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
1547		[b_operator(domain,[FF]),Domain],unknown),WF2),
1548		check_subset_of_wf(FFDomain,Domain,WF2).
1549		% ditto for range
1550		check_range_subset_for_closure_wf(FF,FFRange,Range,WF) :-
1551		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
1552		[b_operator(range,[FF]),Range],unknown),WF2),
1553		check_subset_of_wf(FFRange,Range,WF2).
1554
1555
1556		% try and expand set to AVL and catch enumeration warning exceptions and set OK result value
1557		% if it succeeds with OK = avl_set -> we have an avl_set
1558		% if it fails: it cannot be expanded at the moment
1559		% if it retuns keep_symbolic: expansion cannot be performed and can never be performed; keep set symbolic
1560		expand_and_convert_to_avl_set_catch(R,_AS,_Origin,_Operator,_ResultStatus,_WF) :- var(R),!,fail.
1561		expand_and_convert_to_avl_set_catch(R,_AS,_Origin,_Operator,ResultStatus,_WF) :-
1562		is_infinite_explicit_set(R),!, % we could also use is_infinite_or_symbolic_closure
1563		ResultStatus=keep_symbolic.
1564		expand_and_convert_to_avl_set_catch(R,AS,Origin,Operator,ResultStatus,WF) :-
1565		catch(
1566		(expand_and_convert_to_avl_set(R,AS,Origin,Operator),ResultStatus=avl_set),
1567		enumeration_warning(_,_,_,_,_),
1568		(add_message_wf(Origin,'Attempting symbolic treatment, enumeration warning occured while expanding ARG for ',
1569		Operator,b(value(R),any,[]),WF),
1570		ResultStatus=keep_symbolic)).
1571
1572		expand_and_convert_to_avl_set_warn(R,_AS,_Origin,_Operator,_WF) :- var(R),!,fail.
1573		expand_and_convert_to_avl_set_warn(R,AS,Origin,Operator,WF) :-
1574		% TO DO: check for not fully instantiated closures, like memoization closures where ID not yet known
1575		% it is used before a cut: we need to expand straightaway without choice points
1576		(is_symbolic_closure(R)
1577		-> add_message_wf(Origin,'Expanding symbolic set argument ARG for predicate ',Operator,b(value(R),any,[]),WF)
1578		; true),
1579		% TODO: instead of observe_enumeration_warnings we could push onto the call-stack and pass WF
1580		observe_enumeration_warnings(expand_and_convert_to_avl_set(R,AS,Origin,Operator),
1581		add_message_wf(Origin,'Enumeration warning occured while expanding argument ARG for predicate ',
1582		Operator,b(value(R),any,[]),WF)).
1583		%expand_and_convert_to_avl_set(R,AS,_,Operator,Values) :-
1584		% observe_enumeration_warnings(expand_and_convert_to_avl_set(R,AS,,),
1585		% display_warning_message(Operator,Values)).
1586		%display_warning_message(Operator,Values) :-
1587		% format(user_error,'Enumeration Warning for Operator ~w~n',[Operator]),
1588		% maplist(translate:print_bvalue,Values),nl.
1589
1590		:- block relation_over2(-,?,?,?,?,-,?,?,?).
1591		relation_over2([],_,_,_,_WF,_WFR,_MaxCard,_MaxNrOfRels,_LastPair).
1592		relation_over2(REL,SoFar,Domain,Range,WF,WFR,MaxCard,MaxNrOfRels,LastPair) :-
1593		(var(REL) -> NewLastPair=(X,Y) ; NewLastPair=none), %remember whether we freely chose X,Y
1594		REL = [(X,Y)|T],
1595		(number(MaxCard)
1596		-> MaxCard>0,C1 is MaxCard-1 ,(C1=0 -> T=[] ; true)
1597		; C1=MaxCard),
1598		% TO DO: try to enumerate elements in order
1599		ordered_pair(LastPair,X,Y,not_equal),
1600	?	check_element_of_wf(X,Domain,WF),
1601	?	check_element_of_wf(Y,Range,WF),
1602	?	not_element_of_wf((X,Y),SoFar,WF),
1603		update_waitflag(MaxNrOfRels,WFR,NewWFR,WF),
1604	?	relation_over2(T,[(X,Y)|SoFar],Domain,Range,WF,NewWFR,C1,MaxNrOfRels,NewLastPair).
1605
1606		% check that new pair is greater than previous pair, if that pair was freely chosen
1607		ordered_pair(none,_,_,_).
1608		ordered_pair((LastX,LastY),NewX,NewY,Eq) :- ordered_value(LastX,NewX,EqualX),
1609		check_second_component(EqualX,LastY,NewY,Eq).
1610
1611		:- block check_second_component(-,?,?,?).
1612		check_second_component(equal,X,Y,EqRes) :- ordered_value(X,Y,EqRes).
1613		check_second_component(not_equal,_X,_Y,not_equal). % no need to check 2nd component
1614
1615		:- block ordered_value(-,?,?), ordered_value(?,-,?).
1616		ordered_value(pred_true /* bool_true */,B,Eq) :- !, (B=pred_true /* bool_true */ -> Eq=equal ; Eq=not_equal).
1617		ordered_value(pred_false /* bool_false */,B,Eq) :- !, B=pred_false /* bool_false */, Eq=equal.
1618		ordered_value(int(X),int(Y),Eq) :- !,
1619		kernel_objects:less_than_equal_direct(X,Y), equal_atomic_term(X,Y,Eq).
1620		ordered_value(fd(NrX,T),fd(NrY,T),Eq) :- !,
1621		kernel_objects:less_than_equal_direct(NrX,NrY),
1622		equal_atomic_term(NrX,NrY,Eq).
1623		ordered_value((X1,X2),(Y1,Y2),Eq) :- !, ordered_pair((X1,X2),Y1,Y2,Eq).
1624		ordered_value(string(X),string(Y),Eq) :- !, less_equal_atomic_term(X,Y,Eq).
1625		ordered_value(rec(FX),rec(FY),Eq) :- !,
1626		ordered_fields(FX,FY,Eq).
1627		ordered_value([],Y,Eq) :- !, (Y==[] -> Eq=equal ; Eq=not_equal). % empty set is the smallest set
1628		ordered_value(avl_set(A),Y,Eq) :- !,
1629		(Y==[] -> fail
1630		; Y=avl_set(B) -> (A @< B -> Eq=not_equal ; A@>B -> fail ; Eq=equal)
1631		; print(assuming_strictly_ordered(avl_set(A),Y)),nl,
1632		Eq=not_equal). % TO DO: treat sets better
1633		ordered_value([H|T],Y,Eq) :- !, ordered_value_cons(Y,H,T,Eq).
1634		ordered_value(term(floating(F1)),term(floating(F2)),Eq) :- !,
1635		kernel_reals:real_less_than_equal_wf(term(floating(F1)),term(floating(F2)),no_wf_available),
1636		equal_atomic_term(F1,F2,Eq).
1637		ordered_value(A,B,not_equal) :- print(assuming_strictly_ordered(A,B)),nl.
1638
1639		ordered_value_cons([],_,_,_) :- !,fail.
1640		ordered_value_cons([H2|T2],H,T,Eq) :- !,ordered_pair((H,T),H2,T2,Eq). % Note: order different than for avl_sets!
1641		ordered_value_cons(Y,H,T,not_equal) :- write(assuming_strictly_ordered([H|T],Y)),nl.
1642
1643		:- block ordered_fields(-,?,?).
1644		ordered_fields([],RHS,Eq) :- !,RHS=[], Eq=equal.
1645		ordered_fields([field(Name,ValX)|TX],RHS,Eq) :- !,RHS=[field(Name,ValY)|TY],
1646	?	ordered_value(ValX,ValY,Equal1), check_next_field(Equal1,TX,TY,Eq).
1647		ordered_fields(FX,FY,Eq) :- add_internal_error('Unknown fields: ',ordered_fields(FX,FY,Eq)), Eq=not_equal.
1648
1649		:- block check_next_field(-,?,?,?).
1650	?	check_next_field(equal,TX,TY,EqRes) :- ordered_fields(TX,TY,EqRes).
1651		check_next_field(not_equal,_X,_Y,not_equal). % no need to check next field
1652
1653		:- block less_equal_atomic_term(-,?,?), less_equal_atomic_term(?,-,?).
1654		less_equal_atomic_term(A,B,Res) :- (A==B -> Res=equal ; A @<B, Res=not_equal).
1655
1656		:- block equal_atomic_term(-,?,?), equal_atomic_term(?,-,?).
1657		equal_atomic_term(A,B,Res) :- (A==B -> Res=equal ; Res=not_equal).
1658
1659
1660		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:not_relation_over([(int(1),int(2)),(int(2),int(1))],[int(1),int(2)],[int(2)],_WF) )).
1661		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:not_relation_over([(int(1),int(2))],[],[int(2)],_WF) )).
1662		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:not_relation_over([(int(1),pred_true)],[int(1)],[pred_true],_WF) )).
1663		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:not_relation_over([],[int(1)],[pred_true],_WF) )).
1664		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(2))],[int(3)],[int(1),int(2)],_) ).
1665		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(2))],[int(1)],[int(3)],_) ).
1666		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(3)),(int(1),int(2))],[int(1)],[int(3)],_) ).
1667		:- assert_must_fail( bsets_clp:not_relation_over([(int(1),int(3))],[int(1)],[int(3)],_) ).
1668		:- assert_must_fail( bsets_clp:not_relation_over([],[int(1)],[int(3)],_) ).
1669		:- assert_must_fail( bsets_clp:not_relation_over([],[],[],_) ).
1670		:- block not_relation_over(-,?,?,?).
1671
1672		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
1673		% we do not need the Range; this means we can match more closures (e.g., lambda)
1674		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,_,WF),!,
1675		not_subset_of_wf(FFDomain,Domain,WF).
1676		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),
1677		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,_,WF),!,
1678		not_both_subset_of(FFDomain,FFRange,Domain,Range,WF).
1679		/* could be slightly more efficient: but not clear if warrants additional complexity in code:
1680		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),
1681		check_element_can_be_decided(Domain), % ensures that check_element_of_wf will not block below
1682		check_element_can_be_decided(Range), % ensures that check_element_of_wf will not block below
1683		expand_and_convert_to_avl_set(FF,AER,no_relation_over,''),!,
1684		(is_avl_relation_over_domain(AER,Domain,WF)
1685		-> \+ is_avl_relation_over_range(AER,Range,WF)
1686		; true).
1687		check_element_can_be_decided(X) :- var(X),!,fail.
1688		check_element_can_be_decided(avl_set(_)).
1689		check_element_can_be_decided([]).
1690		check_element_can_be_decided(closure(P,T,B)) :-
1691		custom_explicit_sets:is_interval_closure_or_integerset(closure(P,T,B),Low,Up),
1692		ground(Low), ground(Up).
1693		*/
1694		not_relation_over(R,Dom,Ran,WF) :-
1695		expand_custom_set_to_list_wf(R,ER,_,not_relation_over,WF),
1696		%% print(not_rel(ER,Dom,Ran)),nl,
1697		not_relation_over2(ER,Dom,Ran,WF).
1698
1699
1700		%not_relation_over2(R,_,_) :- when(nonvar(R), (R\=[], R\=[_|_])) . % TYPE ERROR !
1701		:- block not_relation_over2(-,?,?,?).
1702		not_relation_over2([(X,Y)|T],Domain,Range,WF) :-
1703		membership_test_wf(Domain,X,MemRes,WF),
1704		not_relation_over3(MemRes,Y,T,Domain,Range,WF).
1705
1706		:- block not_relation_over3(-,?,?,?,?,?).
1707		not_relation_over3(pred_false,_Y,_T,_Domain,_Range,_WF).
1708		not_relation_over3(pred_true,Y,T,Domain,Range,WF) :-
1709		membership_test_wf(Range,Y,MemRes,WF),
1710		not_relation_over4(MemRes,T,Domain,Range,WF).
1711
1712		:- block not_relation_over4(-,?,?,?,?).
1713		not_relation_over4(pred_false,_T,_Domain,_Range,_WF).
1714		not_relation_over4(pred_true,T,Domain,Range,WF) :-
1715		not_relation_over2(T,Domain,Range,WF).
1716
1717
1718
1719		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf([],[],WF),WF)).
1720		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf([(int(1),int(3))],[int(1)],WF),WF)).
1721		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf(
1722		[(int(0),int(55)),(int(2),int(3)),(int(1),int(3))],[int(1),int(2),int(0)],WF),WF)).
1723		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf(
1724		[(int(99),int(55)),(int(2),int(3)),(int(99),int(4))],[int(2),int(99)],WF),WF)).
1725		:- assert_must_succeed((bsets_clp:domain_wf([],Res,_WF),Res=[])).
1726		:- assert_must_succeed((bsets_clp:domain_wf([(int(1),int(2))],Res,_WF),
1727		kernel_objects:equal_object(Res,[int(1)]))).
1728		:- assert_must_succeed((bsets_clp:domain_wf([(int(1),int(2)),(int(1),int(1))],Res,_WF),
1729		kernel_objects:equal_object(Res,[int(1)]))).
1730		:- assert_must_succeed((bsets_clp:domain_wf([(int(2),int(2)),(int(1),int(2))],Res,_WF),
1731		kernel_objects:equal_object(Res,[int(1),int(2)]))).
1732		:- assert_must_succeed((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(3),int(2)]),
1733		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(3),int(2))]))).
1734		:- assert_must_succeed((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]),
1735		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(1),int(2))]))).
1736		:- assert_must_fail((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]),
1737		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(3),int(2))]))).
1738
1739		:- block domain_wf(-,-,?).
1740		domain_wf(Rel,Res,WF) :- Res == [],!,
1741		empty_set_wf(Rel,WF).
1742		domain_wf(Rel,Res,WF) :- var(Rel),!, % hence Res must me nonvar
1743		(is_custom_explicit_set(Res,domain_wf)
1744		-> expand_custom_set_to_list_wf(Res,Res2,_,propagate_result_to_input2,WF) % avoid expanding twice
1745		; Res2 = Res),
1746		propagate_result_to_input(Res2,Rel,domain,WF),
1747		domain_wf1(Rel,Res2,WF).
1748	?	domain_wf(Rel,Res,WF) :- domain_wf1(Rel,Res,WF).
1749
1750
1751		% propagate result of domain/range back to original relation
1752		propagate_result_to_input(Result,OriginalRel,DomOrRange,WF) :-
1753		propagate_empty_set_wf(Result,result,OriginalRel,WF), % this will trigger before LWF ground
1754		(preferences:preference(use_smt_mode,true)
1755		-> propagate_result_to_input1(Result,OriginalRel,1,DomOrRange)
1756		% hopefully full CHR implementation will avoid the need for this hack
1757		% ; kernel_objects:is_marked_to_be_computed(OriginalRel) -> true % get_last_wait_flag(propagate_result_to_input,WF,LWF)
1758		;
1759		get_wait_flag(2000,propagate_result_to_input,WF,LWF), % TO DO: determine right value for Priority ?
1760		% higher number for data_validation mode seems slightly counterproductive (on private_source_not_available tests)
1761		propagate_result_to_input1(Result,OriginalRel,LWF,DomOrRange) % this slows down test 289 if not guarded, 1088 if guarded
1762		).
1763
1764		:- block propagate_result_to_input1(-,?,?,?), propagate_result_to_input1(?,-,-,?).
1765		% Note: if arg 2 (Rel) is known we will not propagate
1766		propagate_result_to_input1([],Rel,_,_) :- !, empty_set(Rel).
1767		propagate_result_to_input1(Result,Input,LWF,DomOrRange) :-
1768		(kernel_objects:is_marked_to_be_computed(Input) -> true
1769		; propagate_result_to_input2(Result,Input,LWF,DomOrRange)).
1770
1771		%:- block propagate_result_to_input2(-,?).
1772		:- block propagate_result_to_input2(-,?,?,?), propagate_result_to_input2(?,-,-,?).
1773		% maybe do in CHR in future: x:dom(R) => #z.(x,z) : R
1774		% TO DO: make stronger; also support avl_set ...
1775		propagate_result_to_input2([],_Rel,_,_) :- !. % nothing can be said; we could have repeated entries for previous domain elements
1776		propagate_result_to_input2([D|T],Rel,LWF,DomOrRange) :- %print(propagate_result_to_input2([D|T],Rel,LWF,DomOrRange)),nl,
1777		!,
1778		(Rel == [] -> fail % we would need more relation elements to generate the domain/range
1779		; nonvar(Rel) -> true % no propagation
1780		; (DomOrRange=domain -> Rel = [(D,_)|RT] ; Rel = [(_,D)|RT]),
1781		propagate_result_to_input2(T,RT,LWF,DomOrRange)
1782		).
1783		propagate_result_to_input2(CS,Rel,LWF,DomOrRange) :- var(Rel), is_custom_explicit_set(CS),!,
1784		expand_custom_set_to_list(CS,Res,_,propagate_result_to_input2),
1785		propagate_result_to_input2(Res,Rel,LWF,DomOrRange).
1786		propagate_result_to_input2(_1,_2,_LWF,_DomOrRange).
1787
1788		:- block domain_wf1(-,?,?).
1789		domain_wf1(Rel,Res,WF) :- is_custom_explicit_set(Rel,domain_wf),
1790		domain_of_explicit_set_wf(Rel,Dom,WF), !,
1791	?	equal_object_wf(Dom,Res,domain_wf1,WF).
1792		domain_wf1(Rel,Res,WF) :-
1793		expand_custom_set_to_list_wf(Rel,Relation,_,domain_wf,WF),
1794	?	newdomain1(Relation,[],Res,WF),
1795		quick_propagate_domain(Relation,Res,WF).
1796
1797		:- block quick_propagate_domain(-,?,?).
1798		quick_propagate_domain([],_,_WF).
1799		quick_propagate_domain([(X,_)|T],FullRes,WF) :-
1800		quick_propagation_element_information(FullRes,X,WF,FullRes1), % should we use a stronger check ?
1801		quick_propagate_domain(T,FullRes1,WF).
1802
1803		%:- block newdomain1(-,?,-,?). % why was this commented out ?
1804		:- block newdomain1(-,?,?,?).
1805		/* newdomain1(Rel,SoFar,Res,WF) :- var(Rel), !,
1806		domain_propagate_result(Res,Rel,SoFar,WF). */
1807	?	newdomain1(Dom,SoFar,Res,WF) :- newdomain2(Dom,SoFar,Res,WF).
1808
1809		%:- block newdomain2(-,?,?,?).
1810	?	newdomain2([],_SoFar,Res,WF) :- empty_set_wf(Res,WF).
1811		newdomain2([(X,Y)|T],SoFar,Res,WF) :-
1812		(Res==[]
1813		-> MemRes=pred_true, % no new elements can appear, all Xs must already be in SoFar
1814		check_element_of_wf(X,SoFar,WF)
1815		; membership_test_wf(SoFar,X,MemRes,WF),
1816		% now check that card of Relation is greater or equal to Result; if equal set MemRes to pred_false
1817		% if card(Result)=card(dom(Result)) => all elements in Result must be fresh domain elements
1818		card_greater_equal_check([(X,Y)|T],Res,MemRes)
1819		),
1820	?	newdomain3(MemRes,X,T,SoFar,Res,WF).
1821
1822		:- block newdomain3(-,?,?,?,?,?).
1823		newdomain3(pred_true,_,T,SoFar,Res,WF) :- newdomain1(T,SoFar,Res,WF).
1824		newdomain3(pred_false,X,T,SoFar,Res,WF) :-
1825		kernel_objects:mark_as_non_free(X,domain), % X is linked to a particular Y -> it is not free
1826		add_element_wf(X,SoFar,SoFar2,WF),
1827	?	equal_cons_wf(Res,X,Res2,WF),
1828	?	newdomain1(T,SoFar2,Res2,WF).
1829
1830
1831		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_domain_wf(int(2),[(int(2),int(7))],WF),WF)).
1832		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_wf(int(2),[(int(1),int(6)),(int(2),int(7))],WF),WF)). % used to be wfdet; but dom_symbolic can create existential quantifier, not all co-routines/... evaluated in wfdet
1833		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_wf(int(22),[(int(1),int(6)),(int(22),int(7)),(int(33),int(7))],WF),WF)). % used to be wfdet (see above)
1834		:- assert_must_succeed((bsets_clp:in_domain_wf(int(1),[(int(1),int(2))],_))).
1835		:- assert_must_succeed((bsets_clp:in_domain_wf(int(3),[(int(1),int(2)),(int(3),int(4))],_))).
1836		:- assert_must_fail((bsets_clp:in_domain_wf(int(3),[],_))).
1837		:- assert_must_fail((bsets_clp:in_domain_wf(int(3),[(int(1),int(2))],_))).
1838		/* a more efficient version than using element_of and computing domain */
1839
1840		% just like not_empty_set_wf but instantiates with (El,_) as first element
1841		in_domain_wf(El,S,WF) :- var(S),!, force_in_domain_wf(El,S,WF).
1842		in_domain_wf(El,Rel,WF) :- in_domain_wf_lazy(El,Rel,WF).
1843
1844		:- use_module(kernel_non_empty_attr,[mark_var_set_as_non_empty/1]).
1845		% next is also used in apply_to/6
1846		force_in_domain_wf(El,S,WF) :-
1847		(preferences:preference(use_smt_mode,true) -> get_wait_flag0(WF,WF0),
1848		when(ground(WF0),delayed_force_in_domain_wf(El,S,WF))
1849		; % TO DO: non-empty flag
1850		mark_var_set_as_non_empty(S),
1851		get_enumeration_starting_wait_flag(not_empty_domain_wf,WF,LWF), in_domain_lwf(El,S,LWF,WF)).
1852		% delay instantiating S somewhat: it can mess up many other optimisations
1853		% fixes trying to deconstruct infinite set enum warning for test 2022
1854		delayed_force_in_domain_wf(El,S,_WF) :- var(S),!, S=[(El,_)|_]. % TODO: mark _ as irrelevant
1855		delayed_force_in_domain_wf(El,Rel,WF) :- in_domain_wf_lazy(El,Rel,WF).
1856
1857		:- block in_domain_lwf(-,-,-,?).
1858		% was :- block in_domain_lwf(-,?,-,?). but this prevents instantiating El in case Rel becomes known ! see e.g. private_examples/ClearSy/ComparePv10Pv11/DebugPv10/ test 1952, 2270
1859		%:- block in_domain_lwf(-,-,?,?),in_domain_lwf(?,-,-,?). % this annotation fails test 1703
1860		in_domain_lwf(El,Rel,LWF,WF) :- % tools_printing:print_term_summary(in_domain_lwf(El,Rel,LWF)),
1861		(var(Rel) -> ground_value_check(El,GrVal),
1862		in_domain_lwf2(El,Rel,LWF,GrVal,WF) % we could also wait at least until WF0 is fully grounded?
1863		; not_empty_set_unless_closure_wf(Rel,WF),
1864		in_domain_wf_lazy(El,Rel,WF)).
1865
1866		:- block in_domain_lwf2(?,-,-,-,?).
1867		in_domain_lwf2(El,Rel,_LWF,_Grval,WF) :- % tools_printing:print_term_summary(in_domain_lwf2(El,Rel,_LWF,_Grval)),
1868		(var(Rel) -> Rel = [(El,_)|_]
1869		% can create a choice point when unifying with large avl_set:, see rule_Rule_DB_PSR_0003_C
1870		% maybe we should delay even further
1871		; not_empty_set_unless_closure_wf(Rel,WF),
1872		in_domain_wf_lazy(El,Rel,WF)).
1873
1874		not_empty_set_unless_closure_wf(closure(_,_,_),_) :- !. % do not check this; in_domain_wf or other call will find a solution anyway; no need to set up closure constraints twice
1875		not_empty_set_unless_closure_wf(Rel,WF) :- not_empty_set_wf(Rel,WF).
1876
1877		% does not instantiate to [(El,_)|_]
1878		:- block in_domain_wf_lazy(?,-,?).
1879		in_domain_wf_lazy(_DomainElement,[],_WF) :- !,fail.
1880		in_domain_wf_lazy(DomainElement,avl_set(A),_WF) :-
1881		ground_value(DomainElement), !,
1882		check_in_domain_of_avlset(DomainElement,A).
1883		% TO DO: check for infinite closures
1884		in_domain_wf_lazy(DomainElement,ES,WF) :-
1885		is_custom_explicit_set(ES,in_domain_wf_lazy),
1886		domain_of_explicit_set_wf(ES,Dom,WF),!,
1887		check_element_of_wf(DomainElement,Dom,WF).
1888		in_domain_wf_lazy(El,Rel,WF) :-
1889		expand_custom_set_to_list_wf(Rel,Relation,Done,in_domain_wf_lazy,WF),
1890		get_binary_choice_wait_flag(in_domain_wf_lazy(El),WF,LWF), % TO DO: get_pow2_binary_choice_priority(Len,Prio), get_binary_choice_wait_flag_exp_backoff
1891		% if Done == true -> we can use maybe clpfd_inlist or clpfd:element or quick_propagate
1892		quick_propagation_domain_element_list(Done,Relation,El,WF),
1893		in_domain2(El,Relation,WF,LWF).
1894
1895		% a custom implementation of quick_propagation_element_information for checking domain elements and lists only
1896		:- use_module(clpfd_lists,[try_in_fd_value_list_check/4]).
1897		:- block quick_propagation_domain_element_list(-,?,?,?).
1898		quick_propagation_domain_element_list(_,_,_,_) :- preferences:preference(use_clpfd_solver,false),!.
1899		quick_propagation_domain_element_list(_,_,El,_) :- ground(El),!.
1900		quick_propagation_domain_element_list(_,RelList,El,WF) :-
1901		try_in_fd_value_list_check(RelList,(El,_),couple_left(_),WF). % use couple_left to ignore range values
1902
1903
1904		:- block in_domain2(?,-,?,?).
1905		in_domain2(El,[(X,_Y)|T],WF,LWF) :-
1906		(T==[]
1907		-> equal_object_wf(El,X,in_domain2,WF)
1908		; kernel_objects:equality_objects_lwf(El,X,EqRes,LWF,WF),
1909		in_domain3(EqRes,El,T,WF,LWF)
1910		).
1911
1912		:- block in_domain3(-,?,?,?,?).
1913		in_domain3(pred_true,_El,_T,_WF,_LWF).
1914		in_domain3(pred_false,El,T,WF,LWF) :-
1915		get_new_subsidiary_wait_flag(LWF,in_domain2(El,T),WF,NewLWF), % not necessary if T only has single element
1916		in_domain2(El,T,WF,NewLWF).
1917
1918
1919		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[],WF),WF)).
1920		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(2),int(7))],WF),WF)).
1921		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(2),int(7)),(int(4),int(3))],WF),WF)).
1922		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_in_domain_wf(int(4),[(int(2),int(7)),(int(4),int(3))],WF),WF)).
1923		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(1),[(int(1),int(2))],_))).
1924		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2)),(int(3),int(4))],_))).
1925		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[],_))).
1926		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2))],_))).
1927		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2)),(int(2),int(3))],WF),WF)).
1928		/* a more efficient version than using not_element_of and computing domain */
1929
1930
1931		:- block not_in_domain_wf(?,-,?).
1932		not_in_domain_wf(DomainElement,ES,WF) :- is_custom_explicit_set(ES,not_in_domain),
1933		domain_of_explicit_set_wf(ES,Dom,WF),!,
1934		not_element_of_wf(DomainElement,Dom,WF).
1935		not_in_domain_wf(El,Rel,WF) :-
1936		expand_custom_set_to_list_wf(Rel,Relation,_,not_in_domain,WF),
1937		not_in_domain2(Relation,El,WF).
1938		:- block not_in_domain2(-,?,?).
1939		not_in_domain2([],_,_WF).
1940		not_in_domain2([(X,_Y)|T],E,WF) :- not_equal_object_wf(E,X,WF), not_in_domain2(T,E,WF).
1941
1942
1943
1944
1945		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([],[],WF),WF)).
1946		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([(int(1),int(3))],[int(3)],WF),WF)).
1947		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf(
1948		[(int(0),int(55)),(int(2),int(3)),(int(1),int(3))],[int(3),int(55)],WF),WF)).
1949		:- assert_must_succeed((bsets_clp:range_wf([],Res,_WF),Res=[])).
1950		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2))],Res,_WF),
1951		kernel_objects:equal_object(Res,[int(2)]))).
1952		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(1)),(int(2),int(1))],Res,_WF),
1953		kernel_objects:equal_object(Res,[int(1)]))).
1954		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1))],Res,_WF),
1955		kernel_objects:equal_object(Res,[int(1),int(2)]))).
1956		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],Res,_WF),
1957		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1958		:- assert_must_succeed((bsets_clp:range_wf(X,Res,_WF),
1959		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],
1960		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1961		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1962		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(2))])).
1963		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1964		X = [(int(2),int(1)),(int(1),int(2)),(int(2),int(2))])).
1965		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1966		X = [])).
1967		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1968		([int(0),int(1)],[int(0),int(1)]),([int(0),int(2)],[int(0),int(2)]),
1969		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1970		([int(1),int(2)],[int(1),int(2)]),([int(1),int(3)],[int(1),int(3)]),
1971		([int(1),int(4)],[int(1),int(4)]),([int(2)],[int(2)]),([int(2),int(3)],[int(2),int(3)]),
1972		([int(2),int(4)],[int(2),int(4)]),([int(3)],[int(3)]),([int(3),int(4)],
1973		[int(3),int(4)]),([int(4)],[int(4)])],_Res,_WF))).
1974		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1975		([int(0),int(1)],[int(0),int(1)]),
1976		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1977		([int(1),int(2)],[int(1),int(2)])],_Res,_WF))).
1978
1979
1980		:- block range_wf(-,-,?).
1981		range_wf(Rel,Res,WF) :- Res ==[],!, empty_set_wf(Rel,WF).
1982		range_wf(Rel,Res,WF) :- Rel ==[],!, empty_set_wf(Res,WF).
1983	?	range_wf(Rel,Res,WF) :- range_wf1(Rel,Res,WF),
1984		propagate_result_to_input(Res,Rel,range,WF).
1985
1986		:- block range_wf1(-,?,?).
1987		range_wf1(Rel,Res,WF) :-
1988		is_custom_explicit_set(Rel,range_wf1),
1989		range_of_explicit_set_wf(Rel,Range,WF), !,
1990		equal_object_wf(Range,Res,range_wf1,WF).
1991		range_wf1(Rel,Res,WF) :-
1992		% TO DO : propagate information that card of Res <= card of Rel; similar thing for domain
1993		expand_custom_set_to_list_wf(Rel,Relation,_,range_wf1,WF),
1994	?	newrange2(Relation,[],Res,WF),
1995		quick_propagate_range(Relation,Res,WF).
1996
1997
1998		:- block quick_propagate_range(-,?,?).
1999		quick_propagate_range([],_,_WF).
2000		quick_propagate_range([(_,Y)|T],FullRes,WF) :-
2001		quick_propagation_element_information(FullRes,Y,WF,FullRes1), % should we use a stronger check ?
2002		quick_propagate_range(T,FullRes1,WF).
2003
2004		:- block newrange2(-,?,?,?).
2005		newrange2([],_SoFar,Res,WF) :-
2006		empty_set_wf(Res,WF).
2007		newrange2([(X,Y)|T],SoFar,Res,WF) :-
2008		(Res==[]
2009		-> MemRes=pred_true, check_element_of_wf(Y,SoFar,WF)
2010		; membership_test_wf(SoFar,Y,MemRes,WF),
2011		card_greater_equal_check([(X,Y)|T],Res,MemRes), % check that card of Relation is greater or equal to Result; if equal set MemRes to pred_false
2012		(var(MemRes) -> prop_empty_pred_true(Res,MemRes) %,print(delay_range(Y,T)),nl
2013		% TO DO: we could look further in T if we can decide membership for other elements in T ?
2014		; true)
2015		),
2016	?	newrange3(MemRes,Y,T,SoFar,Res,WF).
2017
2018		:- block prop_empty_pred_true(-,?).
2019		prop_empty_pred_true([],R) :- !, R=pred_true.
2020		prop_empty_pred_true(_,_).
2021
2022		% card_greater_equal_check(Set1,Set2,EqFlag) : check that cardinality of Set1 is greater or equal to that of Set2; set EqFlag to pred_false if they are equal
2023		% checking is stopped if EqFlag becomes nonvar
2024		% tested by testcase 1061
2025		:- block card_greater_equal_check(-,?,-), card_greater_equal_check(?,-,-).
2026		card_greater_equal_check(_,_,Flag) :- nonvar(Flag),!. % no longer required; even though we could prune failure !? done later in newrange2/newdomain2 ??!!
2027		card_greater_equal_check([],Set2,Flag) :- !,empty_set(Set2),
2028		Flag=pred_false. % Flag set indicates that both sets have same size
2029		card_greater_equal_check(_,[],_) :- !.
2030		card_greater_equal_check([_|T],[_|R],Flag) :- !, card_greater_equal_check(T,R,Flag).
2031		% To do: deal with AVL args as Result + also use efficient_card_for_set for closures
2032		%card_greater_equal_check([_|T],Set,Flag) :- efficient_card_for_set(B,CardB,CodeB),!,
2033		% f: 1..7 -->> 1..n & n>=7 & n<10 still does not work well
2034		% TO DO: can we merge code with check_card_greater_equal
2035		card_greater_equal_check(_,_,_).
2036
2037
2038		:- block newrange3(-,?,?,?,?,?).
2039		newrange3(pred_true,_Y,T,SoFar,Res,WF) :- newrange2(T,SoFar,Res,WF).
2040		newrange3(pred_false,Y,T,SoFar,Res,WF) :-
2041		kernel_objects:mark_as_non_free(Y,range), % Y is linked to a particular X -> it is not free
2042		add_element_wf(Y,SoFar,SoFar2,WF),
2043	?	equal_cons_wf(Res,Y,Res2,WF),
2044		newrange2(T,SoFar2,Res2,WF).
2045
2046
2047		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([],Res,_WF),Res=[])).
2048		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([int(1),int(2)],Res,_WF),
2049		Res=[(int(1),int(1)),(int(2),int(2))])).
2050		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:identity_relation_over_wf([int(2),int(4)],[(int(4),int(4)),(int(2),int(2))],WF),WF)).
2051		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:identity_relation_over_wf([int(1),int(2),int(4)],[(int(4),int(4)),(int(2),int(2)),(int(1),int(1))],WF),WF)).
2052		:- assert_must_fail((bsets_clp:identity_relation_over_wf([int(1)|_],_,_WF),fail)). /* check: no loop */
2053
2054		:- block identity_relation_over_wf(-,?,?).
2055		identity_relation_over_wf(Set1,IDRel,WF) :-
2056		expand_custom_set_to_list_wf(Set1,ESet1,_,identity_relation_over_wf,WF),
2057		identity_relation_over2(ESet1,IDRel,WF).
2058
2059		:- block identity_relation_over2(-,?,?).
2060		identity_relation_over2([],Res,WF) :- empty_set_wf(Res,WF).
2061		identity_relation_over2([X|T1],Res,WF) :- equal_cons_wf(Res,(X,X),T2,WF), % equal_object([(X,X)|T2],Res),
2062		identity_relation_over2(T1,T2,WF).
2063
2064
2065
2066		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_identity((int(1),int(1)),[int(1),int(2)],WF),WF)).
2067		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[int(1),int(2)],_WF))).
2068		:- assert_must_fail((bsets_clp:in_identity((int(3),int(3)),[int(1),int(2)],_WF))).
2069		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[],_WF))).
2070		in_identity((X,Y),Domain,WF) :-
2071		equal_object_wf(X,Y,in_identity,WF), check_element_of_wf(X,Domain,WF).
2072
2073		:- assert_must_fail((bsets_clp:not_in_identity((int(1),int(1)),[int(1),int(2)],_WF))).
2074		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(1),int(2)),[int(1),int(2)],WF),WF)).
2075		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(3),int(3)),[int(1),int(2)],WF),WF)).
2076		:- assert_must_succeed((bsets_clp:not_in_identity((int(1),int(2)),[],_WF))).
2077		not_in_identity((X,Y),Domain,WF) :-
2078		equality_objects_wf(X,Y,Eq,WF),
2079		not_in_id2(Eq,X,Domain,WF).
2080
2081		:- block not_in_id2(-,?,?,?).
2082		not_in_id2(pred_true,X,Domain,WF) :- not_element_of_wf(X,Domain,WF).
2083		not_in_id2(pred_false,_,_,_).
2084
2085
2086		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([(int(1),int(2)),(int(3),int(4)),(int(5),int(6))], [(int(6),int(5)),(int(2),int(1)),(int(4),int(3))],WF),WF)).
2087		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([(int(1),int(2))], [(int(2),int(1))],WF),WF)).
2088		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([], [],WF),WF)).
2089		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [])).
2090		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [(int(2),int(2))])).
2091		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,[(int(1),int(2)),(int(7),int(6))],_WF),
2092		X = [(int(2),int(1)),(int(6),int(7))])).
2093		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],X,_WF),
2094		X = [(int(2),int(1)),(int(6),int(7))])).
2095		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],
2096		[(int(6),int(7)),(int(2),int(1))],_WF))).
2097		:- assert_must_succeed((bsets_clp:invert_relation_wf(closure([a,b],[string,boolean],b(truth,pred,[])),
2098		closure([b,a],[boolean,string],b(truth,pred,[])),_WF))).
2099
2100		:- block invert_relation_wf(-,-,?).
2101		invert_relation_wf(R,IR,WF) :-
2102		% (nonvar(R) -> invert_relation2(R,IR) ; invert_relation2(IR,R)).
2103		invert_relation2(R,IR,WF). % , print_term_summary(invert_relation(R,IR)).
2104		/* Optimization for some types of closures: Instead of expanding the closures, we just
2105		swap the parameters. This does not work with closures wich have only one parameter
2106		wich is a pair */
2107		invert_relation2(CS,R,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2108		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_1,WF).
2109		invert_relation2(R,CS,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2110		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_2,WF).
2111		%invert_relation2(closure([P1,P2],[T1,T2],Clo),closure([P2,P1],[T2,T1],Clo)) :- !.
2112		invert_relation2(R,IR,WF) :- %try_expand_custom_set_wf(R,ER,invert,WF),
2113		% (nonvar(R) -> invert_relation3(R,IR)
2114		% ; invert_relation3(IR,R),(ground(IR)-> true ; invert_relation3(R,IR))).
2115		invert_relation3(R,IR,WF,1), invert_relation3(IR,R,WF,1).
2116
2117		:- block invert_relation3(-,?,?,?).
2118		invert_relation3(closure(P,T,B),Res,WF,_) :- invert_explicit_set(closure(P,T,B),ICS),
2119		equal_object_wf(Res,ICS,invert_relation3_1,WF).
2120		invert_relation3(avl_set(S),Res,WF,_) :- invert_explicit_set(avl_set(S),ICS),
2121		equal_object_wf(Res,ICS,invert_relation3_2,WF).
2122		invert_relation3([],Res,WF,_) :- empty_set_wf(Res,WF).
2123		invert_relation3([(X,Y)|T],Res,WF,Depth) :-
2124		D1 is Depth+1, get_wait_flag(D1,invert_relation3,WF,LWF),
2125		equal_cons_lwf(Res,(Y,X),IT,LWF,WF),
2126		invert_relation3(T,IT,WF,D1).
2127
2128
2129
2130
2131		tuple_of(X,Y,R) :- check_element_of((X,Y),R).
2132		%tuple_of_wf(X,Y,R,WF) :- check_element_of_wf((X,Y),R,WF).
2133
2134
2135		% RELATIONAL COMPOSITION (;)
2136
2137		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2138		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2139		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2140		[(int(11),int(12)),(int(11),int(33))],
2141		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2142		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(12)),
2143		[(int(11),int(12)),(int(11),int(33))],
2144		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2145		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2146		[(int(11),[int(33),int(32)])],
2147		[([int(32),int(33)],int(22))],WF),WF)).
2148		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:in_composition_wf((int(11),int(33)),
2149		[(int(11),int(12)),(int(11),int(33))],
2150		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2151		% check if (X,Y) element of (F ; G)
2152		in_composition_wf((X,Y),F,G,WF) :-
2153		check_element_of_wf((X,Z1),F,WF), % no need to enumerate Z (TODO: check)
2154		equal_object_wf(Z1,Z2,check_element_of_wf,WF),
2155		check_element_of_wf((Z2,Y),G,WF).
2156
2157		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(11),int(33)),
2158		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2159		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(33),int(22)),
2160		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2161
2162		% just evaluates arguments; TODO: improve or at least pass Type (for symbolic composition)
2163		not_in_composition_wf(Couple,F,G,WF) :-
2164		rel_composition_wf(F,G,Comp,_UnknownType,WF),
2165		not_element_of_wf(Couple,Comp,WF).
2166
2167		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([(int(1),int(2)),(int(3),int(4)),(int(5),int(6))], [(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],
2168		[(int(1),int(1)),(int(5),int(7)),(int(3),int(33))]))).
2169		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([], [(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],[]))).
2170		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],[],[]))).
2171		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([],[],[]))).
2172		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2173		[(int(1),int(11))],X),X = [])).
2174		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],[],X),X = [])).
2175		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2176		[(int(2),int(11))],X),
2177		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2178		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(2))],[(int(2),int(11))],X),
2179		ground(X), bsets_clp:equal_object(X,[(int(1),int(11)),(int(7),int(11))]))).
2180		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(5))],
2181		[(int(2),int(11)),(int(2),int(4))],X),
2182		kernel_objects:equal_object(X,[(int(1),int(11)),(int(1),int(4))]))).
2183		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(1),int(5))],
2184		[(int(2),int(11)),(int(5),int(11))],X),
2185		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2186		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),[int(1)]),(int(1),[int(2),int(5)])],
2187		[([int(1),int(2)],int(13)),([int(5),int(2)],int(12))],X),
2188		kernel_objects:equal_object(X,[(int(1),int(12))]))).
2189
2190		rel_composition(Rel1,Rel2,Comp) :- % only used in unit_tests above
2191		init_wait_flags(WF,[rel_composition]),
2192		rel_composition_wf(Rel1,Rel2,Comp,_UnknownType,WF),
2193	?	ground_wait_flags(WF).
2194
2195		:- block rel_composition_wf(-,-,?,?,?).
2196		rel_composition_wf(Rel1,Rel2,Comp,_,WF) :-
2197		(Rel1==[] ; Rel2==[]),
2198		!,
2199		empty_set_wf(Comp,WF).
2200		rel_composition_wf(Rel1,Rel2,Comp,Type,WF) :- rel_composition1(Rel1,Rel2,Comp,Type,WF).
2201
2202		:- use_module(closures,[is_infinite_non_injective_closure/1]).
2203
2204		:- block rel_composition1(-,?,?,?,?),rel_composition1(?,-,?,?,?).
2205		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2206		(Rel1==[] ; Rel2==[]),!, empty_set_wf(Comp,WF).
2207		rel_composition1(Rel1,Rel2,Comp,Type,WF) :- keep_symbolic(Rel1),
2208		(Rel2 = avl_set(_), \+ is_infinite_non_injective_closure(Rel1)
2209		-> SYMBOLIC=false
2210		; SYMBOLIC=symbolic),
2211		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3),
2212		!,
2213	?	equal_object_wf(Comp,Rel3,rel_composition1_0,WF).
2214
2215		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2216		rel_composition_for_explicit_set(Rel1,Rel2,Res),!, % treats finite Rel1 and avl_set for Rel2
2217		equal_object_wf(Res,Comp,rel_composition1_1,WF).
2218		rel_composition1(Rel1,Rel2,Comp,Type,WF) :- Rel2=closure(_,_,_),
2219		keep_symbolic(Rel2),
2220		% we know keep_symbolic(Rel1) is false
2221		(dom_for_specific_closure(Rel2,Domain,function(_),WF) % TO DO: also deal with relations; in SYMBOLIC mode this may be counter productive; see function_composition ast cleanup rule
2222		-> !,
2223		opt_push_wait_flag_call_stack_info(WF,b_operator_call(composition,[Rel1,Rel2],unknown),WF2),
2224		on_enumeration_warning(expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1,WF2),R=failed),
2225		(R==failed % expansion of Rel1 failed; use symbolic composition
2226		-> symbolic_composition(Rel1,Rel2,true,Type,Rel3),
2227		equal_object_optimized(Rel3,Comp,rel_composition1_4)
2228		; rel_compose_with_inf_fun(Relation1,Domain,Rel2,Comp,WF2)
2229		% this is like map Rel2 over Rel1 in functional programmming
2230		)
2231		; symbolic_composition(Rel1,Rel2,false,Type,Rel3),
2232		!,
2233		expand_custom_set_wf(Rel3,CRes,rel_composition,WF),% do we need to expand ?
2234	?	equal_object_optimized(CRes,Comp,rel_composition1_4)
2235		).
2236		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2237		opt_push_wait_flag_call_stack_info(WF,b_operator_call(composition,[Rel1,Rel2],unknown),WF2),
2238		expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1_2,WF2),
2239		expand_custom_set_to_list_wf(Rel2,Relation2,_,rel_composition1_3,WF2),
2240	?	rel_compose2(Relation1,Relation2,Comp,WF2).
2241
2242
2243		:- use_module(btypechecker, [l_unify_types_strict/2]).
2244		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3) :-
2245		get_set_type(Type,couple(TX,TZ)),
2246		mnf_get_relation_types(Rel1,TX1,TY1),
2247		mnf_get_relation_types(Rel2,TY2,TZ2),
2248		(l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2]) -> true
2249		; add_internal_error('Could not unify range/domain types: ',l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2])),
2250		fail
2251		),
2252		ground((TX1,TY1,TZ)), % avoid creating a closure with non-ground type list
2253		rel_comp_closure(Rel1,Rel2,TX1,TY1,TZ,SYMBOLIC,Rel3).
2254		% generate a closure for {xx,zz | #(yy).(xx|->yy : Rel1 & yy|->zz : Rel2)}
2255		% TO DO: maybe detect special cases: Rel1 is a function/cartesian product, e.g., (((0 .. 76) * (0 .. 76)) * {FALSE}) ; {(FALSE|->0),(TRUE|->1)}
2256		:- use_module(bsyntaxtree, [conjunct_predicates_with_pos_info/3,update_used_ids/3 ]).
2257		rel_comp_closure(Rel1,Rel2,TX,TY,TZ,SYMBOLIC,closure(Args,Types,CBody)) :-
2258		Args = ['_rel_comp1','_rel_comp2'], Types = [TX,TZ],
2259		couple_member_pred('_rel_comp1',TX,'_zzzz_unary',TY,Rel1, Pred1),
2260		couple_member_pred('_zzzz_unary',TY,'_rel_comp2',TZ,Rel2, Pred2),
2261		UsedIds = ['_rel_comp1','_rel_comp2','_zzzz_unary'], % avoid having to call find_identifier_uses
2262		%conjunct_predicates([Pred1,Pred2],P12a), bsyntaxtree:check_computed_used_ids(P12a,UsedIds),
2263		%safe_create_texpr(conjunct(Pred1,Pred2),pred,[used_ids(UsedIds)],P12),
2264		conjunct_predicates_with_pos_info(Pred1,Pred2,P12a),
2265		update_used_ids(P12a,UsedIds,P12),
2266		%b_interpreter_components:create_unsimplified_exists([b(identifier('_zzzz_unary'),TY,[])],P12,Body),
2267		bsyntaxtree:create_exists_opt_liftable([b(identifier('_zzzz_unary'),TY,[])],P12,Body), % cf Thales_All/rule_zcpa2 test 2287
2268		(SYMBOLIC==symbolic
2269		-> mark_bexpr_as_symbolic(Body,CBody)
2270		; CBody=Body).
2271
2272		% generate predicate for X|->Y : Rel
2273		couple_member_pred(X,TX,Y,TY,Rel, Pred) :-
2274		Pred = b(member(b(couple(b(identifier(X),TX,[]),
2275		b(identifier(Y),TY,[])),couple(TX,TY),[]),
2276		b(value(Rel),set(couple(TX,TY)),[])),pred,[]).
2277
2278
2279
2280		:- block rel_compose2(-,?,?,?).
2281		rel_compose2([],_,Out,WF) :- empty_set_wf(Out,WF).
2282		rel_compose2([(X,Y)|T],Rel2,Out,WF) :-
2283	?	rel_extract(Rel2,X,Y,OutXY,[],WF),
2284		% rel_extract(Rel2,X,Y,Out,OutRem),
2285	?	rel_compose2(T,Rel2,OutRem,WF),
2286	?	union_wf(OutRem,OutXY,Out,WF). % used to call union wihout wf; makes test 1394 fail
2287
2288		:- block rel_extract(-,?,?,?,?,?).
2289		rel_extract([],_,_,Rem,Rem,_WF). % should we use equal_object here ?????
2290		rel_extract([(Y1,Z)|T],X,Y,Res,Rem,WF) :-
2291	?	rel_extract(T,X,Y,CT,Rem,WF),
2292	?	equality_objects_wf(Y1,Y,EqRes,WF),
2293		rel_extract2(EqRes,Z,X,CT,Res).
2294
2295		:- block rel_extract2(-,?,?,?,?).
2296		rel_extract2(pred_true, Z, X,CT,Res) :- add_element((X,Z),CT,Res).
2297		rel_extract2(pred_false,_Z,_X,CT,Res) :- Res = CT.
2298
2299
2300		% relational composition of a finite relation with an infinite or symbolic function
2301		rel_compose_with_inf_fun(R,Dom,Fun,CompRes,WF) :- !,
2302		rel_compose_with_inf_fun_acc(R,Dom,Fun,[],CompRes,WF).
2303		:- block rel_compose_with_inf_fun_acc(-,?,?,?,?,?).
2304		rel_compose_with_inf_fun_acc([],_Dom,_Rel2,Acc,Comp,WF) :-
2305		equal_object_wf(Comp,Acc,rel_compose_with_inf_fun_acc,WF).
2306		rel_compose_with_inf_fun_acc([(X,Y)|T],Dom,Fun,Acc,CompRes,WF) :-
2307		membership_test_wf(Dom,Y,MemRes,WF), % check if Y is in the domain of the symbolic relation
2308		rel_compose_with_inf_fun_acc_aux(MemRes,X,Y,T,Dom,Fun,Acc,CompRes,WF).
2309
2310		:- block rel_compose_with_inf_fun_acc_aux(-,?,?,?, ?,?,?,?, ?).
2311		rel_compose_with_inf_fun_acc_aux(pred_true,X,Y,T,Dom,Fun,Acc,CompRes,WF) :-
2312		apply_to(Fun,Y,FY,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
2313		add_element_wf((X,FY),Acc,NewAcc,WF),
2314		rel_compose_with_inf_fun_acc(T,Dom,Fun,NewAcc,CompRes,WF).
2315		rel_compose_with_inf_fun_acc_aux(pred_false,_X,_Y,T,Dom,Fun,Acc,Comp,WF) :-
2316		rel_compose_with_inf_fun_acc(T,Dom,Fun,Acc,Comp,WF).
2317
2318		% TO DO: if we obtain a list such as [(int(1),X),...] in Acc rather than an avl_set,
2319		% we may still be able to sort and avoid quadratic comparisons if e.g.
2320		% first component is a data-type where equality can be decided by unification (integer, bool, global(GS), ...)
2321		% we could put the optimisation into add_element_wf ?
2322		% TO DO: special version for avl_set as relation?
2323
2324		/*
2325		Note: old version; has performance problem, 2021/02_Feb/CDS
2326		the add_element_wf calls below can only construct/instantiate result when empty_set_wf reached
2327		and a lot of pending co-routines pile up for long relation lists
2328
2329		:- block rel_compose_with_inf_fun(-,?,?,?,?).
2330		rel_compose_with_inf_fun([],_Dom,_Rel2,Comp,WF) :- empty_set_wf(Comp,WF).
2331		rel_compose_with_inf_fun([(X,Y)|T],Dom,Fun,CompRes,WF) :-
2332		membership_test_wf(Dom,Y,MemRes,WF), rel_compose_with_inf_fun_aux(MemRes,X,Y,T,Dom,Fun,CompRes,WF).
2333
2334		:- block rel_compose_with_inf_fun_aux(-,?,?,?, ?,?,?,?).
2335		rel_compose_with_inf_fun_aux(pred_true,X,Y,T,Dom,Fun,CompRes,WF) :-
2336		apply_to(Fun,Y,FY,WF),
2337		add_element_wf((X,FY),CT,CompRes,WF),
2338		rel_compose_with_inf_fun(T,Dom,Fun,CT,WF).
2339		rel_compose_with_inf_fun_aux(pred_false,_X,_Y,T,Dom,Fun,Comp,WF) :-
2340		rel_compose_with_inf_fun(T,Dom,Fun,Comp,WF).
2341		*/
2342
2343		:- assert_must_abort_wf(bsets_clp:rel_iterate_wf([],int(-1),_R,set(couple(integer,integer)),WF),WF).
2344		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([], int(2),[],set(couple(integer,integer)),_WF))).
2345		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([(int(1),int(2)),(int(3),int(4)),(int(5),int(6))], int(1),[(int(1),int(2)),(int(3),int(4)),(int(5),int(6))],set(couple(integer,integer)),_WF))).
2346		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([(pred_true,pred_true)], int(0),
2347		[(pred_true,pred_true),(pred_false,pred_false)],set(couple(boolean,boolean)),_WF))).
2348		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:rel_iterate_wf([(int(1),int(2)),
2349		(int(2),int(4)),(int(4),int(6))], int(2),[(int(1),int(4)),(int(2),int(6))],
2350		set(couple(integer,integer)),WF),WF)).
2351		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF), R=[],
2352		bsets_clp:equal_object(X,R))).
2353		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF),
2354		R=[(int(1),int(2)),(int(2),int(3))],
2355		bsets_clp:equal_object(X,R))).
2356		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(2),X,set(couple(integer,integer)),_WF),
2357		R=[(int(1),int(2)),(int(2),int(3))],
2358		bsets_clp:equal_object(X,[(int(1),int(3))]))).
2359		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2360		R=[(int(1),int(2)),(int(2),int(3))],
2361		bsets_clp:equal_object(X,[]))).
2362		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2363		R=[(int(1),int(2)),(int(2),int(3)),(int(1),int(1))],
2364		bsets_clp:equal_object(X,[(int(1),int(1)),(int(1),int(2)),(int(1),int(3))]))).
2365
2366		rel_iterate_wf(Rel,int(Nr),Res,Type,WF) :-
2367		opt_push_wait_flag_call_stack_info(WF,b_operator_call(iterate,
2368		[Nr,Rel],unknown),WF2),
2369		rel_iterate1(Nr,Rel,Res,Type,WF2).
2370
2371		:- block rel_iterate1(-,?,?,?,?).
2372		rel_iterate1(X,Rel,Res,Type,WF) :-
2373		%value_variables(Rel,GrV),
2374		rel_iterate2(X,Rel,Res,Type,WF).
2375
2376		rel_iterate2(X,Rel,Res,Type,WF) :-
2377		( X=1 -> equal_object_wf(Res,Rel,rel_iterate2,WF)
2378		; X>1 -> X1 is X-1,
2379		rel_iterate2(X1,Rel,R1,Type,WF),
2380		rel_composition_wf(Rel,R1,Res,Type,WF)
2381		; X=0 -> rel_iterate0(Rel,Type,Res,WF)
2382		; add_wd_error('negative index in iterate',X,WF)
2383		).
2384
2385		:- use_module(bsyntaxtree,[get_set_type/2]).
2386		:- block rel_iterate0(?,-,?,?).
2387		rel_iterate0(_Rel,EType,Res,WF) :-
2388		get_set_type(EType,couple(Type,Type)),
2389		event_b_identity_for_type(Type,Res,WF).
2390
2391		:- use_module(typing_tools,[is_infinite_type/1]).
2392		event_b_identity_for_type(Type,Res,WF) :-
2393		create_texpr(identifier('_zzzz_unary'),Type,[],TIdentifier1), % was [generated]
2394		create_texpr(identifier('_zzzz_binary'),Type,[],TIdentifier2), % was [generated]
2395		(is_infinite_type(Type) -> Info = [prob_annotation('SYMBOLIC')] ; Info =[]),
2396		create_texpr(equal(TIdentifier1,TIdentifier2),pred,Info,TPred),
2397		construct_closure(['_zzzz_unary','_zzzz_binary'],[Type,Type],TPred,CRes),
2398		% for small types we could do: all_objects_of_type(Type,All), identity_relation_over_wf(All,CRes,WF)
2399		%, print(constructed_eventb_identity(Res)),nl
2400		equal_object_wf(Res,CRes,WF).
2401
2402
2403		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([],[(int(1),int(11))],[],_WF))).
2404		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2405		[(int(1),int(11))],[(int(1),(int(2),int(11)))],_WF))).
2406		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2407		[(int(2),int(11))],[],_WF))).
2408		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2409		[(int(2),int(11))],X,_WF),
2410		X = [])).
2411		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2412		[(int(1),int(11))],X,_WF),
2413		kernel_objects:equal_object(X,[(int(1),(int(2),int(11)))]))).
2414		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(1),int(6))],
2415		[(int(1),int(11))],X,_WF),
2416		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(6),int(11)))]))).
2417		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2418		[(int(1),int(11)),(int(1),int(12))],X,_WF),
2419		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))]))).
2420		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2421		[(int(1),int(11)),(int(1),int(12))],
2422		[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))],_WF))).
2423		:- assert_must_succeed((bsets_clp:direct_product_wf(avl_set(node((fd(1,'Name'),fd(2,'Name')),true,1,empty,node((fd(2,'Name'),fd(3,'Name')),true,0,empty,empty))),
2424		avl_set(node((fd(1,'Name'),fd(2,'Name')),true,1,empty,node((fd(2,'Name'),fd(3,'Name')),true,0,empty,empty))),
2425		avl_set(node((fd(1,'Name'),fd(2,'Name'),fd(2,'Name')),true,1,empty,node((fd(2,'Name'),fd(3,'Name'),fd(3,'Name')),true,0,empty,empty)))
2426		,_WF))).
2427
2428		:- block direct_product_wf(-,?,?,?),direct_product_wf(?,-,?,?).
2429		direct_product_wf(Rel1,Rel2,Prod,WF) :-
2430		try_expand_and_convert_to_avl_with_check(Rel1,E1,direct_product), % to do: try_expand_and_convert_to_avl_unless_large_wf(Rel1,E1,WF),
2431		try_expand_and_convert_to_avl_with_check(Rel2,E2,direct_product),
2432	?	direct_product_wf1(E1,E2,Prod,WF).
2433
2434		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2435		direct_product_explicit_set(Rel1,Rel2,Res),!,
2436		equal_object_wf(Prod,Res,direct_product_wf1,WF).
2437		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2438		expand_custom_set_to_list_wf(Rel1,Relation1,_,direct_product_wf1_1,WF),
2439		expand_custom_set_to_list_wf(Rel2,Relation2,_,direct_product_wf1_2,WF),
2440	?	direct_product2(Relation1,Relation2,Prod,WF),
2441	?	direct_product_backwards(Relation1,Relation2,Prod,WF).
2442
2443		:- block direct_product2(-,?,?,?).
2444		direct_product2([],_,Out,WF) :- equal_object_wf(Out,[],direct_product2,WF).
2445		direct_product2([(X,Y)|T],Rel2,Out,WF) :-
2446	?	direct_product_tuple(Rel2,X,Y,Out,OutRem,WF),
2447		direct_product2(T,Rel2,OutRem,WF).
2448
2449		:- block direct_product_tuple(-,?,?,?,?,?).
2450		direct_product_tuple([],_,_,Res,Rem,WF) :- equal_object_optimized_wf(Res,Rem,direct_product_tuple,WF).
2451		direct_product_tuple([(X2,Z)|T],X,Y,Res,Rem,WF) :-
2452		direct_product_tuple(T,X,Y,CT,Rem,WF),
2453		equality_objects_wf(X2,X,EqRes,WF),
2454	?	direct_product_tuple3(EqRes,X,Y,Z,CT,Res,WF).
2455
2456		:- block direct_product_tuple3(-,?,?,?,?,?,?).
2457		direct_product_tuple3(pred_true,X,Y,Z,CT,Res,WF) :-
2458	?	equal_cons_wf(Res,(X,(Y,Z)),CT,WF). /* no need for add_element as output uniquely determines X,Y,Z !?*/
2459		direct_product_tuple3(pred_false,_X,_Y,_Z,CT,Res,WF) :- equal_object_optimized_wf(Res,CT,direct_product_tuple3,WF).
2460
2461		:- block direct_product_backwards(?,?,-,?).
2462		% Propagate information backwards from result to arguments
2463		direct_product_backwards(R1,R2,Prod,WF) :-
2464		((ground_value(R1) ; ground_value(R2)) -> true
2465		; expand_custom_set_to_list_wf(Prod,ProdList,_,direct_product_backwards,WF),
2466	?	direct_product_propagate_back(ProdList,R1,R2,WF)
2467		).
2468
2469		:- block direct_product_propagate_back(-,?,?,?).
2470		direct_product_propagate_back([],_,_,_WF).
2471		direct_product_propagate_back([(X,(Y,Z))|T],R1,R2,WF) :-
2472	?	check_element_of_wf((X,Y),R1,WF), check_element_of_wf((X,Z),R2,WF),
2473		direct_product_propagate_back(T,R1,R2,WF).
2474
2475		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([],[(int(3),int(4))],[]))).
2476		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([(int(1),int(2))],
2477		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2478		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2479		[(int(3),int(4))],X), ground(X),
2480		equal_object(X,[((int(1),int(3)),(int(2),int(4)))]))).
2481		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2482		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2483		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))], [],X),X == [])).
2484		:- assert_must_succeed((bsets_clp:parallel_product([], [(int(3),int(4))],X),X == [])).
2485
2486	?	parallel_product(Rel1,Rel2,Prod) :- parallel_product_wf(Rel1,Rel2,Prod,no_wf_available).
2487
2488		:- block parallel_product_wf(-,?,?,?),parallel_product_wf(?,-,?,?).
2489		% NOTE: we now have in_parallel_product; as such parallel products are kept symbolic
2490		%parallel_product_wf(Rel1,Rel2,Prod,WF) :- (keep_symbolic(Rel1) -> true ; keep_symbolic(Rel2)),
2491		% print_term_summary(parallel_product(Rel1,Rel2,Prod)),nl,
2492		%% % TO DO: generate closure
2493		% %{xy,mn|#(x,y,m,n).(xy=(x,y) & mn=(m,n) & (x,m):S & (y,n):R)}
2494		% fail.
2495		parallel_product_wf(Rel1,Rel2,Prod,WF) :-
2496		expand_custom_set_to_list_wf(Rel1,Relation1,_,parallel_product_1,WF),
2497		expand_custom_set_to_list_wf(Rel2,Relation2,_,parallel_product_2,WF),
2498		parallel_product2(Relation1,Relation2,ProdRes,WF),
2499	?	equal_object_optimized_wf(ProdRes,Prod,parallel_product,WF).
2500
2501		:- use_module(kernel_equality,[conjoin_test/4]).
2502		%(Rel1||Rel2) = {(x,y),(m,n)| (x,m):Rel1 & (y,n):Rel2}
2503
2504		% TO DO: use this in b_interpreter_check:
2505		in_parallel_product_test(((X,Y),(M,N)),Rel1,Rel2,Result,WF) :-
2506	?	conjoin_test(MemRes1,MemRes2,Result,WF),
2507	?	membership_test_wf(Rel1,(X,M),MemRes1,WF),
2508		membership_test_wf(Rel2,(Y,N),MemRes2,WF).
2509
2510		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_parallel_product_wf(((int(1),int(2)),(int(11),int(22))),[(int(1),int(11))],[(int(2),int(22))],WF),WF)).
2511
2512		in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2513		in_parallel_product_test(El,Rel1,Rel2,pred_true,WF).
2514
2515
2516		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_in_parallel_product_wf(((int(1),int(11)),(int(2),int(22))),[(int(1),int(11))],[(int(2),int(22))],_WF))).
2517
2518		not_in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2519	?	in_parallel_product_test(El,Rel1,Rel2,pred_false,WF).
2520
2521
2522		:- block parallel_product2(-,?,?,?).
2523		parallel_product2([],_,Out,WF) :- empty_set_wf(Out,WF).
2524		parallel_product2([(X,Y)|T],Rel2,Out,WF) :-
2525		parallel_product_tuple(Rel2,X,Y,Out,Tail,WF),
2526		parallel_product2(T,Rel2,Tail,WF).
2527
2528		:- block parallel_product_tuple(-,?,?,?,?,?).
2529		parallel_product_tuple([],_,_,Tail1,Tail2,WF) :- equal_object_wf(Tail1,Tail2,parallel_product_tuple,WF).
2530		parallel_product_tuple([(X2,Y2)|T],X,Y,Rel2,Tail,WF) :-
2531		equal_object_wf(Rel2,[((X,X2),(Y,Y2))|RT],parallel_product_tuple,WF),
2532		parallel_product_tuple(T,X,Y,RT,Tail,WF).
2533
2534
2535		% -------------------------------------------------
2536
2537		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6)),(int(2),int(7))],[int(1)],[int(7),int(6)],WF),WF)). %% with wf_det leads to residue custom_explicit_sets:b_not_test_closure_enum
2538		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(8),int(6)],WF),WF)).
2539		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2540		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6)),(int(1),int(7))],[int(1)],[int(7),int(6)],WF),WF)).
2541		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2542		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_partial_function([(int(1),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2543		:- assert_must_fail((bsets_clp:not_partial_function([],[int(1)],[int(7)],_WF))).
2544		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[int(7)],_WF),
2545		X = [(int(1),int(7))])).
2546		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7)],_WF),
2547		X = [(int(2),int(7)),(int(1),int(7))])).
2548		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2549		[int(7),int(6)],_WF),
2550		X = [([(int(1),int(2))],int(7)),
2551		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2552		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2553		[int(7),int(6)],_WF),
2554		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2555		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2556		[int(7),int(6)],_WF),
2557		X = [([(int(1),int(2))],int(7)),
2558		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2559		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[[int(7),int(6)]],_WF),
2560		X = [(int(1),[int(6),int(7)])])).
2561		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2562		X = [(int(2),int(7)),(int(1),int(7))])).
2563		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2564		X = [(int(2),int(7)),(int(2),int(6))])).
2565		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2566		X = [(int(2),int(7)),(int(1),int(2))])).
2567		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2568		X = [(int(2),int(7)),(int(3),int(6))])).
2569		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2570		X = [(int(2),int(7)),(int(2),int(5))])).
2571		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2572		X = [(int(1),int(7)),(int(2),int(6)),(int(2),int(7))])).
2573		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2574		X = [(int(1),int(7)),(int(5),int(75))])).
2575		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2576		X = [(int(1),int(7)),(int(0),int(7))])).
2577		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2578		X = [(int(1),int(7)),(int(-1),int(7))])).
2579		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2580		X = [(int(1),int(7)),(int(0),int(7))])).
2581		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('Name'),global_set('Code'),_WF),
2582		X = [(fd(1,'Name'),fd(1,'Code'))])).
2583		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2584		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2585		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2586		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2587		:- assert_must_succeed((bsets_clp:not_partial_function([(fd(1,'Code'),int(1)),(fd(1,'Code'),int(2))],
2588		global_set('Code'),global_set('NAT1'),_WF) )).
2589
2590		:- block not_partial_function(-,?,?,?).
2591		not_partial_function([],_Domain,_Range,_WF) :- !,fail.
2592		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
2593		% we do not need the Range; this means we can match more closures (e.g., lambda)
2594		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2595		not_subset_of_wf(FFDomain,Domain,WF).
2596		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),
2597		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2598		not_both_subset_of(FFDomain,FFRange,Domain,Range,WF).
2599		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2600		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
2601		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2602		get_range_id_expression(P,T,TRangeID),!,
2603		subset_test(FFDomain,Domain,SubRes,WF),
2604		when(nonvar(SubRes),
2605		(SubRes=pred_false -> true % not a subset -> it is not a partial function over the domain
2606		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
2607		not_partial_function(R,Domain,Range,WF) :-
2608		expand_and_convert_to_avl_set_warn(R,AER,not_partial_function,'ARG /: ? +-> ?',WF),!,
2609		% TO DO: expand_and_convert_to_avl_set_catch and provide symbolic treatment similar to partial_function
2610		% e.g., to support f = NATURAL1 * {22,33} & not(f: NATURAL1 +-> NATURAL)
2611		is_not_avl_partial_function(AER,Domain,Range,WF).
2612		not_partial_function(R,Domain,Range,WF) :-
2613		expand_custom_set_to_list_wf(R,ER,_,not_partial_function,WF),
2614		not_pf(ER,[],Domain,Range,WF).
2615
2616		is_not_avl_partial_function(AER,Domain,Range,WF) :-
2617		(is_avl_partial_function(AER)
2618		-> is_not_avl_relation_over_domain_range(AER,Domain,Range,WF)
2619		; true
2620		).
2621
2622		:- block not_pf(-,?,?,?,?).
2623		not_pf([],_,_,_,_) :- fail.
2624		not_pf([(X,Y)|T],SoFar,Dom,Ran,WF) :-
2625		membership_test_wf_with_force(SoFar,X,MemRes,WF),
2626		not_pf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2627
2628		:- block not_pf2(-,?,?,?,?,?,?,?).
2629		not_pf2(pred_true,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* then not a function */
2630		not_pf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
2631		membership_test_wf_with_force(Dom,X,MemRes,WF), % creates a choice point in SMT mode
2632		not_pf2a(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2633
2634		:- block not_pf2a(-,?,?,?,?,?,?,?).
2635		not_pf2a(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* function, but domain wrong */
2636		not_pf2a(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
2637		remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,_LWF,Done), %% provide _LWF ??
2638		not_pf2b(Done,X,Y,T,SoFar,Dom2,Ran,WF).
2639
2640		:- block not_pf2b(-, ?,?,?, ?,?,?, ?).
2641		not_pf2b(_Done, X,Y,T, SoFar,Dom2,Ran, WF) :-
2642		add_element_wf(X,SoFar,SoFar2,WF),
2643		(T==[] -> not_element_of_wf(Y,Ran,WF)
2644		; membership_test_wf_with_force(Ran,Y,MemRes,WF),
2645		prop_empty_pred_false(T,MemRes), % if T=[] -> Y must not be in Ran
2646		not_pf3(MemRes,T,SoFar2,Dom2,Ran,WF)).
2647
2648		:- block prop_empty_pred_false(-,?).
2649		prop_empty_pred_false([],R) :- !, R=pred_false.
2650		prop_empty_pred_false(_,_).
2651
2652		:- block not_pf3(-,?,?,?,?,?).
2653		not_pf3(pred_false,_T,_SoFar,_Dom2,_Ran,_WF). /* illegal range */
2654		not_pf3(pred_true,T,SoFar,Dom2,Ran,WF) :-
2655		not_pf(T,SoFar,Dom2,Ran,WF).
2656
2657		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2658		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_wf([(int(1),int(1)),(int(2),int(1))],global_set('NATURAL'),global_set('NATURAL'),WF),WF)).
2659		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_wf([(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2660		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:partial_function_wf([(int(2),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2661		:- assert_must_succeed((bsets_clp:partial_function([],[int(1)],[int(7)]))).
2662		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1)],[int(7)]),
2663		X = [(int(1),int(7))])).
2664		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),
2665		equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2666		:- assert_must_succeed((findall(X,bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),L),
2667		length(L,Len), Len >= 4,
2668		(preferences:get_preference(convert_comprehension_sets_into_closures,true) -> true ; Len=4) )).
2669		:- assert_must_succeed((bsets_clp:partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2670		[int(7),int(6)]),
2671		equal_object(X,[([(int(1),int(2))],int(7)),
2672		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
2673		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2674		[int(7),int(6)],_WF),
2675		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2676		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2677		[int(7),int(6)],_WF),
2678		X = [([(int(1),int(2))],int(7)),
2679		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2680		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[int(1)],[[int(7),int(6)]],_WF),
2681		X = [(int(1),[int(6),int(7)])])).
2682		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2683		X = [(int(1),int(7)),(int(5),int(75))])).
2684		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2685		X = [(int(1),int(7)),(int(0),int(7))])).
2686		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2687		X = [(int(1),int(7)),(int(-1),int(7))])).
2688		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2689		X = [(int(1),int(7)),(int(0),int(7))])).
2690		:- assert_must_fail((bsets_clp:partial_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
2691		X = [(int(2),int(7)),(int(2),int(6))])).
2692		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('Name'),global_set('Code'),_WF),
2693		X = [(fd(1,'Name'),fd(1,'Code'))])).
2694		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2695		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2696		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2697		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2698
2699		partial_function(R,Domain,Range) :- init_wait_flags(WF,[partial_function]),
2700		partial_function_wf(R,Domain,Range,WF),
2701	?	ground_wait_flags(WF).
2702
2703		:- use_module(kernel_equality,[get_cardinality_powset_wait_flag/5]).
2704		:- use_module(closures,[is_lambda_value_domain_closure/5]).
2705		:- block partial_function_wf(-,-,?,?).
2706		partial_function_wf(R,_Domain,_Range,_WF) :- R==[], !.
2707		partial_function_wf(R,Domain,Range,WF) :- (Domain==[] ; Range==[]), !, empty_set_wf(R,WF).
2708		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2709		custom_explicit_sets:is_definitely_maximal_set(Range),
2710		% we do not need the Range; this means we can match more closures (e.g., lambda)
2711		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2712		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF).
2713		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2714		% TODO: this will fail if is_definitely_maximal_set was true above !
2715		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2716		% same as for total_function_wf check
2717		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2718		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
2719		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2720		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2721		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2722		get_range_id_expression(P,T,TRangeID),
2723		!,
2724		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2725		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2726		[b_operator(range,[FF]),Range],unknown),WF3),
2727		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF3). % we could use symbolic_range_subset_check
2728		partial_function_wf(R,Domain,Range,WF) :-
2729		expand_and_convert_to_avl_set_catch(R,AER,partial_function_wf,'ARG : ? +-> ?',ResultStatus,WF),!,
2730		(ResultStatus=avl_set
2731	?	-> is_avl_partial_function_over(AER,Domain,Range,WF)
2732		; % keep symbolic
2733		(debug_mode(off) -> true ; print('SYMBOLIC +-> check : '),translate:print_bvalue(R),nl),
2734		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
2735		symbolic_domain_subset_check(R,Domain,WF),
2736		symbolic_range_subset_check(R,Range,WF),
2737		symbolic_functionality_check(R,WF)
2738		).
2739		partial_function_wf(R,Domain,Range,WF) :-
2740		get_cardinality_powset_wait_flag(Domain,partial_function_wf,WF,Card,CWF),
2741		% probably we should compute real cardinality of set of partial functions over Domain +-> Range ?
2742		% the powset waitflag uses 2^Card as priority; is the number of partial functions when Range contains just a single element
2743		% slows down test 1088: TO DO investigate
2744		% get_cardinality_partial_function_wait_flag(Domain,Range,partial_function_wf,WF,Card,_,CWF),
2745		%% Maybe we should only enumerate partial functions for domain variables ; e.g., not f <+ {x |-> y} : T +-> S
2746		%% print_bt_message(pf_dom_card(Card)),nl, %%%
2747		% probably we should use a special version when R is var
2748		propagate_empty_set_wf(Domain,dom_pf,R,WF),
2749		propagate_empty_set_wf(Range,ran_pf,R,WF),
2750	?	(var(R) -> pf_var_r(R,var,Domain,Range,Card,WF,CWF) ; pf_var_r(R,nonvar,Domain,Range,Card,WF,CWF)).
2751
2752		% symbolic dom(R) <: Domain check for closures
2753		symbolic_domain_subset_check(R,Domain,WF) :-
2754		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2755		[b_operator(domain,[R]),Domain],unknown),WF2),
2756		domain_subtraction_wf(Domain,R,Res,WF2), % works symbolically
2757		(debug_mode(off) -> true ; print('Domain Violations: '),translate:print_bvalue(Res),nl),
2758		empty_set_wf(Res,WF2). % empty_set does a symbolic treatment calling gen_typed_ids and b_not_test_exists:
2759		% symbolic ran(R) <: Range check for closures
2760		symbolic_range_subset_check(R,Range,WF) :-
2761		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2762		[b_operator(range,[R]),Range],unknown),WF2),
2763		range_subtraction_wf(R,Range,Res,WF2), % works symbolically
2764		(debug_mode(off) -> true ; print('Range Violations: '),translate:print_bvalue(Res),nl),
2765		empty_set_wf(Res,WF2). % works symbolically
2766		symbolic_functionality_check(Closure,WF) :-
2767		custom_explicit_sets:symbolic_functionality_check_closure(Closure,ViolationsClosure),!,
2768		(debug_mode(off) -> true ; print('FUNCTIONALITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2769		empty_set_wf(ViolationsClosure,WF). % works symbolically
2770		symbolic_functionality_check(R,WF) :-
2771		add_error_wf(symbolic_functionality_check,'Could not check functionality of:',R,R,WF).
2772
2773		symbolic_injectivity_check(Closure,WF) :-
2774		custom_explicit_sets:symbolic_injectivity_check_closure(Closure,ViolationsClosure),!,
2775		(debug_mode(off) -> true ; print('INJECTIVITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2776		empty_set_wf(ViolationsClosure,WF). % works symbolically
2777		symbolic_injectivity_check(R,WF) :-
2778		add_error_wf(symbolic_functionality_check,'Could not check injectivity of:',R,R,WF).
2779
2780
2781		is_avl_partial_function_over(AER,Domain,Range,WF) :-
2782		is_avl_partial_function(AER),
2783	?	is_avl_relation_over_domain(AER,Domain,WF),
2784	?	is_avl_relation_over_range(AER,Range,WF).
2785
2786		% symbolically check that the range of lambda closure is a subset of a given Range
2787		% TRangeID is obtained by calling get_range_id_expression(P,T,TRangeID)
2788		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2789		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2790		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2791		% CHECK not(#P.(Pred & TRangeID /: Range))
2792		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2793		is_empty_closure_wf(P,T,Pred2,WF2). % do we need to rename _lambda_result_ using rename_lambda_result_id ?
2794		% now the negation thereof:
2795		check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2796		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
2797		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2798		% CHECK (#P.(Pred & TRangeID /: Range))
2799		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2800		is_non_empty_closure_wf(P,T,Pred2,WF2).
2801		test_lambda_closure_range(P,T,Pred,TRangeID,Range,Res,WF) :-
2802		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset, % it is actually a reify check
2803		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2804		% reify not(#P.(Pred & TRangeID /: Range))
2805		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2806		test_empty_closure_wf(P,T,Pred2,Res,WF2).
2807
2808		get_not_in_range_pred_aux(Pred,TRangeID,Range,NewPred) :- % construct (Pred & TRangeID /: Range)
2809		ExpectedRange = b(value(Range),set(RanT),[]),
2810		get_texpr_type(TRangeID,RanT),
2811		safe_create_texpr(not_member(TRangeID,ExpectedRange),pred,NotMemCheck),
2812		conjunct_predicates([Pred,NotMemCheck],NewPred).
2813
2814
2815		% if first argument is empty, second argument must also be empty
2816		:- block propagate_empty_set_wf(-,?,?,?).
2817		propagate_empty_set_wf([],_PP,A,WF) :- !, %print(prop_empty(_PP,A)),nl,
2818		kernel_objects:empty_set_wf(A,WF). % TO DO: add WF
2819		propagate_empty_set_wf(_,_,_,_).
2820
2821		:- block pf_var_r(-,?,?,?,?,?,-).
2822		pf_var_r(R,var,Domain,Range,_Card,WF,_CWF) :- % if R was var: see if it is now an AVL set; otherwise we have already checked it
2823		expand_and_convert_to_avl_set_warn(R,AER,pf_var_r,'ARG : ? +-> ?',WF),!,
2824	?	is_avl_partial_function_over(AER,Domain,Range,WF).
2825		pf_var_r(R,_,Domain,Range,Card,WF,CWF) :-
2826		expand_custom_set_to_list_wf(R,ER,_,partial_function_wf,WF),
2827		%get_last_wait_flag(partial_fun(Domain),WF,LWF),
2828	?	pf_w(ER,[],Domain,Range,Card,_Large,WF,CWF).
2829
2830		pf_w(T,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2831		(Card==0 -> T=[]
2832	?	; pf(T,SoFar,Dom,Ran,Card,Large,WF,LWF)).
2833
2834		:- block pf(-,?,?,?,?,?,?,-).
2835		pf(LIST,_,_,_,_,_WF,_,_LWF) :- LIST==[],!. % avoid leaving choicepoint
2836		pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF) :- nonvar(AVL),AVL=avl_set(_A),
2837		add_internal_error('AVL arg: ',pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF)),fail.
2838		pf([],_,_,_,_,_WF,_,_LWF).
2839		pf(LIST,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2840		(var(LIST) -> ListWasVar = true ; ListWasVar = false), % is ListWasVar = true we are doing the enumeration driven by LWF being ground
2841		LIST = [(X,Y)|T],
2842		dec_card(Card,NC),/* Card ensures we do not build too big lists */
2843		Dom \== [],
2844	?	remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done),
2845	?	check_element_of_wf(Y,Ran,WF),
2846	?	pf1(Done, X,Y,T,SoFar,Dom2,Ran,NC,Large,WF,LWF).
2847
2848		:- block dec_card(-,?).
2849		dec_card(inf,NewC) :- !, NewC=inf.
2850		dec_card(inf_overflow,NewC) :- !, NewC=inf_overflow.
2851		dec_card(C,NewC) :- C>0, NewC is C-1.
2852
2853		:- block pf1(-, ?,?,?,?,?,?,?,?,?,?).
2854		pf1(_Done, X,_Y,T,SoFar,Dom2,Ran,Card,Large,WF,LWF) :-
2855		not_element_of_wf(X,SoFar,WF), /* check that it is a function */
2856		%% check_element_of_wf(Y,Ran,WF), % this check is now done above in pf
2857		add_new_element_wf(X,SoFar,SoFar2,WF),
2858	?	pf_w(T,SoFar2,Dom2,Ran,Card,Large,WF,LWF).
2859
2860		remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done) :- compute_large(Dom,Large),
2861		((ListWasVar==true,var(X),var(Y),Large==false,
2862		preference(convert_comprehension_sets_into_closures,false), % not in symbolic mode
2863		ground_value(Dom))
2864		-> %% (X, Y are free and we drive the enumeration: we can influence which element is taken from Dom
2865		remove_a_minimal_element(X,Dom,Dom2,WF,Done) %%%%%%%%%% added Jul 15 2008
2866	?	; remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,LWF,Done)
2867		).
2868		compute_large(Dom,Large) :- % check if the domain is large; ensure that we compute this only once
2869		(nonvar(Large) -> true
2870		; var(Dom) -> true
2871		; dont_expand_this_explicit_set(Dom) -> Large=large
2872		; Large=false).
2873
2874		:- assert_must_succeed(( bsets_clp:remove_a_minimal_element(X,[int(1)],R,_WF,Done),
2875		X==int(1), Done==true, R=[] )).
2876		:- assert_must_succeed(( init_wait_flags(WF), bsets_clp:remove_a_minimal_element(X,[int(1),int(2),int(3)],R,WF,Done), ground_wait_flags(WF),
2877		X==int(2), Done==true, R=[int(3)] )).
2878		:- assert_must_succeed(( init_wait_flags(WF), bsets_clp:remove_a_minimal_element(X,[int(1),int(2),int(3)],R,WF,Done), ground_wait_flags(WF),
2879		X==int(1), R=[int(2),int(3)], Done==true )).
2880		:- assert_must_succeed(( init_wait_flags(WF), bsets_clp:remove_a_minimal_element(X,[int(1),int(2),int(3)],R,WF,Done), ground_wait_flags(WF),
2881		X==int(3), R=[], Done==true )).
2882		:- assert_must_succeed(( init_wait_flags(WF), CL=closure(['_zzzz_binary'],[integer],b(member( b(identifier('_zzzz_binary'),integer,[]),
2883		b(interval(b(value(int(1)),integer,[]),b(value(int(10)),integer,[])),set(integer),[])),pred,[])),
2884		bsets_clp:remove_a_minimal_element(X,CL,R,WF,Done), ground_wait_flags(WF),
2885		X=int(9), Done==true, kernel_objects:equal_object(R,[int(10)]) )).
2886
2887		/* usage: restrict number of possible choices if element to remove is free */
2888		/* select one element; and disallow all elements appearing before it in the list */
2889		remove_a_minimal_element(X,Set,Res,WF,Done) :-
2890		expand_custom_set_to_list_wf(Set,ESet,EDone,remove_a_minimal_element,WF),
2891		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done).
2892
2893		:- use_module(kernel_equality,[get_cardinality_wait_flag/4]).
2894		:- block remove_a_minimal_element2(?,?,-,?,?,?).
2895		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done) :- var(ESet),
2896		% should not happen as we wait for EDone
2897		add_internal_error('Illegal call: ',remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done)),
2898		fail.
2899		remove_a_minimal_element2(X,ESet,_EDone,Res,WF,Done) :-
2900		ESet \= [],
2901		(ESet = [El]
2902		-> X=El, empty_set_wf(Res,WF), Done=true % only one choice
2903		; get_cardinality_wait_flag(ESet,remove_a_minimal_element2,WF,CWF),
2904		remove_a_minimal_element3(X,ESet,Res,WF,Done,CWF)
2905		).
2906
2907		:- block remove_a_minimal_element3(?,?,?,?,?,-).
2908		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- var(Res), !,
2909		append(_,[X|TRes],ESet), % WHAT IF Res has been instantiated in the meantime ???
2910		equal_object_wf(Res,TRes,remove_a_minimal_element2_2,WF),Done=true.
2911		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- %print(remove_min_nonvar_res(Res)),nl,
2912		equal_cons_wf(ESet,X,Res,WF), Done=true.
2913
2914
2915		% reified version of partial function test partial_function_wf:
2916		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([],[int(1),int(2)],[int(7),int(6)],pred_true,WF),WF)).
2917		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],pred_true,WF),WF)).
2918		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],pred_true,WF),WF)).
2919		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([(int(2),int(8))],[int(1),int(2)],[int(7),int(6)],pred_false,WF),WF)).
2920		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([(int(3),int(7))],[int(1),int(2)],[int(7),int(6)],pred_false,WF),WF)).
2921		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_function_test_wf([(int(1),int(7)),(int(1),int(6))],[int(1),int(2)],[int(7),int(6)],pred_false,WF),WF)).
2922
2923		:- use_module(kernel_equality,[subset_test/4]).
2924		:- block partial_function_test_wf(-,?,?,-,?), partial_function_test_wf(?,-,-,-,?).
2925		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_true,!,
2926	?	partial_function_wf(FF,Domain,Range,WF).
2927		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_false,!,
2928		not_partial_function(FF,Domain,Range,WF). % TO DO: remove not_partial_function to use check_is_partial_function?
2929		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2930		custom_explicit_sets:is_definitely_maximal_set(Range),
2931		% we do not need the Range; this means we can match more closures (e.g., lambda)
2932		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2933		subset_test(FFDomain,Domain,Res,WF).
2934		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2935		% TODO: this will fail if is_definitely_maximal_set was true above !
2936		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2937		% same as for total_function_wf check
2938		subset_test(FFDomain,Domain,DomainOk,WF),
2939		(DomainOk==pred_false -> Res = pred_false
2940		; conjoin_test(DomainOk,RangeOk,Res,WF),
2941		subset_test(FFRange,Range,RangeOk,WF)).
2942		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2943		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2944		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2945		get_range_id_expression(P,T,TRangeID),
2946		!,
2947		subset_test(FFDomain,Domain,DomainOk,WF),
2948		(DomainOk == pred_false -> Res=pred_false
2949		; conjoin_test(DomainOk,RangeOk,Res,WF),
2950		test_lambda_closure_range(P,T,Pred,TRangeID,Range,RangeOk,WF)
2951		).
2952		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2953		expand_and_convert_to_avl_set_warn(R,AER,partial_function_test_wf,'ARG : ? +-> ?',WF),!,
2954		% TO DO: use expand_and_convert_to_avl_set_catch
2955		(is_avl_partial_function(AER)
2956		-> % TO DO: we could do something similar to this instead: is_not_avl_relation_over_domain_range
2957		domain_of_explicit_set_wf(avl_set(AER),FFDomain,WF),
2958		subset_test(FFDomain,Domain,DomainOk,WF),
2959		(DomainOk == pred_false -> Res=pred_false
2960		; range_of_explicit_set_wf(avl_set(AER),FFRange,WF),
2961		conjoin_test(DomainOk,RangeOk,Res,WF),
2962		subset_test(FFRange,Range,RangeOk,WF)
2963		)
2964		; Res=pred_false).
2965		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2966		expand_custom_set_to_list_wf(R,ER,_,partial_function_test_wf,WF),
2967		check_is_partial_function_acc_wf(ER,[],Domain,Range,Res,WF).
2968
2969		:- block check_is_partial_function_acc_wf(-,?,?,?,?,?).
2970		check_is_partial_function_acc_wf([],_,_,_,Res,_WF) :- !, Res=pred_true.
2971		check_is_partial_function_acc_wf([(A,FA)|T],Acc,Dom,Ran,Res,WF) :- !,
2972		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF),
2973		(MemResDomRan==pred_false
2974		-> Res = pred_false
2975		; membership_test_wf(Acc,A,MemResNotFunc,WF),
2976		negate(MemResNotFunc,MemResFunctionality),
2977		conjoin_test(MemResDomRan,MemResFunctionality,PF_Head,WF),
2978		(PF_Head == pred_false -> Res = pred_false
2979		; T==[] -> Res=PF_Head
2980		; add_element_wf(A,Acc,NewAcc,WF),
2981		conjoin_test(PF_Head,PF_Tail,Res,WF),
2982		check_is_partial_function_acc_wf(T,NewAcc,Dom,Ran,PF_Tail,WF))
2983		).
2984
2985		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF) :-
2986		membership_test_wf(Dom, A,MemResDom,WF), % use membership_test_wf_with_force for SMT mode ??
2987		(MemResDom == pred_false -> MemResDomRan = pred_false
2988		; membership_test_wf(Ran,FA,MemResRan,WF),
2989		conjoin_test(MemResDom,MemResRan,MemResDomRan,WF)).
2990
2991		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:total_function_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
2992		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[int(7)]),
2993		X = [(int(1),int(7))])).
2994		:- assert_must_succeed((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
2995		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2996		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3))]],[int(7),int(6)]),
2997		kernel_objects:equal_object(X,[([(int(1),int(3))],int(7)),([(int(1),int(2))],int(7))]))).
2998		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2999		[int(7),int(6)]),
3000		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
3001		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
3002		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[[int(7),int(6)]]),
3003		kernel_objects:equal_object(X,[(int(1),[int(6),int(7)])]))).
3004		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3005		[[int(7),int(6)]]),
3006		kernel_objects:equal_object(X,[([(int(1),int(2))],[int(6),int(7)]),
3007		([(int(2),int(3)),(int(1),int(3))],[int(6),int(7)])]))).
3008		:- assert_must_succeed((bsets_clp:total_function(X,[ [(int(1),int(3)),(int(2),int(3))]],
3009		[int(6)]),
3010		kernel_objects:equal_object(X,[ ([(int(2),int(3)),(int(1),int(3))], int(6)) ]))).
3011		:- assert_must_succeed((bsets_clp:total_function(X,global_set('Name'),
3012		[[],[fd(1,'Code'),fd(2,'Code')],[fd(1,'Code')],[fd(2,'Code')]]),
3013		kernel_objects:enumerate_basic_type(X,set(couple(global('Name'),set(global('Code'))))),
3014		kernel_objects:equal_object(X,[(fd(3,'Name'),[fd(2,'Code')]),(fd(1,'Name'),[fd(2,'Code')]),(fd(2,'Name'),[])]))).
3015
3016		%:- assert_must_succeed(( kernel_waitflags:init_wait_flags(WF),bsets_clp:total_function_wf(TF,global_set('Code'),
3017		% closure([zzzz],[set(set(couple(integer,boolean)))],
3018		% member(identifier(zzzz),
3019		% pow_subset(value(closure([zzzz],[set(couple(integer,boolean))],
3020		% member('ListExpression'(['Identifier'(zzzz)]),
3021		% 'Seq'(value([pred_true /* bool_true */,pred_false /* bool_false */])))))))),WF),
3022		% kernel_objects:equal_object(TF,[ (fd(1,'Code'), [[],[(int(1),pred_true /* bool_true */)],[(int(1),pred_true /* bool_true */),(int(2),pred_true /* bool_true */)]]),
3023		% (fd(2,'Code'), [[],[(int(1),pred_true /* bool_true */)],[(int(1),pred_true /* bool_true */),(int(2),pred_true /* bool_true */)]]) ]),
3024		% kernel_waitflags:ground_wait_flags(WF) )).
3025
3026		:- assert_must_succeed((bsets_clp:total_function([],[],[int(7)]))).
3027
3028		:- assert_must_fail((bsets_clp:total_function([],[int(1)],[int(7)]))).
3029		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3030		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
3031		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3032		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(5))]))).
3033		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3034		kernel_objects:equal_object(X,[(int(2),int(7))]))).
3035		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3036		[int(7),int(6)]),
3037		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
3038		([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3039		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3040		[int(7),int(6)]),
3041		kernel_objects:equal_object(X,[([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3042
3043		total_function(R,Domain,Range) :- init_wait_flags(WF,[total_function]),
3044		total_function_wf(R,Domain,Range,WF),
3045	?	ground_wait_flags(WF).
3046
3047
3048		:- assert_must_succeed((bsets_clp:total_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
3049		nonvar(X),X=[(A,B),(C,D)],A==int(1),C==int(2),\+ ground(B),\+ ground(D), B=int(7),D=int(7) )).
3050
3051		:- block total_function_wf(-,-,-,?).
3052		total_function_wf(FF,Domain,_Range,WF) :- FF == [],!,
3053		empty_set_wf(Domain,WF).
3054		total_function_wf(FF,Domain,Range,WF) :-
3055		Range == [],!,
3056		empty_set_wf(FF,WF), empty_set_wf(Domain,WF).
3057		total_function_wf(FF,Domain,Range,WF) :-
3058		% TO DO: if FF or Domain nonvar but \= [] -> check if other variable becomes []
3059	?	total_function_wf1(FF,Domain,Range,WF).
3060
3061		:- block total_function_wf1(?,-,?,?).
3062		total_function_wf1(FF,Domain,_Range,WF) :-
3063		FF==[],!,
3064		empty_set_wf(Domain,WF).
3065		total_function_wf1(FF,Domain,Range,WF) :-
3066		custom_explicit_sets:is_definitely_maximal_set(Range),
3067		% we do not need the Range; this means we can match more closures (e.g., lambda)
3068		(nonvar(FF),
3069		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF)
3070		-> !,
3071		equal_object_wf(FFDomain,Domain,total_function_wf1_1,WF)
3072		; var(FF),
3073		get_wait_flag1(WF,WF1), var(WF1),
3074		\+ (custom_explicit_sets:get_card_for_specific_custom_set(Domain,Card), number(Card)),
3075		% we have a total_function over a possibly infinite domain,
3076		% better wait: maybe a recursive of other closure will be produced for FF
3077		!,
3078		when( (nonvar(FF) ; nonvar(WF1)), total_function_wf1(FF,Domain,Range,WF))
3079		).
3080		total_function_wf1(FF,Domain,Range,WF) :- nonvar(FF),
3081		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3082		equal_object_wf(FFDomain,Domain,total_function_wf1_2,WF),
3083		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
3084		total_function_wf1(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3085		total_function_avl_set(AEF,Domain,Range,WF).
3086		total_function_wf1(FF,Domain,Range,WF) :-
3087		% want to replace FF by closure: needs to be a variable!
3088		var(FF),
3089		% if the total function can not be build up explicitly (i.e. infinite domain)
3090		% TODO: can / should this be relaxed?
3091		custom_explicit_sets:is_infinite_explicit_set(Domain), % get_card_for_specific_custom_set or is_infinite_or_symbolic_closure
3092		% TO DO: delay if Domain infinite or closure and not yet known and range is type
3093		kernel_objects:infer_value_type(Domain,set(DomT)),
3094		kernel_objects:infer_value_type(Range,set(RanT)),
3095		!,
3096		% IDEA : TF = %x.(x:Domain|DEFAULT) <+ SFF, where SFF is partial function and DEFAULT is some default value
3097		% build up a partial function instead (fulfilling all constraints)
3098		% better? : %x.(x:Domain|IF x:dom(SFF) THEN SFF(x) ELSE DEFAULT)?
3099		partial_function_wf(SFF,Domain,Range,WF),
3100		% next, build up a total function mapping everything to a default value
3101		% this function will be overriden by the partial function to fulfilling
3102		% given constraints
3103		% 1. identifiers for closure
3104		create_texpr(identifier('__domid__'),DomT,[],TDomId),
3105		create_texpr(identifier('__ranid__'),RanT,[],TRanId),
3106		% 2. domain identifier might take all values of the domain
3107		create_texpr(member(TDomId,b(value(Domain),set(DomT),[])),pred,[],DomMember),
3108		% 3. pick a single value for the range identifier
3109		check_element_of_wf(RangeElement,Range,WF),
3110		%% external_functions:observe_value(RangeElement,"range"),external_functions:observe_value(SFF,"pf"),
3111		create_texpr(equal(TRanId,b(value(RangeElement),RanT,[])),pred,[],RanMember),
3112		% 4. conjunct and form closure (should be treated symbolically)
3113		conjunct_predicates([RanMember,DomMember],Pred),
3114		Default = closure(['__domid__','__ranid__'],[DomT,RanT],Pred),
3115		% 5. override default values where needed
3116		override_relation(Default,SFF,FF,WF),
3117		get_last_wait_flag(enum_symb_tf,WF,LastWF),
3118		when(nonvar(LastWF), % if we enum too early test 1619 fails; see also test 2022
3119		% as partial_function_wf does not fully enumerate the new variable SFF we may have to enumerate SFF; see test 2328
3120		(enumerate_basic_type_wf(RangeElement,RanT,WF),
3121		enumerate_basic_type_wf(SFF,set(couple(DomT,RanT)),WF)
3122		)).
3123		total_function_wf1(R,Domain,Range,WF) :-
3124		try_expand_and_convert_to_avl_with_check(Domain,EDomain,keep_intervals(1000),total_function), % avoid multiple expansions, but useless when dom_for_lambda_closure case triggers below ! TO DO: fix
3125		% TO DO: maybe avoid converting intervals which are not fully instantiated ?
3126		% TODO: done by clause above? % TO DO ?: if Range singleton set {R} and Domain infinite: return %x.(x:Domain|R); if Range not empty choose one element
3127		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
3128		propagate_empty_set_wf(Range,tf_range,ER,WF), % if the range of a total function is empty then the function must be empty
3129	?	total_function_wf2(ER,EDomain,Range,WF).
3130
3131		:- block total_function_wf2(?,-,?,?).
3132		total_function_wf2(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3133		total_function_avl_set(AEF,Domain,Range,WF).
3134		total_function_wf2(R,Domain,Range,WF) :-
3135		cardinality_as_int_wf(Domain,int(Card),WF),
3136	?	total_function_wf3(R,Card,Domain,Range,WF).
3137
3138		:- use_module(kernel_card_arithmetic,[is_inf_or_overflow_card/1]).
3139		total_function_wf3(FF,Card,Domain,Range,WF) :-
3140		nonvar(FF),
3141		(number(Card) -> (Card >= 1000 -> true ; is_symbolic_closure(FF)) ; true),
3142		% note: we can have symbolic closures with a finite domain: /*@symbolic */ %p.(p:BOOL|(%t.(t:NATURAL|t+100)))
3143		custom_explicit_sets:dom_for_lambda_closure(FF,FFDomain),
3144		% we have a lambda closure where we cannot determine the range,
3145		% otherwise dom_range_for_specific_closure would have succeeded
3146		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 --> NATURAL
3147		FF = closure(P,T,Pred),
3148		get_range_id_expression(P,T,TRangeID),
3149		!,
3150		equal_object_wf(FFDomain,Domain,total_function1_closure,WF),
3151		% CHECK not(#P.(Pred & P /: Range))
3152		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF).
3153		total_function_wf3(R,Card,Domain,Range,WF) :- nonvar(Card),is_inf_or_overflow_card(Card),!,
3154		when(nonvar(R), total_function_symbolic(R,Domain,Range,WF)).
3155		total_function_wf3(R,Card,Domain,Range,WF) :-
3156		card_convert_int_to_peano(Card,PeanoCard),
3157		((nonvar(R);ground(PeanoCard))
3158		-> true
3159		; get_last_wait_flag(total_fun(Domain),WF,WF1)),
3160	?	when((nonvar(R);ground(PeanoCard);
3161		(nonvar(PeanoCard),nonvar(WF1))), /* mal 12/5/04: changed , into ; 17/3/2008: added WF1 */
3162		/* reason for delaying nonvar(Card): Card grounded bit by bit by cardinality; avoid
3163		triggering too early and missing tf_var */
3164		total_function1(R,Card,PeanoCard,Domain,Range,WF
3165		)).
3166
3167		:- use_module(library(lists),[last/2]).
3168		% for a closure get the identifier or proj expression that represents range values
3169		get_range_id_expression([PairID],[Type],Res) :- !,
3170		Type = couple(_,TX),
3171		TP = b(identifier(PairID),Type,[]),
3172		safe_create_texpr(second_of_pair(TP),TX,Res). % prj2(PairID) ,
3173		%TO DO: test this e.g. with f = /*@symbolic*/ {x|x:NATURAL1*INTEGER & prj2(INTEGER,INTEGER)(x)=prj1(INTEGER,INTEGER)(x)+1} & f: NATURAL1 --> NATURAL
3174		% but currently lambda closure detection in dom_for_lambda_closure cannot handle such closures anyway
3175		get_range_id_expression(P,T,b(identifier(ID),Type,[])) :- last(P,ID), last(T,Type).
3176
3177		total_function_avl_set(AEF,Domain,Range,WF) :-
3178		(Domain = avl_set(Dom) -> is_avl_total_function_over_domain(AEF,Dom)
3179		; is_avl_partial_function(AEF),
3180		domain_of_explicit_set_wf(avl_set(AEF),AEF_Domain,WF),
3181		equal_object_wf(AEF_Domain,Domain,total_function_avl_set,WF)
3182		),
3183		is_avl_relation_over_range(AEF,Range,WF).
3184
3185		total_function_symbolic(FF,Domain,Range,WF) :-
3186		(debug_mode(off) -> true ; print('SYMBOLIC --> check : '),translate:print_bvalue(FF),nl),
3187		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
3188		domain_wf(FF,Domain,WF),
3189		symbolic_range_subset_check(FF,Range,WF),
3190		symbolic_functionality_check(FF,WF).
3191
3192		total_function1(FF,Card,PeanoCard,Domain,Range,WF) :- nonvar(Card),is_inf_or_overflow_card(Card),
3193		nonvar(PeanoCard),is_inf_or_overflow_card(PeanoCard),!,
3194		total_function_symbolic(FF,Domain,Range,WF).
3195		total_function1(FF,_,_,Domain,Range,WF) :-
3196		expand_and_convert_to_avl_set_catch(FF,AEF,total_function1,'ARG : ? --> ?',ResultStatus,WF),!,
3197		(ResultStatus=avl_set -> total_function_avl_set(AEF,Domain,Range,WF)
3198		; % keep symbolic
3199		% TO DO: ensure no pending co-routine infinite_peano in card_convert_int_to_peano
3200		total_function_symbolic(FF,Domain,Range,WF)
3201		).
3202		total_function1(R,_,Card,Domain,Range,WF) :-
3203		try_expand_custom_set_wf(R,ER,total_function1,WF),
3204	?	total_function2(ER,Card,Domain,Range,WF).
3205
3206		total_function2(ER,Card,Domain,Range,WF) :-
3207		var(ER),ground(Card),!,
3208		tf_var(TotalFunction,[],Card,Domain,Range,WF),
3209		ER=TotalFunction.
3210		total_function2(ER,Card,Domain,Range,WF) :-
3211		(ground(Card)
3212		-> get_wait_flag(0,tot_fun,WF,LWF) % we seem to know the domain exactly now; see e.g. test 1316
3213		; get_wait_flag(2,total_function2,WF,LWF)), % ensure we don't start binding function as soon as Card is bound; important for test 1393; should we use another priority ?
3214	?	tf(ER,[],Card,Domain,Range,WF,LWF).
3215
3216		:- block tf(-,?,-,?,?,?,?),tf(-,?,?,?,?,?,-).
3217		tf([],_,0,Dom,_,WF,_) :- empty_set_wf(Dom,WF).
3218		tf(FUN,SoFar,s(Card),Dom,Ran,WF,LWF) :- var(FUN),nonvar(Dom), % try setting up skeleton for total fun
3219		remove_exact_first_element(X,Dom,Dom2),not_element_of_wf(X,SoFar,WF),var(FUN),!,
3220	?	FUN = [(X,Y)|T], tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3221		tf([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF,LWF) :-
3222		not_element_of_wf(X,SoFar,WF),
3223		remove_element_wf(X,Dom,Dom2,WF), %mal: 17/3/08 changed to _wf version
3224	?	tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3225		tf(CS,SoFar,Card,Dom,Ran,WF,LWF) :- nonvar(CS), is_custom_explicit_set(CS),
3226		expand_custom_set_to_list_wf(CS,ER,_,tf,WF),
3227		tf(ER,SoFar,Card,Dom,Ran,WF,LWF).
3228		tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF) :-
3229		check_element_of_wf(Y,Ran,WF),
3230		%when((nonvar(T);nonvar(Card)), /* mal 12/5/04: changed , into ; */
3231		add_new_element_wf(X,SoFar,SoFar2,WF), %%% try_expand_and_convert_to_avl
3232	?	tf(T,SoFar2,Card,Dom2,Ran,WF,LWF).
3233
3234		:- block tf_var(-,?,-,?,?,?).
3235		tf_var(F,_,Card,Dom,_,WF) :- Card==0,!,F=[],empty_set_wf(Dom,WF). % avoid choice point
3236		tf_var([],_,0,Dom,_,WF) :- empty_set_wf(Dom,WF).
3237		tf_var([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF) :-
3238		/* supposes that X + Y are unbound */
3239		/* TO DO: rewrite like enumerate <-------------------------- */
3240		((var(X),var(Y)) -> true ; (print_message(warning,'Nonvar in tf_var: '),
3241		print_message(warning,((X,Y))))),
3242		remove_exact_first_element(X,Dom,Dom2),
3243		not_element_of_wf(X,SoFar,WF),
3244		check_element_of_wf(Y,Ran,WF),
3245		add_new_element_wf(X,SoFar,SoFar2,WF),
3246		tf_var(T,SoFar2,Card,Dom2,Ran,WF).
3247
3248
3249
3250		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1)],[int(7)]),
3251		X = [(int(1),int(7))])).
3252		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3253		kernel_objects:equal_object(X,[(int(2),int(8)),(int(1),int(7))]))).
3254		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1)],[int(7),int(3)]),
3255		X = [(int(1),int(7))])).
3256		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(3)]),
3257		X = [(int(1),int(3)),(int(2),int(3))])).
3258		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3259		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
3260		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3261		X = [(int(1),int(7)),(int(1),int(8))])).
3262
3263
3264
3265		total_bijection(R,Domain,Range) :- init_wait_flags(WF,[total_bijection]),
3266		total_bijection_wf(R,Domain,Range,WF),
3267	?	ground_wait_flags(WF).
3268
3269		:- block total_bijection_wf(?,-,?,?).
3270		total_bijection_wf(FF,Domain,Range,WF) :- nonvar(FF),
3271		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3272		equal_object_wf(FFDomain,Domain,total_bijection_wf_1,WF),
3273		equal_object_wf(FFRange,Range,total_bijection_wf_2,WF).
3274		%(R,Domain,Range,WF) :- Domain==Range,!, print(eq_domain_range),nl, total_injection_wf(R,Domain,Range,WF).
3275		total_bijection_wf(R,Domain,Range,WF) :-
3276		same_cardinality_wf(Domain,Range,WF),
3277		total_injection_wf2(R,Domain,Range,WF). % TO DO: use cardinality_as_int_wf ? makes test 1194 fail
3278
3279		%Note: we used to call custom code: total_bijection_wf2(R,Domain,Card,Range,WF).
3280		% total_injection_wf2 gives a considerable performance boost, e.g., for test 1222 ClearSy/alloc_large.mch or NQueens with >->>
3281
3282		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_function([(int(1),int(6)),(int(2),int(7))],[int(1),int(2),int(3)],[int(7),int(6)],WF),WF)).
3283		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_function([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(8),int(6)],WF),WF)).
3284		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_function([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
3285		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_function([(int(1),int(6)),(int(1),int(7))],[int(1)],[int(7),int(6)],WF),WF)).
3286		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1)],[int(7)],_WF),
3287		X = [(int(1),int(7))])).
3288		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3289		X = [(int(2),int(7)),(int(1),int(7))])).
3290		:- assert_must_succeed((bsets_clp:not_total_function([],[int(1)],[int(7)],_WF))).
3291		:- assert_must_succeed((bsets_clp:not_total_function([],[global_set('NAT1')],[global_set('Name')],_WF))).
3292		:- assert_must_succeed((bsets_clp:not_total_function([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3293		:- assert_must_succeed((bsets_clp:not_total_function([(int(1),int(7)), (int(2),int(1))],
3294		[int(1),int(2)],[int(7)],_WF))).
3295		:- assert_must_succeed((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3296		X = [(int(2),int(7)),(int(2),int(6))])).
3297
3298		:- block not_total_function(-,?,?,?), not_total_function(?,-,?,?).
3299		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
3300		% we do not need the Range; this means we can match more closures (e.g., lambda)
3301		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
3302		not_equal_object_wf(FFDomain,Domain,WF).
3303		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),
3304		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3305		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
3306		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
3307		not_total_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
3308		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
3309		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
3310		get_range_id_expression(P,T,TRangeID),!,
3311		equality_objects_wf(FFDomain,Domain,SubRes,WF), % compare: subset_test for not_partial_function
3312		when(nonvar(SubRes),
3313		(SubRes=pred_false -> true % not equal -> it is not a total function over the domain
3314		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
3315		not_total_function(R,Domain,Range,WF) :-
3316		try_expand_and_convert_to_avl_with_check(R,ER,not_total_function_range),
3317		try_expand_and_convert_to_avl_unless_large_wf(Range,ERange,WF),
3318		not_total_function2(ER,Domain,ERange,WF).
3319
3320		% repeat block, in case Domain or R is a closure
3321		:- block not_total_function2(-,?,?,?), not_total_function2(?,-,?,?).
3322		not_total_function2(R,Domain,Range,WF) :-
3323		expand_and_convert_to_avl_set_warn(R,AER,not_total_function2,'ARG /: ? --> ?',WF),
3324		!,
3325		not_total_function_avl(AER,Domain,Range,WF).
3326		not_total_function2(R,Domain,ERange,WF) :-
3327		expand_custom_set_to_list_wf(R,ER,_,not_total_function2,WF),
3328		try_expand_and_convert_to_avl_with_check(Domain,EDomain,keep_intervals(1000),not_total_function_domain),
3329		not_tf(ER,[],EDomain,ERange,WF).
3330
3331		not_total_function_avl(_AER,Domain,_Range,_WF) :- is_infinite_explicit_set(Domain),!,
3332		true. % a finite AVL set cannot be a total function over an infinite domain
3333		not_total_function_avl(AER,Domain,Range,WF) :-
3334		expand_and_convert_to_avl_set_warn(Domain,ADom,not_total_function2,'? /: ARG --> ?',WF),
3335		!,
3336		(is_avl_total_function_over_domain(AER,ADom)
3337		->
3338		is_not_avl_relation_over_range(AER,Range,WF)
3339		; true
3340		).
3341		not_total_function_avl(AER,EDomain,ERange,WF) :-
3342		expand_custom_set_to_list_wf(avl_set(AER),ER,_,not_total_function_avl,WF),
3343		not_tf(ER,[],EDomain,ERange,WF).
3344
3345
3346		:- use_module(kernel_equality,[membership_test_wf_with_force/4]).
3347
3348		:- block not_tf(-,?,?,?,?).
3349		not_tf([],_,Domain,_,WF) :- not_empty_set_wf(Domain,WF).
3350		not_tf([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf_with_force(SoFar,X,MemRes,WF),
3351		not_tf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3352
3353		:- block not_tf2(-,?,?,?, ?,?,?,?). %, not_tf2(?,?,?,?, -,?,?), not_tf2(?,?,?,?, ?,-,?).
3354		not_tf2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).% :- check_element_of_lazy(X,SoFar,WF).
3355		not_tf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3356		%not_element_of_wf(X,SoFar,WF),
3357		membership_test_wf_with_force(Dom,X,MemRes,WF),
3358		not_tf3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3359
3360		:- block not_tf3(-, ?,?,?,?, ?,?,?).
3361		not_tf3(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF).
3362		not_tf3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3363		remove_element_wf(X,Dom,Dom2,WF),
3364		membership_test_wf_with_force(Ran,Y,MemRes,WF),
3365		not_tf4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3366
3367		:- block not_tf4(-, ?,?,?,?, ?,?,?).
3368		not_tf4(pred_false,_X,_Y,_T,_SoFar,_Dom2,_Ran,_WF).
3369		not_tf4(pred_true,X,_Y,T,SoFar,Dom2,Ran,WF) :-
3370		%check_element_of_wf(Y,Ran,WF), %DO WE NEED THIS ????
3371		add_new_element_wf(X,SoFar,SoFar2,WF),
3372		not_tf(T,SoFar2,Dom2,Ran,WF).
3373
3374
3375
3376		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_bijection([(int(1),int(6)),(int(2),int(7))],[int(1),int(2),int(3)],[int(7),int(6)],WF),WF)).
3377		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_bijection([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(8),int(6)],WF),WF)).
3378		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_bijection([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
3379		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1)],[int(7)],_WF),
3380		X = [(int(1),int(7))])).
3381		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3382		X = [(int(2),int(7)),(int(1),int(6))])).
3383		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3384		X = [(int(1),int(7)),(int(2),int(6))])).
3385		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(3)],_WF),
3386		X = [(int(1),int(3)),(int(2),int(3))])).
3387		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3388		X = [(int(2),int(7)),(int(1),int(7))])).
3389		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1)],[int(7),int(8)],_WF),
3390		X = [(int(1),int(7))])).
3391		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7)],_WF),
3392		X = [(int(2),int(7))])).
3393		:- assert_must_succeed((bsets_clp:not_total_bijection([],[int(1)],[int(7)],_WF))).
3394		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3395		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(1),int(7)), (int(2),int(1))],
3396		[int(1),int(2)],[int(7)],_WF))).
3397		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3398		X = [(int(2),int(7)),(int(2),int(6))])).
3399
3400		:- block not_total_bijection(-,?,?,?), not_total_bijection(?,-,?,?).
3401		not_total_bijection(FF,Domain,Range,WF) :-
3402		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3403		not_equal_object_wf((FFDomain,FFRange),(Domain,Range),WF).
3404		not_total_bijection(avl_set(_),Domain,_Range,_WF) :-
3405		is_infinite_explicit_set(Domain),!.
3406		% a finite set cannot be a total bijection over an infinite domain, see test 1641
3407		not_total_bijection(R,Domain,Range,WF) :-
3408		try_expand_custom_set_wf(R,ER,not_total_bijection,WF),
3409		not_tot_bij(ER,[],Domain,Range,WF).
3410
3411		:- block not_tot_bij(-,?,?,?,?).
3412		not_tot_bij([],_,Domain,Range,WF) :- empty_not_tot_bij(Domain,Range,WF).
3413		not_tot_bij([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf(SoFar,X,MemRes,WF),
3414		not_tot_bij2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3415
3416		:- use_module(kernel_equality,[empty_set_test_wf/3]).
3417		:- block empty_not_tot_bij(-,?,?).
3418		empty_not_tot_bij(Domain,Range,WF) :-
3419		empty_set_test_wf(Domain,EqRes,WF),
3420		empty_not_tot_bij2(EqRes,Range,WF).
3421		:- block empty_not_tot_bij2(-,?,?).
3422		empty_not_tot_bij2(pred_false,_,_).
3423		empty_not_tot_bij2(pred_true,Range,WF) :- not_empty_set_wf(Range,WF).
3424
3425		:- block not_tot_bij2(-,?,?,?,?,?,?,?).
3426		not_tot_bij2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).
3427		not_tot_bij2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3428		membership_test_wf(Dom,X,MemRes,WF),
3429		not_tot_bij3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3430
3431		:- block not_tot_bij3(-,?,?,?,?,?,?,?).
3432		not_tot_bij3(pred_false,_X,_,_T,_SoFar,_Dom,_Ran,_WF). % X not a member of domain
3433		not_tot_bij3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3434		remove_element_wf(X,Dom,Dom2,WF),
3435		membership_test_wf(Ran,Y,MemRes,WF),
3436		not_tot_bij4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3437
3438		:- block not_tot_bij4(-,?,?,?,?,?,?,?).
3439		not_tot_bij4(pred_false,_X,_,_T,_SoFar,_Dom2,_Ran,_WF). % Y not a member of range
3440		not_tot_bij4(pred_true,X,Y,T,SoFar,Dom2,Ran,WF) :-
3441		remove_element_wf(Y,Ran,Ran2,WF),
3442		add_element_wf(X,SoFar,SoFar2,WF),
3443		not_tot_bij(T,SoFar2,Dom2,Ran2,WF).
3444
3445
3446
3447		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_restriction_wf([(int(1),int(2)),(int(2),int(3))],[int(3)],[(int(2),int(3))],WF),WF)).
3448		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_restriction_wf([(int(1),int(2)),(int(2),int(3))],[int(2),int(3)],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3449		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_restriction_wf([],[int(2),int(3)],[],WF),WF)).
3450		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[int(1)],[],_WF))).
3451		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[],[],_WF))).
3452		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(1)],[],_WF))).
3453		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(2)],[(int(1),int(2))],_WF))).
3454		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3455		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3456		kernel_objects:equal_object(X,R))).
3457		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,Y,R,_WF),
3458		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],Y=global_set('Name'),
3459		kernel_objects:equal_object(X,R))).
3460		:- assert_must_fail((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3461		X = [(int(1),fd(3,'Name')),(int(2),fd(1,'Name'))],
3462		kernel_objects:equal_object(X,R))).
3463
3464		:- block range_restriction_wf(-,?,?,?),range_restriction_wf(?,-,-,?).
3465
3466		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3467		ok_to_try_restriction_explicit_set(S,R,Res),
3468		range_restriction_explicit_set_wf(R,S,SR,WF),!,
3469		equal_object_wf(SR,Res,range_restriction,WF).
3470		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3471		expand_custom_set_to_list_wf(R,ER,_,range_restriction,WF),
3472	?	relation_restriction_wf(ER,S,Res,pred_true,range,WF).
3473
3474		% heuristic: should we try restriction_explicit_set or
3475		% is relation_restriction with its stronger constraint propagation better
3476		ok_to_try_restriction_explicit_set(S,R,Res) :-
3477		nonvar(S),
3478		(var(Res) -> true
3479		; S=avl_set(_),
3480		nonvar(R), R=avl_set(_) % otherwise constraint propagation from normal relation_restriction better
3481		).
3482
3483		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([],[int(2)],[],WF),WF)).
3484		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([(int(1),int(2)),(int(2),int(3))],[int(2)],[(int(2),int(3))],WF),WF)).
3485		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([(int(1),int(2)),(int(2),int(3))],[],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3486		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([(int(1),int(2)),(int(2),int(3))],[int(1)],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3487
3488		:- block range_subtraction_wf(-,?,?,?),range_subtraction_wf(?,-,-,?).
3489		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3490		S==[],!,
3491		equal_object_wf(R,Res,range_subtraction1,WF).
3492		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3493		ok_to_try_restriction_explicit_set(S,R,Res),
3494		range_subtraction_explicit_set_wf(R,S,SR,WF),!,
3495		equal_object_wf(SR,Res,range_subtraction2,WF).
3496		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3497		expand_custom_set_to_list_wf(R,ER,_,range_subtraction,WF),
3498	?	relation_restriction_wf(ER,S,Res,pred_false,range,WF).
3499
3500		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_range_restriction_wf((int(2),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(33),int(3)],WF),WF)).
3501		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_range_restriction_wf((int(1),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(3)],WF),WF)).
3502
3503		:- block in_range_restriction_wf(-,-,-,?).
3504		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3505	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3506		; preference(convert_comprehension_sets_into_closures,true)),
3507		!,
3508		Rel \== [], % avoid setting up check_element_of for X then
3509		% x |-> y : Rel |>> Set <=> x|->y : Rel & y: Set
3510		check_element_of_wf(Pair,Rel,WF),
3511		Pair = (_,P2),
3512		check_element_of_wf(P2,Set,WF).
3513		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3514		range_restriction_wf(Rel,Set,Res,WF),
3515		check_element_of_wf(Pair,Res,WF).
3516
3517		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_range_restriction_wf((int(2),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(1),int(2)],WF),WF)).
3518		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_range_restriction_wf((int(11),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(33),int(2)],WF),WF)).
3519
3520		:- block not_in_range_restriction_wf(-,-,-,?).
3521		not_in_range_restriction_wf(Pair,Rel,Set,WF) :-
3522		range_restriction_wf(Rel,Set,Res,WF),
3523		not_element_of_wf(Pair,Res,WF).
3524
3525		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_range_subtraction_wf((int(2),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(33),int(1)],WF),WF)).
3526		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_range_subtraction_wf((int(1),int(3)),[(int(2),int(3)),(int(1),int(3))],[],WF),WF)).
3527
3528		:- block in_range_subtraction_wf(-,-,-,?).
3529		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3530	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3531		; preference(convert_comprehension_sets_into_closures,true)),
3532		!,
3533		Rel \== [], % avoid setting up check_element_of for X then
3534		% x |-> y : Rel |>> Set <=> x|->y : Rel & y/: Set
3535		check_element_of_wf(Pair,Rel,WF),
3536		Pair = (_,P2),
3537		not_element_of_wf(P2,Set,WF).
3538		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3539		range_subtraction_wf(Rel,Set,Res,WF),
3540		check_element_of_wf(Pair,Res,WF).
3541
3542		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_range_subtraction_wf((int(2),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(3),int(2)],WF),WF)).
3543		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_range_subtraction_wf((int(11),int(3)),[(int(2),int(3)),(int(1),int(3))],[int(33),int(2)],WF),WF)).
3544
3545		:- block not_in_range_subtraction_wf(-,-,-,?).
3546		not_in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3547		range_subtraction_wf(Rel,Set,Res,WF),
3548		not_element_of_wf(Pair,Res,WF).
3549
3550
3551
3552		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_restriction_wf((int(2),int(3)),[int(33),int(2)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3553		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_restriction_wf((int(1),int(3)),[int(1)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3554
3555		:- block in_domain_restriction_wf(-,-,-,?).
3556		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3557	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3558		; preference(convert_comprehension_sets_into_closures,true)),
3559		!,
3560		Rel \== [], % avoid setting up check_element_of for X then
3561		% x |-> y : Set <| Rel <=> x|->y : Rel & x: Set
3562		check_element_of_wf(Pair,Rel,WF),
3563		Pair = (P1,_),
3564		check_element_of_wf(P1,Set,WF).
3565		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3566		domain_restriction_wf(Set,Rel,Res,WF),
3567		check_element_of_wf(Pair,Res,WF).
3568
3569		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_domain_restriction_wf((int(2),int(3)),[int(33),int(1)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3570		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_domain_restriction_wf((int(11),int(3)),[int(11),int(2)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3571
3572		:- block not_in_domain_restriction_wf(-,-,-,?).
3573		not_in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3574		domain_restriction_wf(Set,Rel,Res,WF),
3575		not_element_of_wf(Pair,Res,WF).
3576
3577		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_restriction_wf([int(2),int(4)],[(int(1),int(4)),(int(2),int(3))],[(int(2),int(3))],WF),WF)).
3578		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_restriction_wf([int(1),int(2)],[(int(1),int(2)),(int(2),int(3))],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3579		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_restriction_wf([int(2),int(3)],[],[],WF),WF)).
3580		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],[],_WF))).
3581		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],R,_WF), R==[])).
3582		:- assert_must_fail((bsets_clp:domain_restriction_wf(_,[],R,_WF), R=[int(_)|_])).
3583		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2)],[(int(1),int(2))],[],_WF))).
3584		:- assert_must_succeed((bsets_clp:domain_restriction_wf([],[(int(1),int(2))],[],_WF))).
3585		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3586		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2)),(int(2),_)],_,_WF))).
3587		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2),int(1)],X,R,_WF),
3588		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3589		kernel_objects:equal_object(X,R))).
3590
3591
3592		:- block domain_restriction_wf(?,-,?,?),domain_restriction_wf(-,?,-,?).
3593		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3594		ok_to_try_restriction_explicit_set(S,R,Res),
3595		domain_restriction_explicit_set_wf(S,R,SR,WF),!,
3596		equal_object_wf(SR,Res,domain_restriction,WF).
3597		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3598		expand_custom_set_to_list_wf(R,ER,_,domain_restriction,WF),
3599	?	relation_restriction_wf(ER,S,Res,pred_true,domain,WF).
3600
3601		% a predicate to compute domain/range restriction/subtraction
3602		:- block relation_restriction_wf(?,-,- ,?,?,?),
3603		relation_restriction_wf(-,?,? ,?,?,?).
3604		relation_restriction_wf([],_S,Res,_AddWhen,_DomOrRange,WF) :-
3605	?	empty_set_wf(Res,WF).
3606		relation_restriction_wf([(X,Y)|T],S,Res,AddWhen,DomOrRange,WF) :-
3607		(DomOrRange=domain
3608		-> membership_test_wf(S,X,MemRes,WF) % TO DO: pass WF !
3609		; membership_test_wf(S,Y,MemRes,WF)),
3610		(nonvar(MemRes)
3611		%MemRes==AddWhen % MemRes already set; we will ensure that (X,Y) in Res below; this slows down Alstom Compilation Regle !
3612		% doing the membership_test on the result Res if MemRes\==AddWhen only makes sense if we cannot fully compute the restriction ?? i.e. if T is not a closed list ?
3613		-> true %,(MemRes==AddWhen -> true ; print_term_summary(relation_restriction([(X,Y)|T],S,Res,AddWhen,DomOrRange)),nl)
3614		; (AddWhen=pred_true -> InResult=MemRes
3615		; negate(InResult,MemRes)), % from bool_pred
3616	?	membership_test_wf(Res,(X,Y),InResult,WF)
3617		% TO DO: same for explicit version; gets called e.g. if S = 1..n (1..n <| [1,2,3] = [1,2])
3618		% can now solve e.g. {x|x <| [1,2,3] = [1,2] & card(x)=2} = {{1,2}}
3619		% or x <| s = [1,2,3] \/ {29|->29} & x <: 1..100 & s = %i.(i:1..50|i)
3620		),
3621	?	relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF).
3622		:- block relation_restriction_aux(-,?,?,?,?,?, ?,?,?).
3623		relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF) :-
3624		MemRes==AddWhen,!, % (X,Y) should be added to result
3625		% TO DO: collect result until we delay ? and then do equal_object ?
3626	?	equal_cons(Res,(X,Y),RT), % was : equal_object([(X,Y)|RT],Res),
3627		%equal_cons_wf(Res,(X,Y),RT,WF), % makes tests 982, 1302, 1303 fail; TO DO: investigate
3628		%when(nonvar(RT), % causes problem for test 982
3629	?	relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3630		relation_restriction_aux(_MemRes,_X,_,T,S,RT,AddWhen,DomOrRange,WF) :-
3631		% the couple is filtered out
3632	?	relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3633
3634
3635		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([int(1),int(3)],[(int(1),int(4)),(int(2),int(3))],[(int(2),int(3))],WF),WF)).
3636		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([int(3),int(4)],[(int(1),int(2)),(int(2),int(3))],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3637		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([int(1)],[],[],WF),WF)).
3638		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([],[(int(11),int(21))],[(int(11),int(21))],WF),WF)).
3639		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2))],[],_WF))).
3640		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(3)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3641		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2)),(int(2),int(X))],R,_WF),
3642		R=[(int(2),int(YY))], YY==X)).
3643		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(5),int(3)],X,R,_WF),
3644		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3645		kernel_objects:equal_object(X,R))).
3646		:- block domain_subtraction_wf(?,-,?,?),domain_subtraction_wf(-,?,-,?).
3647		domain_subtraction_wf(S,R,Res,WF) :- S==[],!,
3648		equal_object_wf(R,Res,domain_subtraction1,WF).
3649		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3650		ok_to_try_restriction_explicit_set(S,R,Res),
3651		domain_subtraction_explicit_set_wf(S,R,SR,WF),!,
3652		equal_object_wf(SR,Res,domain_subtraction2,WF).
3653		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3654		expand_custom_set_to_list_wf(R,ER,_,domain_subtraction,WF),
3655		try_expand_and_convert_to_avl_with_check(S,AS,keep_intervals(500),domain_subtraction),
3656		% (ground(ER) -> domain_subtraction_acc(ER,AS,[],Res) ;
3657	?	relation_restriction_wf(ER,AS,Res,pred_false,domain,WF)
3658		% )
3659		.
3660
3661
3662
3663		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_subtraction_wf((int(2),int(3)),[int(33),int(1)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3664		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_subtraction_wf((int(2),int(3)),[],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3665
3666		:- block in_domain_subtraction_wf(-,-,-,?).
3667
3668		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3669	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3670		; preference(convert_comprehension_sets_into_closures,true)),
3671		!,
3672		Rel \== [], % avoid setting up check_element_of for X then
3673		% x |-> y : Set <<| Rel <=> x|->y : Rel & x/: Set
3674		check_element_of_wf(Pair,Rel,WF),
3675		Pair = (P1,_),
3676		not_element_of_wf(P1,Set,WF).
3677		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3678		domain_subtraction_wf(Set,Rel,Res,WF),
3679		check_element_of_wf(Pair,Res,WF).
3680
3681
3682		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_domain_subtraction_wf((int(2),int(3)),[int(33),int(2)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3683		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_domain_subtraction_wf((int(11),int(3)),[int(33),int(2)],[(int(2),int(3)),(int(1),int(3))],WF),WF)).
3684
3685		:- block not_in_domain_subtraction_wf(-,-,-,?).
3686		not_in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3687		domain_subtraction_wf(Set,Rel,Res,WF),
3688		not_element_of_wf(Pair,Res,WF).
3689
3690		% similar to kernel_objects, but adds case for [_|_]
3691		treat_arg_symbolically(X) :- var(X),!.
3692		treat_arg_symbolically([H|T]) :- \+ ground(H) ; treat_arg_symbolically(T).
3693		treat_arg_symbolically(global_set(_)).
3694		treat_arg_symbolically(freetype(_)).
3695		treat_arg_symbolically(closure(P,T,B)) :- \+ kernel_objects:small_interval(P,T,B).
3696
3697		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override_relation([(int(1),int(2))],[(int(1),int(3))],[(int(1),int(3))],WF),WF)).
3698		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override_relation([(int(1),int(2))],[(int(2),int(3))],[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3699		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(1),int(3))],X,_WF),
3700		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(3))]))).
3701		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(3),int(6))],X,_WF),
3702		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(2)),(int(3),int(6))]))).
3703
3704		:- block override_relation(-,-,?,?). % overwrite AST node
3705		override_relation(R,S,Res,WF) :- R==[],!, equal_object_wf(S,Res,override_relation1,WF).
3706		override_relation(R,S,Res,WF) :- S==[],!, equal_object_wf(R,Res,override_relation2,WF).
3707		override_relation(R,S,Res,WF) :- Res==[],!, empty_set_wf(S,WF), empty_set_wf(R,WF).
3708		override_relation(R,S,Res,WF) :- /* R <+ S */
3709		override_custom_explicit_set_wf(R,S,ORes,WF),!,
3710		equal_object_wf(ORes,Res,override_relation3,WF).
3711		override_relation(R,S,Res,WF) :- /* R <+ S */
3712		domain_wf(S,DS,WF),
3713		domain_subtraction_wf(DS,R,DSR,WF),
3714		union_wf(DSR,S,Res,WF). % in principle we could call disjoint_union_wf, but fails 1112, 1751
3715
3716		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_override_relation_wf((int(1),int(2)),[(int(1),int(2))],[(int(2),int(3))],WF),WF)).
3717		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_override_relation_wf((int(2),int(3)),[(int(1),int(2))],[(int(2),int(3))],WF),WF)).
3718		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_override_relation_wf((int(2),int(3)),[(int(1),int(2)),(int(2),int(4))],[(int(2),int(3))],WF),WF)).
3719		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:in_override_relation_wf((int(2),int(4)),[(int(1),int(2)),(int(2),int(4))],[(int(2),int(3))],WF),WF)).
3720
3721		:- block in_override_relation_wf(-,-,-,?).
3722		in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3723		check_element_of_wf(Pair,Rel1,WF).
3724		in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3725		check_element_of_wf(Pair,S,WF).
3726		in_override_relation_wf((X,Y),Rel1,S,WF) :-
3727	?	(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3728		; preference(convert_comprehension_sets_into_closures,true)),
3729		!,
3730		domain_wf(S,DS,WF),
3731		membership_test_wf(DS,X,MemRes,WF),
3732		in_override_aux(MemRes,X,Y,Rel1,S,WF).
3733		in_override_relation_wf(Pair,Rel1,S,WF) :-
3734		override_relation(Rel1,S,Res,WF),
3735		check_element_of_wf(Pair,Res,WF).
3736
3737		:- block in_override_aux(-,?,?,?,?,?).
3738		in_override_aux(pred_true,X,Y,_R,S,WF) :-
3739		check_element_of_wf((X,Y),S,WF).
3740		in_override_aux(pred_false,X,Y,R,_S,WF) :-
3741		check_element_of_wf((X,Y),R,WF).
3742
3743		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_in_override_relation_wf((int(2),int(3)),[(int(1),int(2)),(int(2),int(4))],[(int(2),int(3))],WF),WF)).
3744		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_override_relation_wf((int(2),int(4)),[(int(1),int(2)),(int(2),int(4))],[(int(2),int(3))],WF),WF)).
3745
3746		:- block not_in_override_relation_wf(-,-,-,?).
3747		not_in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3748		not_element_of_wf(Pair,Rel1,WF).
3749		not_in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3750		not_element_of_wf(Pair,S,WF).
3751		not_in_override_relation_wf((X,Y),Rel1,S,WF) :-
3752	?	(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3753		; preference(convert_comprehension_sets_into_closures,true)),
3754		!,
3755		domain_wf(S,DS,WF),
3756		membership_test_wf(DS,X,MemRes,WF),
3757		not_in_override_aux(MemRes,X,Y,Rel1,S,WF).
3758		not_in_override_relation_wf(Pair,Rel1,S,WF) :-
3759		override_relation(Rel1,S,Res,WF),
3760		not_element_of_wf(Pair,Res,WF).
3761
3762		:- block not_in_override_aux(-,?,?,?,?,?).
3763		not_in_override_aux(pred_true,X,Y,_R,S,WF) :-
3764		not_element_of_wf((X,Y),S,WF).
3765		not_in_override_aux(pred_false,X,Y,R,_S,WF) :-
3766		not_element_of_wf((X,Y),R,WF).
3767
3768		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override([],int(1),int(3),[(int(1),int(3))],WF),WF)).
3769		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override([(int(1),int(2)),(int(2),int(6))],int(1),int(3),[(int(1),int(3)),(int(2),int(6))],WF),WF)).
3770		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override([(int(1),int(2)),(int(2),int(6))],int(2),int(3),[(int(1),int(2)),(int(2),int(3))],WF),WF)).
3771
3772		% override for a single pair
3773		:- block override(-,?,?,?,?), override(?,-,?,?,?),
3774		override(?,?,-,?,?). % also wait on Y; try to generate avl if possible; can only be used in substitution anyway
3775		/* R <+ {X |-> Y} as used by substitution R(X) := Y */
3776		override(R,X,Y,Res,WF) :-
3777		override_pair_explicit_set(R,X,Y,ORes),!,
3778		equal_object_wf(ORes,Res,override1,WF).
3779		override(R,X,Y,Res,WF) :-
3780		if(try_expand_custom_set_to_list(R,ER,_,override),
3781		(
3782		override2(ER,X,Y,[(X,Y)],ORes,WF),
3783	?	equal_object_wf(ORes,Res,override2,WF)),
3784		( %print_term_summary(exception(R)), % Virtual Timeout exception occured
3785		override_relation(R,[(X,Y)],Res,WF)
3786		)).
3787
3788		:- block override2(-,?,?,?,?,?).
3789		override2([],_X,_Y,Remainder,Res,WF) :- equal_object_optimized_wf(Remainder,Res,override2,WF). %equal_object(Remainder,Res).
3790		override2([(V,W)|T],X,Y,Remainder,Res,WF) :-
3791		equality_objects_wf(V,X,EqRes,WF),
3792		override2c(EqRes,V,W,T,X,Y,Remainder,Res,WF).
3793
3794		:- block override2c(-, ?,?,?, ?,?,?,?,?).
3795		override2c(pred_true,_V,_W,T,X,Y,_Remainder,Res,WF) :-
3796		equal_cons_wf(Res,(X,Y),T2,WF),
3797		override2(T,X,Y,[],T2,WF). /* set remainder to [], we have already added (X,Y) */
3798		override2c(pred_false,V,W,T,X,Y,Remainder,Res,WF) :-
3799		equal_cons_wf(Res,(V,W),T2,WF),
3800		override2(T,X,Y,Remainder,T2,WF).
3801
3802
3803
3804		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],WF),WF)).
3805		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2)),(int(3),int(3))],[int(1),int(2)],[int(2)],WF),WF)).
3806		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2)),(int(1),int(3)),(int(4),int(4))],[int(1),int(2)],[int(2),int(3)],WF),WF)).
3807		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2)),(int(1),int(3)),(int(4),int(4))],[int(2)],[int(2),int(3)],WF),WF)).
3808		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],_WF)).
3809		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(2)],[],_WF)).
3810		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(3)],[],_WF)).
3811		:- assert_must_succeed((bsets_clp:image_wf([(int(1),int(2)),(int(1),int(3))],
3812		[int(X)],R,_WF), X=1, kernel_objects:equal_object(R,[int(2),int(3)]))).
3813		:- assert_must_succeed((bsets_clp:image_wf([([int(1),int(2)],int(6)),
3814		([int(1),int(2),int(3)],int(7)),
3815		([int(2),int(1)],int(8))],
3816		[[int(X),int(1)]],R,_WF), X=2,
3817		kernel_objects:equal_object(R,[int(6),int(8)]))).
3818		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2))],[int(1),int(2)],[int(2)],_WF)).
3819		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(1)],_WF)).
3820		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[],_WF)).
3821
3822
3823		:- block image_wf(-,?,?,?).
3824		image_wf(Rel,_,Res,WF) :- Rel==[],!,empty_set_wf(Res,WF).
3825		image_wf(Rel,S,Res,WF) :-
3826		image_for_id_closure(Rel,S,Img),!, % we don't require S to be known here
3827		equal_object_wf(Img,Res,image_wf_id_closure,WF).
3828		image_wf(Rel,S,Res,WF) :-
3829		image_wf0(Rel,S,Res,WF).
3830
3831		:- block image_wf0(?,-,?,?).
3832		image_wf0(Rel,S,Res,WF) :- /* Res = Rel[S] */
3833		(S==[] -> empty_set_wf(Res,WF)
3834		; opt_push_wait_flag_call_stack_info(WF,b_operator_call(image,[Rel,S],unknown),WF2),
3835		image1(Rel,S,Res,WF2) ).
3836
3837		keep_symbolic(R) :- var(R),!,fail.
3838		keep_symbolic(closure(_,_,_)) :- preferences:get_preference(convert_comprehension_sets_into_closures,true),!.
3839		keep_symbolic(R) :- dont_expand_this_explicit_set(R).
3840
3841		:- block image1(-,?,?,?).
3842		image1(Rel,S,Res,WF) :-
3843		image_for_explicit_set(Rel,S,Img,WF),!,
3844		equal_object_wf(Img,Res,image1_1,WF),
3845		quick_propagate_subset_range(Res,Rel,WF).
3846		%image1(Rel,S,Res,WF) :- expand_custom_set_to_list(S,ES),!, image_of_set(ES,Rel,Res,WF).
3847		image1(Rel,Set,Res,WF) :-
3848		keep_symbolic(Rel),
3849		(preferences:get_preference(convert_comprehension_sets_into_closures,true), % in this case keep_symbolic is always true
3850		nonvar(Set),is_infinite_explicit_set(Set) % in this case we have to expand Rel below; what if Rel also infinite ?? --> TO DO : symbolic treatment
3851		-> debug_println(9,infinite_for_image1(Set)),
3852		fail
3853		; true),
3854		( dom_for_specific_closure(Rel,Domain,function(_),WF)
3855		-> !,
3856		expand_custom_set_to_list_wf(Set,ESet,_,image1,WF), % TO DO: what if keep_symbolic(Set)
3857		image_for_inf_fun(ESet,Domain,Rel,[],Res,WF)
3858		; get_relation_types(Rel,DomType,RangeType),!,
3859		image_symbolic(Set,Rel,DomType,RangeType,Res,WF)
3860		).
3861		image1(Rel,S,Res,WF) :-
3862		on_enumeration_warning(expand_custom_set_to_list_wf(Rel,Relation,_,image1_2,WF), R=failed),
3863		% bad if Rel is a big closure ! image_for_list_relation(Relation,S,Res).
3864		(R==failed -> write(failed),nl,
3865		mnf_get_relation_types(Rel,DomType,RangeType),% must succeed, as Rel is a closure with types
3866		image_symbolic(S,Rel,DomType,RangeType,Res,WF) % does not treat special case image_for_inf_fun
3867		; propagate_singleton_image(Relation,S,Res,WF),
3868		% TO DO: we could propagate cardinality constraints about Relation,S and Res
3869		% we could also try to infer all_different constraints in case card(S)=card(Res) and f is a function
3870		image_for_list_relation(Relation,S,[],Res,WF)
3871		).
3872
3873		image_symbolic(Set,Rel,DomType,RangeType,Res,WF) :-
3874		expand_custom_set_to_list_wf(Set,ESet,_,image1_2,WF),
3875		(is_symbolic_closure(Rel)
3876		-> Symbolic=symbolic_try_expand, ground_value_check((Rel,ESet),GRel) % also wait for ESet to be ground so that we can catch enumeration warning exceptions, cf. test 2428 when theorem and foralls not expanded
3877		; Symbolic=expand, ground_value_check(Rel,GRel)
3878		),
3879		when(nonvar(GRel), image_for_large_relation(ESet,Rel,Symbolic,DomType,RangeType,[],Res,WF)).
3880		% Alternative: We could compute closure by calculating {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3881		% image_closure(Set,Rel,DomType,RangeType,Closure ),
3882
3883		% propagate that f[{x}] = {r1,...,rk} => x|->ri : f (or {x}*{r1,...,rk} <: f); see test 1532
3884		propagate_singleton_image(R,S,Res,_) :-
3885		(var(S) ; var(Res) ; nonvar(R), is_custom_explicit_set(R,psi)), !.
3886		propagate_singleton_image(Relation,S,avl_set(Res),WF) :-
3887		custom_explicit_sets:singleton_set(S,El), % we have the image by a singleton set {El}
3888		expand_custom_set_to_list_wf(avl_set(Res),LR,_,prop_singleton,WF),
3889		!,
3890		l_check_element_of(LR, El, Relation, WF). % propagate x|->ri : f (will force membership)
3891		propagate_singleton_image(_,_,_,_).
3892
3893		l_check_element_of([],_,_,_).
3894		l_check_element_of([H|T],El,Relation,WF) :-
3895		check_element_of_wf((El,H),Relation,WF),
3896		l_check_element_of(T,El,Relation,WF).
3897
3898		% quick_propagate_in_range(Set, Relation,WF) : propagate that Set <: ran(Relation)
3899		:- block quick_propagate_subset_range(-,?,?).
3900		quick_propagate_subset_range(avl_set(_),_,_) :- !.
3901		quick_propagate_subset_range([],_,_) :- !.
3902		quick_propagate_subset_range([H|T],Relation,WF) :- is_custom_explicit_set(Relation,range_wf1),
3903		range_of_explicit_set_wf(Relation,Range,WF), !,
3904		quick_propagation_element_information(Range,H,WF,NewRange),
3905		quick_propagate_subset_range2(T,NewRange,WF).
3906		quick_propagate_subset_range(_,_,_).
3907
3908		:- block quick_propagate_subset_range2(-,?,?).
3909		quick_propagate_subset_range2([H|T],NewRange,WF) :- !,
3910		quick_propagation_element_information(NewRange,H,WF,NewRange1),
3911		quick_propagate_subset_range2(T,NewRange1,WF).
3912		quick_propagate_subset_range2(_,_,_).
3913
3914		:- use_module(btypechecker, [unify_types_strict/2]).
3915		get_relation_types(Value,Domain,Range) :-
3916		kernel_objects:infer_value_type(Value,VT),
3917		unify_types_strict(VT,set(couple(Domain,Range))). % deal also with seq types
3918		% VT=set(couple(Domain,Range)).
3919		% a version that must not fail:
3920		mnf_get_relation_types(Value,Domain,Range) :-
3921		(get_relation_types(Value,Domain,Range) -> true
3922		; add_internal_error('Failed: ',get_relation_types(Value,Domain,Range)),
3923		Domain=any, Range=any).
3924
3925		:- block image_for_large_relation(-,?,?,?,?,?,?,?), image_for_large_relation(?,?,?,?,?,-,?,?).
3926	?	image_for_large_relation([],_,_,_,_,Acc,Res,WF) :- equal_object_wf(Acc,Res,WF).
3927		image_for_large_relation([XX|T],Rel,Symbolic,DomType,RangeType,Acc,Res,WF) :-
3928		get_image_singleton_closure(XX,DomType,RangeType,Rel, Par,TPara,Body),
3929		expand_closure_direct_if_possible(Symbolic,Par,TPara,Body,ImagesForXX,WF),
3930		union_wf(Acc,ImagesForXX,NewAcc,WF),
3931		(T == [] -> equal_object_wf(NewAcc,Res,WF)
3932		; image_for_large_relation(T,Rel,Symbolic,DomType,RangeType,NewAcc,Res,WF)).
3933
3934		get_image_singleton_closure(XX,DomType,RangeType,Rel, [yy], [RangeType], Body) :-
3935		Body = b(member(b(couple(b(value(XX),DomType,[]),
3936		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3937		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]).
3938		% TO DO: simplify above if we have Rel = closure(P,T,B); which we usually will
3939
3940		expand_closure_direct_if_possible(symbolic_try_expand,Par,Types,Body,Result,WF) :- !,
3941		catch_enumeration_warning_exceptions(
3942		custom_explicit_sets:expand_normal_closure_direct(Par,Types,Body,Result,_Done,WF),
3943		(mark_bexpr_as_symbolic(Body,SBody),
3944		Result = closure(Par,Types,SBody) % TODO: we could set definitely_symbolic for next iteration
3945		),
3946		false,
3947		ignore(image_for_large_relation)).
3948		expand_closure_direct_if_possible(definitely_symbolic,Par,Types,Body,Result,_WF) :- !,
3949		mark_bexpr_as_symbolic(Body,SBody),
3950		Result = closure(Par,Types,SBody).
3951		expand_closure_direct_if_possible(_,Par,Types,Body,Result,WF) :-
3952		% do not memoize this (many different values):
3953		custom_explicit_sets:expand_normal_closure_direct(Par,Types,Body,Result,_Done,WF).
3954
3955
3956		/* no longer used
3957		% construct a closure for {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3958		image_closure(Set,Rel,DomType,RangeType,Closure ) :- custom_explicit_sets:singleton_set(Set,XX),!,
3959		% do not set up existential quantifier if Set is singleton set
3960		Closure = closure([yy],[RangeType],Body),
3961		Body = b(member(b(couple(b(value(XX),DomType,[]),
3962		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3963		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]).
3964		image_closure(Set,Rel,DomType,RangeType,Closure ) :-
3965		Closure = closure([yy],[RangeType],Body),
3966		couple_member_pred(xx,DomType,yy,RangeType,Rel, Predxxyy),
3967		Body = b(exists([b(identifier(xx),DomType,[])],
3968		b(conjunct(
3969		b(member(b(identifier(xx),DomType,[]),b(value(Set),set(DomType),[])),pred,[]), % TO DO : force evaluation !
3970		Predxxyy),
3971		pred,[])),pred,[used_ids([yy])]).
3972		*/
3973
3974		% very similar to rel_compose_with_inf_fun, indeed f[S] = ran((id(S);f))
3975		:- block image_for_inf_fun(-,?,?,?,?,?).
3976		image_for_inf_fun([],_Dom,_Rel2,Acc,Comp,WF) :- equal_object_wf(Acc,Comp,WF).
3977		image_for_inf_fun([X|T],Dom,Fun,Acc,CompRes,WF) :-
3978		membership_test_wf(Dom,X,MemRes,WF),
3979		image_for_inf_fun_aux(MemRes,X,T,Dom,Fun,Acc,CompRes,WF).
3980
3981		:- block image_for_inf_fun_aux(-,?,?, ?,?,?,?,?).
3982		image_for_inf_fun_aux(pred_true,X,T,Dom,Fun,Acc,CompRes,WF) :-
3983		apply_to(Fun,X,FX,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
3984		add_element_wf(FX,Acc,NewAcc,WF), % will block until Acc Known !!
3985		% TO DO USE: equal_cons_wf(CompRes,FX,CT,WF) + accumulator !,
3986		image_for_inf_fun(T,Dom,Fun,NewAcc,CompRes,WF).
3987		image_for_inf_fun_aux(pred_false,_X,T,Dom,Fun,Acc,Comp,WF) :-
3988		image_for_inf_fun(T,Dom,Fun,Acc,Comp,WF).
3989
3990
3991		/*
3992		:- block image_of_set(-,?,?,?,?), image_of_set(?,?,-,?,?).
3993		image_of_set([],Rel,ImageSoFar,Res,WF) :- equal_object(ImageSoFar,Res).
3994		image_of_set([H|T],Rel,ImageSoFar,Res,WF) :-
3995		image_of_element(Rel,H,ImageSoFar,SF2,WF),
3996		image_of_set(T,Rel,SF2,Res,WF).
3997
3998		image_of_element([],_,Acc,Res,WF) :- equal_object(Acc,Res).
3999		image_of_element([(A,B)|T],H,Acc,Res,WF) :- equality....
4000		image_of_element(avl_set(),H,Acc,Res,WF) :- ....
4001		image_of_element(closure(),....
4002		*/
4003
4004		% Computing the image of a relation which is stored as a list: traverse the relation
4005		:- block image_for_list_relation(-,?,?,?,?).
4006	?	image_for_list_relation([],_,_,Res,WF) :- empty_set_wf(Res,WF).
4007		image_for_list_relation([(X,Y)|T],S,ImageSoFar,Res,WF) :-
4008		((T==[], definitely_not_empty(Res))
4009		-> MemRes=pred_true, % we need at least one more element for Res
4010		check_element_of_wf(X,S,WF)
4011		; (Res==[],ImageSoFar==[]) -> MemRes=pred_false, not_element_of_wf(X,S,WF) % Result empty: X cannot be in S
4012		; membership_test_wf(S,X,MemRes,WF)
4013		),
4014	?	image4(MemRes,Y,T,S,ImageSoFar,Res,WF).
4015
4016		definitely_not_empty(Set) :- nonvar(Set), Set \== [], \+ functor(Set,closure,3). % Set \= closure(_,_,_).
4017
4018		:- block image4(-, ?,?,?, ?,?,?).
4019		image4(pred_true, Y,T,S, ImageSoFar,Res,WF) :-
4020		(Res==[]
4021		-> MemRes=pred_true, check_element_of_wf(Y,ImageSoFar,WF)
4022		; membership_test_wf(ImageSoFar,Y,MemRes,WF)
4023		),
4024	?	image5(MemRes,Y,T,S,ImageSoFar,Res,WF).
4025		image4(pred_false, _Y,T,S, ImageSoFar,Res,WF) :-
4026	?	image_for_list_relation(T,S,ImageSoFar,Res,WF).
4027
4028		:- block image5(-, ?,?,? ,?,?,?).
4029		image5(pred_true,_Y,T,S,ImageSoFar,Res,WF) :- /* we have already added Y to the image */
4030		image_for_list_relation(T,S,ImageSoFar,Res,WF).
4031		image5(pred_false,Y,T,S,ImageSoFar,Res,WF) :-
4032		add_element_wf(Y,ImageSoFar,ImageSoFar2,WF),
4033		kernel_objects:mark_as_non_free(Y,image), % Y has been added to image, no longer freely choosable
4034		equal_cons_wf(Res,Y,Res2,WF),
4035	?	image_for_list_relation(T,S,ImageSoFar2,Res2,WF).
4036
4037
4038
4039		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_for_closure1_wf([(int(1),int(2)),(int(2),int(1)),(int(3),int(3))],[int(2)],[int(1),int(2)],WF),WF)).
4040		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_for_closure1_wf([(int(1),int(2)),(int(2),int(1)),(int(3),int(3))],[],[],WF),WF)).
4041		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_for_closure1_wf([(int(1),int(2)),(int(2),int(1)),(int(3),int(3))],[int(3)],[int(3)],WF),WF)).
4042		% version for computing closure1(Rel)[S]
4043		:- block image_for_closure1_wf(-,?,?,?),image_for_closure1_wf(?,-,?,?).
4044		image_for_closure1_wf(Rel,S,Res,WF) :- (Rel==[] ; S==[]),!,empty_set_wf(Res,WF).
4045		image_for_closure1_wf(Rel,Set,Res,WF) :-
4046		try_expand_and_convert_to_avl_unless_large_wf(Set,ESet,WF),
4047	?	image_for_closure1_wf_aux(Rel,ESet,Res,WF).
4048
4049		:- use_module(library(avl),[avl_height/2]).
4050		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
4051		((nonvar(S),S=avl_set(_))
4052		-> closure1_for_explicit_set_from(Rel,S,Closure1Rel),!,
4053		% if S is known: start from S (currently only deals with Rel=avl_set(_)
4054		range_wf(Closure1Rel,Res,WF)
4055		; Rel=avl_set(AR), avl_height(AR,AR_Height),
4056		((set_smaller_than(S,4),AR_Height>4)
4057		-> !, % TO DO: we could do the same for small S if Rel is large
4058		when(ground(S), (expand_and_convert_to_avl_set(S,ES,image_for_closure1_wf_aux,'closure1(ARG)[?]') ->
4059		closure1_for_explicit_set_from(Rel,avl_set(ES),Closure1Rel),
4060		range_wf(Closure1Rel,Res,WF)
4061		; image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S))
4062		))
4063		; % Don't do this if avl_height too large; then it is probably better to compute the image for S only
4064		AR_Height < 13, % how big should we make this magic constant; or should we time-out ? 2^14=16384
4065		closure1_for_explicit_set(Rel,Closure1Rel),!, % we can compute it effiently; don't use code below
4066		image_wf(Closure1Rel,S,Res,WF)
4067		)
4068		).
4069		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
4070	?	propagate_result_in_range(Rel,S,Res,WF),
4071	?	image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S)).
4072
4073		% no need to treat avl_sets; already covered as special case above
4074		set_smaller_than([],_).
4075		set_smaller_than([_|T],N) :- N>1, nonvar(T), N1 is N-1, set_smaller_than(T,N1).
4076
4077		image_for_closure1_iterate(Rel,S,Acc,Res,WF,FIRST) :-
4078		image_wf0(Rel,S,Res1,WF),
4079		ground_value_check(Res1,RV),
4080	?	image_for_closure1_check_fix(RV,Rel,Acc,Res1,Res,WF,FIRST).
4081
4082		:- block image_for_closure1_check_fix(-,?,?,?,?,?,?).
4083		image_for_closure1_check_fix(_,Rel,Acc,Res1,Res,WF,FIRST) :-
4084		%try_expand_and_convert_to_avl_unless_large_wf(Res1,ERes1,WF),
4085		difference_set(Res1,Acc,New),
4086		try_expand_and_convert_to_avl(New,ENew), % we compute difference_set below; we most definitely will need an explicit finite representation
4087		(not_empty_set_wf(ENew,WF),
4088		union(ENew,Acc,Acc1), % Note: we do not call union_wf - should we do this
4089		% upon first iteration remove also S from New -> New2 and pass New2 to image_for_closure1_iterate
4090		% TO DO: investigate whether this also makes sense for further iterations; always remove S
4091		(FIRST=first_iteration(S) -> difference_set(ENew,S,New2) ; New2=ENew),
4092	?	image_for_closure1_iterate(Rel,New2,Acc1,Res,WF,not_first)
4093		;
4094	?	empty_set_wf(ENew,WF),equal_object_optimized_wf(Acc,Res,image_for_closure1_check_fix,WF)).
4095
4096		% propagate information that if closure1(Rel)[.] = Res => Res <: range(Rel)
4097		% x: 1..n --> 1..n & closure1(x)[{1}] = {} & n=100
4098		:- block propagate_result_in_range(?,?,-,?).
4099		propagate_result_in_range(Rel,_S,_Res,_WF) :-
4100		ground_value(Rel),!. % no propagation required
4101		propagate_result_in_range(Rel,S,[],WF) :- !,
4102		domain_wf(Rel,Domain,WF),
4103		not_subset_of_wf(S,Domain,WF).
4104		propagate_result_in_range(Rel,_,Res,WF) :-
4105		range_wf(Rel,Range,WF),
4106	?	check_subset_of_wf(Res,Range,WF).
4107
4108		:- use_module(probsrc(avl_tools),[avl_height_less_than/2]).
4109
4110		% version for computing iterate(K,Rel)[S]
4111		% iteration
4112		:- block image_for_iterate_wf(?,-,?,?,?,?), image_for_iterate_wf(?,?,-,?,?,?).
4113		image_for_iterate_wf(_Rel,_K,S,Res,_,WF) :- S==[],!,empty_set_wf(Res,WF).
4114		image_for_iterate_wf(Rel,int(K),S,Res,Type,WF) :-
4115		image_for_iterate_k(K,Rel,S,Res,Type,WF).
4116
4117		:- block image_for_iterate_k(-,?,?,?,?,?).
4118		image_for_iterate_k(K,Rel,S,Res,Type,WF) :-
4119		nonvar(Rel),
4120		Rel=avl_set(AVL),
4121		(var(S) -> avl_height_less_than(AVL,11) ; avl_height_less_than(AVL,3)),
4122		!, % compute the iteration once; possibly better constraint propagation and performance if S enumerated
4123		% e.g. x:{1,10,20} & iterate({1|->10,20|->1,10|->20},2)(x) = 20
4124		rel_iterate_wf(Rel,int(K),RelIterated,Type,WF),
4125		image_wf(RelIterated,S,Res,WF).
4126		image_for_iterate_k(K,Rel,S,Res,_,WF) :-
4127		image_for_iterate_k_loop(K,Rel,S,Res,WF).
4128
4129		:- block image_for_iterate_k_loop(?,?,-,?,?).
4130		image_for_iterate_k_loop(0,_Rel,Acc,Result,WF) :- !,
4131		equal_object_optimized_wf(Acc,Result,image_for_iterate_k,WF).
4132		image_for_iterate_k_loop(K,Rel,Acc,Result,WF) :-
4133		image_wf0(Rel,Acc,Acc1,WF), % we could try and detect fix point if K> some limit or time for iteration is measurable
4134		if((K>10, K mod 10 =:= 0, % check for fixpoint every 10 iterations
4135		nonvar(Acc1), Acc1=avl_set(_), quick_custom_explicit_set_approximate_size(Acc1,Size1),
4136		quick_custom_explicit_set_approximate_size(Acc,Size0),
4137		Size0=Size1, % only check for equality if approximate sizes match
4138		equal_explicit_sets_wf(Acc,Acc1,WF)),
4139		K1=0, % fixpoint found, no need to continue iterating
4140		K1 is K-1),
4141		image_for_iterate_k_loop(K1,Rel,Acc1,Result,WF).
4142
4143		special_operator_for_image(b(Rel,Type,_),Kind,Args) :- special_image_aux(Rel,Type,Kind,Args).
4144		special_image_aux(closure(Rel),_,closure,[Rel]). % we have closure1(Rel)[Set] -> avoid computing full closure
4145		special_image_aux(iteration(Rel,K),Type,iteration(Type),[Rel,K]).
4146		% TODO: reflexive closure, id_closure (this will probably be more natural as special case for a value)
4147
4148		image_for_special_operator(closure,[Rel],S,Res,WF) :- image_for_closure1_wf(Rel,S,Res,WF).
4149		image_for_special_operator(iteration(Type),[Rel,K],S,Res,WF) :-
4150		image_for_iterate_wf(Rel,K,S,Res,Type,WF).
4151
4152		:- use_module(kernel_objects,[singleton_set_element/4]).
4153		apply_fun_for_special_operator(Kind,EArgs,FunArg,Res,WF,Span) :-
4154		InitialSet = [FunArg], % TODO: try convert to AVL, note: closure1 not really useful in fun. application context
4155		image_for_special_operator(Kind,EArgs,InitialSet,SetRes,WF),
4156		singleton_set_element(SetRes,Res,Span,WF).
4157
4158		% iterate(%x.(x:NATURAL|x+2),2000)(20) much faster this way, 15 ms vs 4 seconds
4159		% iterate(%x.(x:NATURAL|x+2),2000)[{20}]: ditto
4160
4161
4162		% -----------------------------------
4163
4164		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:apply_to([(int(2),int(22))],int(2),int(22),WF),WF)).
4165		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:apply_to([(int(1),int(22)),(int(3),int(33)),(int(4),int(44))],int(3),int(33),WF),WF)). % used to be wfdet (see in_domain_wf above)
4166		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:apply_to([(int(1),[int(22)]),(int(3),[int(32),int(33)]),(int(4),[int(44)])],int(3),[int(32),int(33)],WF),WF)). % used to be wfdet (see in_domain_wf above)
4167		:- assert_must_succeed(bsets_clp:apply_to([(int(1),int(2))],int(1),int(2),_WF)).
4168		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),int(2),_WF),F=[(int(3),int(2)),(int(2),int(1))])).
4169		:- assert_must_succeed((bsets_clp:apply_to(F,X,int(1),_WF),F=[(int(3),int(2)),(int(2),int(1))],X=int(2))).
4170		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),_,_WF),F=[(int(3),[int(2),int(3)]),(int(2),[])])).
4171
4172		:- assert_must_fail(bsets_clp:apply_to([(int(1),int(2)),(int(1),int(3))],int(1),int(3),_WF)).
4173		/* input not a function */
4174		apply_to(R,X,Y,WF) :- apply_to(R,X,Y,unknown,unknown,WF).
4175	?	apply_to(R,X,Y,Span,WF) :- apply_to(R,X,Y,unknown,Span,WF).
4176
4177		% comment in to perform profiling at function call level; can lead to big slowdowns
4178		%:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4179		%:- use_module(source_profiler,[opt_add_source_location_hits/2]).
4180		%apply_to(_R,_X,_Y,_FunctionType,Span,_WF) :- opt_add_source_location_hits(Span,1),fail.
4181
4182		:- block apply_to(-,-,-,?,?,?).
4183		apply_to(R,X,Y,_FunctionType,Span,WF) :-
4184		% we could check if WD condition discharged in Span
4185		(\+ preferences:preference(find_abort_values,false) ; preference(data_validation_mode,true)),
4186		!,
4187		apply_to_var_block_abort(R,X,Y,R,Span,WF). % we have to know R before we can do anything
4188		apply_to(R,X,Y,FunctionType,Span,WF) :-
4189		(var(R),var(X) -> force_in_domain_wf(X,R,WF) ; true),
4190	?	apply_to1(R,X,Y,R,FunctionType,Span,WF).
4191
4192
4193
4194		:- use_module(preferences,[preference/2]).
4195		:- use_module(clpfd_tables,[can_translate_function_to_element_constraint/2,check_apply_with_element_constraint/5]).
4196		:- block apply_to1(-,-,?,?,?,?,?).
4197		apply_to1(R,X,Y,InitialRel,FunctionType,Span,WF) :-
4198		(var(R) -> apply_to_var(R,X,Y,InitialRel,Span,WF)
4199		; R\=[], can_translate_function_to_element_constraint(R,FunctionType) ->
4200		check_apply_with_element_constraint(R,X,Y,FunctionType,WF)
4201	?	; apply_to_nonvar(R,X,Y,InitialRel,Span,WF),
4202		propagate_range_membership(R,Y)
4203		).
4204		:- block apply_to2(-,-,?,?,?,?).
4205		apply_to2(R,X,Y,InitialRel,Span,WF) :-
4206		(var(R)
4207		-> apply_to_var(R,X,Y,InitialRel,Span,WF)
4208	?	; apply_to_nonvar(R,X,Y,InitialRel,Span,WF)
4209		).
4210
4211		:- use_module(clpfd_lists,[get_finite_fdset_information/2,combine_fdset_information/3,
4212		assert_fdset_information/2,get_fdset_information/2]).
4213		% tested in test 1478; initially slows down NQueens
4214		%:- block propagate_range_membership(-,?). % not necessary
4215		propagate_range_membership([(_,RanEl)|T],X) :- nonvar(RanEl),
4216		preferences:preference(use_clpfd_solver,true),
4217		preferences:preference(find_abort_values,false),
4218		get_finite_fdset_information(RanEl,Info), % TO DO: try and detect if we can apply element/3 from clpfd
4219		\+ ground(X),
4220		get_fdset_information(X,InfoX),
4221		Info \= InfoX, % avoids NQueens slowdown; TO DO: check if more precise than InfoX; otherwise no use in collecting info
4222		!,
4223		propagate_range_membership(T,Info,X).
4224		propagate_range_membership(_,_).
4225		:- block propagate_range_membership(-,?,?).
4226		propagate_range_membership([],Info,El) :- !,
4227		% note: the information for the first few elements might have become more precise; TO DO: wait until list known and then propagate ?+ keep on propagating ??
4228		assert_fdset_information(Info,El).
4229		propagate_range_membership([(_,RanEl)|T],Acc,X) :-
4230		nonvar(RanEl), % otherwise we have no info: we may just as well stop
4231		get_finite_fdset_information(RanEl,RInfo),
4232		combine_fdset_information(Acc,RInfo,NewAcc),
4233		NewAcc \= no_fdset_info,
4234		!,
4235		propagate_range_membership(T,NewAcc,X).
4236		propagate_range_membership(_,_,_).
4237
4238
4239		apply_to_var(R,X,Y,InitialRel,Span,WF) :-
4240		mark_var_set_as_non_empty(R),
4241		get_wait_flag(1.0,apply_to_var,WF,WF1), % see tests 1393, 1562??
4242		% was: get_wait_flag0(WF,WF1), but see test 1706 (in conjunction for improvement for test 2033)
4243		when(((nonvar(WF1),ground(X));nonvar(R)), % only instantiate R when X sufficiently instantiated (TO DO: maybe use some for of equality_objects with existing relation R set up so far ??)
4244		(var(R) ->
4245		R=[(X,Y)|Tail],
4246		optional_functionality_check(Tail,X,WF)
4247		; apply_to_nonvar(R,X,Y,InitialRel,Span,WF))).
4248
4249		:- block apply_to_var_block_abort(-,?,?,?,?,?).
4250		apply_to_var_block_abort(R,X,Y,InitialRel,Span,WF) :-
4251		apply_to_nonvar(R,X,Y,InitialRel,Span,WF).
4252
4253		optional_functionality_check(Tail,X,WF) :-
4254		preferences:preference(disprover_mode,true),!,
4255		not_in_domain_wf(X,Tail,WF). % we assert that R is a function ; when disproving we can assume well-definedness
4256		% Note: this can cut down the search space ; see e.g. test 1230 (but e.g. it will not find a problem with test 1169, RULE_r967_1)
4257		optional_functionality_check(_,_X,_WF). % TO DO: maybe lazily check if we have other elements with X as first arg if find_abort_values is true
4258
4259
4260		:- use_module(closures,[is_recursive_closure/3]).
4261		:- use_module(memoization,[is_memoization_closure/4,apply_to_memoize/8]).
4262		:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4263		:- if(\+ environ(no_wd_checking,true)).
4264		apply_to_nonvar([],X,_Y,InitialRel,Span,WF) :-
4265		\+ preferences:preference(find_abort_values,false),
4266		add_wd_error_span('function applied outside of domain (#2): ', '@fun'(X,InitialRel),Span,WF).
4267		:- endif.
4268		apply_to_nonvar([(X2,Y2)|T],X,Y,InitialRel,Span,WF) :-
4269		equality_objects_wf(X2,X,EqRes,WF),
4270		% this check on Y2 below is important if both Y and Y2 are instantiated but X,X2 not yet
4271		% example: aload_R07_cbc.mch (Savary) or cbc_sequence check for R08_ByteArray for aload_R07 event (test 1349)
4272		% however: slows down test 583 !
4273		(var(EqRes) -> equality_objects_wf(Y2,Y,EqResY,WF),
4274		prop_apply_eqxy(EqResY,EqRes) % propagate: if Y/=Y2 => X/=X2
4275		; EqResY=not_called),
4276	?	apply_to4(EqRes,EqResY,Y2,T,X,Y,InitialRel,Span,WF).
4277		apply_to_nonvar(avl_set(A),X,Y,_InitialRel,Span,WF) :-
4278		apply_to_avl_set(A,X,Y,Span,WF).
4279		apply_to_nonvar(closure(P,T,B),X,Y,_InitialRel,Span,WF) :-
4280		%is_custom_explicit_set(Closure,apply), % should also work for avl_set,...
4281		(is_memoization_closure(P,T,B,MemoID)
4282		% Function application with memoization; currently enabled by add /*@desc memo */ pragma to abstract constant
4283		-> apply_to_memoize(MemoID,P,T,B,X,Y,Span,WF)
4284		; is_recursive_closure(P,T,B) % TO DO: maybe we should do the same for functions marked as memoize symbolic/uni-directional/computed ? (although we have new rule for check_element_of_function_closure which makes this redundant ??)
4285		-> % print_term_summary(apply_recursive_closure(X,P,T,B)),
4286		%hit_profiler:add_profile_hit(rec_apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4287		ground_value_check(X,XV), block_apply_closure_to_nonvar_groundx(XV,X,Y,P,T,B,Span,WF)
4288		; %hit_profiler:add_profile_hit(apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4289		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)).
4290
4291
4292		:- block block_apply_closure_to_nonvar_groundx(-,?,?, ?,?,?, ?,?).
4293		block_apply_closure_to_nonvar_groundx(_,X,Y, P,T,B, Span,WF) :- apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF).
4294
4295		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4296		kernel_tools:ground_bexpr(B),
4297		!, % then if the element of function succeeds there is no need to check WD
4298		if(check_element_of_function_closure(X,Y,P,T,B,WF),
4299		true, % No need to check for well-definedness; no pending choice points
4300		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF) % here we need to check; it could be that the result Y was instantiated
4301		).
4302		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4303		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF).
4304
4305		% if we first check preferences:preference(find_abort_values,false) to avoid a choice
4306		% point, we get a big slow-down on Alstom models; e.g., vesg_Mar12
4307		% WARNING: This choice point can be set up in WF0 !
4308		apply_closure_to_nonvar(X,Y,P,T,B,_,WF) :-
4309		(preferences:preference(find_abort_values,true) -> true ; !), % slow down ???!
4310		check_element_of_function_closure(X,Y,P,T,B,WF) .
4311		apply_closure_to_nonvar(X,_,P,T,B,Span,WF) :- % removing this clause doubles runtime of COMPUTE_GRADIENT_CHANGE
4312		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF).
4313
4314		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF) :-
4315		\+ preferences:preference(find_abort_values,false),
4316		not_in_domain_wf(X,closure(P,T,B),WF),
4317		when((ground(X),ground(closure(P,T,B))),
4318		add_wd_error_span('function applied outside of domain (#3): ', '@fun'(X,closure(P,T,B)),Span,WF)).
4319
4320
4321		% propagate equality_objects between range and domain elements for function application:
4322		:- block prop_apply_eqxy(-,-).
4323		prop_apply_eqxy(Eqy,Eqx) :- var(Eqy),!, (Eqx = pred_true -> Eqy = pred_true ; true).
4324		prop_apply_eqxy(pred_false,pred_false).
4325		prop_apply_eqxy(pred_true,_).
4326
4327		:- block apply_to4(-,?,?, -,?,?,?,?,?).
4328		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4329		var(EqResX),!, % Tail bound
4330		(Tail == []
4331		-> (preferences:preference(find_abort_values,false)
4332		-> EqResX = pred_true,
4333		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4334		; apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4335		)
4336		; Tail = avl_set(_) -> apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) % TO DO: improve ! (e.g., expand to list if small or check if X can be in domain,...)
4337		; Tail = closure(_,_,_) -> apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4338		; Tail \= [_|_] -> add_internal_error('Illegal Tail: ',apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)),fail
4339		; Tail = [(X3,Y3)|T3], % setup equality check with X3, purpose: detect, e.g., when no other element in tail can match we can force EqResX to pred_true
4340	?	apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3)
4341		).
4342		apply_to4(pred_true,EqResY,Y2, Tail,X,Y,_InitialRel,_,WF) :-
4343	?	(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to4,WF) ; EqResY = pred_true),
4344		optional_functionality_check(Tail,X,WF).
4345	?	apply_to4(pred_false,_EqResY,_Y2,T,X,Y,InitialRel,Span,WF) :- apply_to2(T,X,Y,InitialRel,Span,WF).
4346
4347		% we delay setting up equality_objects until X3 is at least partially known, see test 1715 Alstom_essai2_boucle1
4348		% TO DO: we could check if X3==X above
4349		:- block apply_to4_call5(-,?,?, ?,?,?,?,?,?, -,?,?).
4350		apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, _X3,_Y3,_T3) :- nonvar(EqResX),!,
4351		apply_to4(EqResX,EqResY,Y2,Tail,X,Y,InitialRel,Span,WF).
4352		apply_to4_call5(EqResX,EqResY,Y2, _Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3) :- % X3 must now be bound
4353		equality_objects_wf(X3,X,EqRes3,WF),
4354	?	apply_to5(EqResX,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, InitialRel,Span,WF).
4355
4356		% version which wait suntil first argument known
4357		:- block apply_to4_block(-,?,?, ?,?,?,?,?,?).
4358		apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4359		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF).
4360
4361
4362		% apply_to5: implements a watched-literal style treatment of function application
4363		% we watch whether X unifies with two elements of the function, if only one element left we can force equality
4364		% TEST:
4365		% f : 11..23 +-> 1..10 & f = {a|->2, b|->3, c|->4} & card({a,b,c})=3 & f(x)=r & a>b & b>c & x>b
4366		:- block apply_to5(-,?,-, ?,?,?,?, ?,?, ?,?,?),apply_to5(-,?,?, ?,?,?,-, ?,?, ?,?,?).
4367		apply_to5(EqRes,EqResY,EqRes3, Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4368		var(EqRes),!,
4369		% EqRes3 and T3 must be known; TO DO: improve predicate so that we have to wait on T3 only when EqRes3=pred_false
4370		(EqRes3 = pred_false -> % we cannot match next element, move tail one forward
4371		(T3 = [] -> EqRes=pred_true ; true),
4372		apply_to4(EqRes,EqResY,Y2,T3,X,Y,InitialRel,Span,WF)
4373		; /* EqRes3 = pred_true */
4374		% we match the next entry in the list; discard Y2 and jump to (X3,Y3) and return as solution
4375	?	equal_object_wf(Y3,Y,apply_to6,WF), optional_functionality_check(T3,X,WF),
4376		% TO DO: we could also do equality_objects if necessary between Y and Y3, as in apply_to4 for Y and Y2
4377		opt_force_false(EqRes)
4378		).
4379		apply_to5(pred_true,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, _InitialRel,_Span,WF) :-
4380		(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to5,WF) ; EqResY = pred_true),
4381		opt_force_false(EqRes3),
4382		optional_functionality_check([(X3,Y3)|T3],X,WF).
4383		apply_to5(pred_false,_EqResY,EqRes3, _Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4384		(var(EqRes3) -> % it can be that EqRes3 is about to be triggered
4385		equality_objects_wf(Y3,Y,EqResY3,WF),
4386		prop_apply_eqxy(EqResY3,EqRes3) % propagate: if Y/=Y3 => X/=X3
4387		; EqResY3=not_called),
4388		apply_to4(EqRes3,EqResY3,Y3, T3,X,Y,InitialRel,Span,WF).
4389
4390		opt_force_false(EqRes) :-
4391		(preference(find_abort_values,false) -> EqRes=pred_false
4392		; true). % TO DO: if EqRes becomes pred_true: raise abort_error as the relation was not a function
4393
4394
4395
4396		/********************************************/
4397		/* surjection_relation(R,Domain,Range) */
4398		/* R : Domain <->> Range */
4399		/********************************************/
4400		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:surjection_relation_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)). % used to be wfdet (see in_domain_wf above)
4401		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:surjection_relation_wf([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(6),int(7)],WF),WF)).
4402
4403		surjection_relation_wf(R,Domain,Range,WF) :-
4404		is_surjective(R,Range,WF),
4405		% TODO: is not optimal since ran(R)<:Range is already implied by is_surjective and
4406		% checked a second time by relation_over_wf/4
4407	?	relation_over_wf(R,Domain,Range,WF).
4408
4409		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_surjection_relation_wf([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(6),int(7)],WF),WF)).
4410
4411		not_surjection_relation_wf(R,Domain,Range,WF) :-
4412		expand_custom_set_to_list_wf(R,ER,Done,not_surjection_relation_wf,WF),
4413		not_tot_surj_rel(ER,Done,[],Domain,Range,Range,WF).
4414
4415		/*********************************************/
4416		/* total_surjection_relation(R,Domain,Range) */
4417		/* R : Domain <<->> Range */
4418		/*********************************************/
4419		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:total_surjection_relation_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4420		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:total_surjection_relation_wf([(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4421
4422
4423		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1)],[int(11),int(12)]),L),
4424		lists:maplist(sort,L,SL), sort(SL,SSL), % added May15th due to change in domain_wf (bsets_clp:propagate_result_to_input); TO DO: see if we can go back to just one solution
4425		length(SSL,1))).
4426		%:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11),int(12)]),L), length(L,7))).
4427		% the new domain predicate also instantiates from result; meaning that duplicate solutions are now generated
4428		:- assert_must_succeed((findall(SR,(bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11),int(12)]),sort(R,SR)),L), sort(L,SL),length(SL,7))).
4429		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11)]),L),
4430		length(L,1))).
4431
4432		total_surjection_relation(R,Domain,Range) :- init_wait_flags(WF,[total_surjection_relation]),
4433	?	total_surjection_relation_wf(R,Domain,Range,WF), ground_wait_flags(WF).
4434
4435		total_surjection_relation_wf(R,Domain,Range,WF) :-
4436	?	relation_over_wf(R,Domain,Range,WF),
4437		check_relation_is_total(R,Domain,WF), % calls domain which now instantiates R if Domain known
4438		check_relation_is_surjective(R,Range,WF).
4439
4440
4441		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_relation_wf([(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4442		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_relation_wf([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4443		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_total_surjection_relation_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4444
4445		not_total_surjection_relation_wf(R,Domain,Range,WF) :-
4446		expand_custom_set_to_list_wf(R,ER,Done,not_total_surjection_relation_wf,WF),
4447	?	not_tot_surj_rel(ER,Done,Domain,Domain,Range,Range,WF).
4448
4449
4450		/********************************************/
4451		/* partial_surjection(R,DomType,RangeType) */
4452		/* R : DomType +->> RangeType */
4453		/********************************************/
4454
4455		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)). % used to be wfdet (see in_domain_wf above)
4456		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6),int(2)],WF),WF)).
4457		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4458		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4459		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],global_set('Name')),
4460		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4461		:- assert_must_succeed((bsets_clp:partial_surjection_wf(X,[int(1),int(2),int(3)],global_set('Name'),_WF),
4462		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4463		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4464		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4465		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4466		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))]))). /* mult. */
4467		:- assert_must_succeed((X=[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))],
4468		bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]))).
4469		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4470		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4471		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4472		X = [(int(2),int(7)),(int(1),int(7))])).
4473		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4474		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4475		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(_X,
4476		[int(1),int(2),int(3),int(4),int(5),int(6),int(7)],[int(2),int(3),int(4)]) )).
4477		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4478		X = [(int(2),int(7)),(int(2),int(6))])).
4479
4480		partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[partial_surjection]),
4481		partial_surjection_wf(R,Domain,Range,WF),
4482	?	ground_wait_flags(WF).
4483
4484		:- block partial_surjection_wf(-,-,?,?).
4485		partial_surjection_wf(R,Domain,Range,WF) :-
4486		check_card_greater_equal(Domain,geq,Range,CardDom,CardRange),
4487		(surjection_has_to_be_total_injection(CardDom,CardRange)
4488		% LAW: card(setX) = card(setY) => ff: setX +->> setY <=> ff: setX >-> setY
4489	?	-> total_function_wf(R,Domain,Range,WF),
4490		injective(R,WF)
4491		; is_surjective(R,Range,WF),
4492		partial_function_wf(R,Domain,Range,WF)
4493		).
4494
4495
4496		% check_card_greater_equal(A,B) : quick check that card(A) >= card(B); also works with infinite cardinality
4497		% TO DO: replace by a better constraint propagating predicate (also working for partially instantiated lists,...)
4498		% compared with computing card and setting up < constraint: will only compute card if it can be done efficiently + deals with inf
4499		% check_card_greater_equal(SetA,EQ,SetB) ; EQ=eq or geq
4500		:- block check_card_greater_equal(-,?,?,?,?).
4501		check_card_greater_equal([],_,R,0,0) :- !, empty_set(R).
4502		check_card_greater_equal(A,EQ,B,CA,CB) :- check_card_greater_equal2(A,EQ,B,CA,CB).
4503
4504		:- use_module(inf_arith,[block_inf_greater_equal/2]).
4505		:- block check_card_greater_equal2(?,?,-,?,?).
4506		check_card_greater_equal2(A,EQ,B,CardA,CardB) :-
4507		efficient_card_for_set(A,CardA,CodeA),
4508		efficient_card_for_set(B,CardB,CodeB),!,
4509		call(CodeA), call(CodeB),
4510		(EQ=eq -> CardA=CardB ; block_inf_greater_equal(CardA,CardB)).
4511		check_card_greater_equal2(_A,_,_B,'?','?').
4512
4513
4514		:- block is_surjective(-,-,?).
4515		is_surjective(R,Range,WF) :-
4516		(var(R) -> setup_surj_range(Range,R,WF)
4517		; range_wf(R,Range,WF)).
4518
4519		setup_surj_range(Range,R,WF) :-
4520		setup_range(Range,Res,DONE,WF),
4521		equal_when_done(Res,R,DONE).
4522		:- block equal_when_done(?,?,-).
4523	?	equal_when_done(Res,R,_DONE) :- equal_object(Res,R).
4524
4525
4526		:- block setup_range(-,?,?,?).
4527		setup_range(global_set(G),Res,DONE,WF) :-
4528		expand_custom_set_wf(global_set(G),ES,setup_range,WF),
4529		setup_range(ES,Res,DONE,WF).
4530		setup_range(freetype(ID),Res,DONE,WF) :-
4531		expand_custom_set_wf(freetype(ID),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4532		setup_range(avl_set(S),Res,DONE,WF) :-
4533		expand_custom_set_wf(avl_set(S),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4534		setup_range(closure(P,T,B),Res,DONE,WF) :-
4535		expand_custom_set_wf(closure(P,T,B),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4536		setup_range([],_,done,_WF).
4537		setup_range([H|T],[(_,H)|ST],DONE,WF) :- setup_range(T,ST,DONE,WF).
4538
4539
4540
4541		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],
4542		[int(1),int(2)],[int(7),int(6)],WF),WF)).
4543		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],
4544		[int(7),int(6),int(2)],WF),WF)).
4545		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4546		X = [(int(2),int(7)),(int(1),int(6))])).
4547		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4548		X = [(int(2),int(7)),(int(2),int(6))])).
4549		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4550		X = [(int(2),int(7)),(int(1),int(6))])).
4551		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4552		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4553		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4554		X = [(int(2),int(7)),(int(1),int(7))])).
4555		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4556		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4557
4558
4559
4560		/* /: Domain +->> Range */
4561		not_partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[not_partial_surjection]),
4562		not_partial_surjection_wf(R,Domain,Range,WF),
4563		ground_wait_flags(WF).
4564
4565		:- block not_partial_surjection_wf(-,?,?,?).
4566		not_partial_surjection_wf(R,DomType,RangeType,WF) :-
4567		partial_surjection_test_wf(R,DomType,RangeType,pred_false,WF).
4568
4569
4570		%not_surjective_relation_wf(R,DomType,RType,WF) :-
4571		% invert_relation_wf(R,IR,WF),
4572		% not_total_relation_wf(IR,RType,DomType,WF).
4573
4574
4575		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:partial_surjection_test_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],pred_true,WF),WF)).
4576		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:partial_surjection_test_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6),int(2)],pred_false,WF),WF)).
4577
4578		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF) :-
4579		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4580		(IsPF==pred_false -> PredRes=pred_false
4581		; range_wf(R,RelRan,WF),
4582	?	conjoin_test(IsPF,IsSurjective,PredRes,WF),
4583	?	subset_test(RangeType,RelRan,IsSurjective,WF)
4584		).
4585
4586
4587		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:total_relation_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4588		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:total_relation_wf([(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4589
4590		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4591		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4592		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4593		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4594		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4595		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(7))]))).
4596		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4597		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4598		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4599		kernel_objects:equal_object(X,[(int(2),int(7))]))).
4600		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4601		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(8))]))).
4602		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4603		kernel_objects:equal_object(X,[(int(2),int(7)),(int(3),int(6)),(int(1),int(7))]))).
4604		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4605		kernel_objects:equal_object(X,[]))).
4606
4607		/****************************************/
4608		/* total_relation_wf(R,Domain,Range,WF) */
4609		/* R : Domain <<-> Range */
4610		/****************************************/
4611
4612	?	total_relation_wf(R,Domain,Range,WF) :- relation_over_wf(R,Domain,Range,WF),
4613		check_relation_is_total(R,Domain,WF).
4614
4615		% this predicates assume that the relation's range and domain have already been checked
4616		check_relation_is_total(Relation,Domain,WF) :- domain_wf(Relation,Domain,WF).
4617		check_relation_is_surjective(Relation,Range,WF) :-
4618		range_wf(Relation,Range,WF). % we could also call is_surjective (which does setup_surj_range) ?
4619
4620		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_total_relation_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4621		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_relation_wf([(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4622		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4623		X = [(int(2),int(7)),(int(1),int(6))])).
4624		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4625		X = [(int(2),int(7)),(int(1),int(7))])).
4626		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4627		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4628		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4629		X = [(int(2),int(7)),(int(2),int(6))])).
4630		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4631		X = [(int(2),int(7))])).
4632		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4633		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(8))])).
4634		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4635		X = [(int(2),int(7)),(int(3),int(6)),(int(1),int(7))])).
4636
4637		:- block not_total_relation_wf(-,?,?,?).
4638		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
4639		% we do not need the Range; this means we can match more closures (e.g., lambda)
4640		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
4641		not_equal_object_wf(FFDomain,Domain,WF).
4642		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),
4643		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
4644		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
4645		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
4646		not_total_relation_wf(R,Domain,Range,WF) :-
4647		expand_custom_set_to_list_wf(R,ER,Done,not_total_relation_wf,WF),
4648	?	not_tot_surj_rel(ER,Done,Domain,Domain,[],Range,WF). % empty DelRange means we don't do surjective test
4649
4650		% can be used to check not total, not surj, not total surj relation
4651		:- block not_tot_surj_rel(-,?,?,?,?,?,?).
4652		not_tot_surj_rel([],_,DelDomain,_,DelRange,_,WF) :-
4653	?	at_least_one_set_not_empty(DelDomain,DelRange,WF).
4654		not_tot_surj_rel([_|_],Done,DelDom,Dom,_DelRan,_Ran,_WF) :- nonvar(Done),
4655		Done \= no_check_to_be_done,
4656		nonvar(DelDom),DelDom \= [],
4657		nonvar(Dom),is_infinite_explicit_set(Dom),
4658		!. % a finite expanded list can never be a total relation over an infinite domain
4659		not_tot_surj_rel([(X,Y)|T],_Done,DelDom,Dom,DelRan,Ran,WF) :-
4660		membership_test_wf(Dom,X,MemRes,WF),
4661	?	not_tr2(MemRes,X,Y,T,DelDom,Dom,DelRan,Ran,WF).
4662
4663		% check if one of the two sets is non-empty
4664		at_least_one_set_not_empty(Set1,Set2,_) :- (Set=Set1 ; Set=Set2),
4665		nonvar(Set),
4666		(Set=avl_set(_) ; Set=[_|_]), % we can avoid leaving choice point
4667		!.
4668		at_least_one_set_not_empty(Set1,_,WF) :- not_empty_set_wf(Set1,WF).
4669		at_least_one_set_not_empty(Set1,Set2,WF) :- empty_set_wf(Set1,WF),not_empty_set_wf(Set2,WF).
4670
4671		:- block not_tr2(-,?,?,?,?,?,?,?,?).
4672		not_tr2(pred_false,_X,_Y,_T,_DelDom,_Dom,_DelRan,_Ran,_WF).
4673		not_tr2(pred_true,X,Y,T,DelDom,Dom,DelRan,Ran,WF) :-
4674		delete_element_wf(X,DelDom,DelDom2,WF), % set DelDom initially to [] to avoid totality check
4675		membership_test_wf(Ran,Y,MemRes,WF),
4676	?	not_tr3(MemRes,Y,T,DelDom2,Dom,DelRan,Ran,WF).
4677
4678		:- block not_tr3(-,?,?,?,?,?,?,?).
4679		not_tr3(pred_false,_Y,_T,_DelDom2,_Dom,_DelRan,_Ran,_WF).
4680		not_tr3(pred_true,Y,T,DelDom2,Dom,DelRan,Ran,WF) :-
4681		delete_element_wf(Y,DelRan,DelRan2,WF), % set DelRan initially to [] to avoid surjection check
4682	?	not_tot_surj_rel(T,no_check_to_be_done,DelDom2,Dom,DelRan2,Ran,WF).
4683
4684		/******************************************/
4685		/* total_surjection(R,DomType,RangeType) */
4686		/* R : DomType -->> RangeType */
4687		/******************************************/
4688
4689		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:total_surjection_wf([(int(2),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)). % used to be wfdet (see in_domain_wf above)
4690		:- assert_must_succeed(exhaustive_kernel_succeed_check((bsets_clp:total_surjection_wf([(int(2),int(6)),(int(1),int(7)),(int(3),int(7))],[int(1),int(2),int(3)],[int(7),int(6)],WF),kernel_waitflags:ground_det_wait_flag(WF)))). %% TO DO: get rid of multiple solutions
4691		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1)],[int(7)]),
4692		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4693		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4694		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4695		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7)]),
4696		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4697		:- assert_must_fail((bsets_clp:total_surjection([],[int(1)],[int(7)]))).
4698		:- assert_must_fail((bsets_clp:total_surjection([(int(7),int(7))],[int(1)],[int(7)]))).
4699		:- assert_must_fail((bsets_clp:total_surjection([(int(1),int(7)), (int(2),int(1))],
4700		[int(1),int(2)],[int(7)]))).
4701		:- assert_must_fail((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4702		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4703
4704
4705		total_surjection(R,Domain,Range) :- init_wait_flags(WF),
4706		total_surjection_wf(R,Domain,Range,WF),
4707	?	ground_wait_flags(WF).
4708
4709		:- block total_surjection_wf(-,-,?,?).
4710		total_surjection_wf(R,DomType,RangeType,WF) :-
4711		check_card_greater_equal(DomType,geq,RangeType,CardDom,CardRange),
4712	?	total_function_wf(R,DomType,RangeType,WF),
4713		% setup_surj_range(RangeType,R,WF).
4714		(surjection_has_to_be_total_injection(CardDom,CardRange)
4715		% LAW: card(setX) = card(setY) => ff: setX -->> setY <=> ff: setX >-> setY
4716		-> injective(R,WF) % if domain and range have same cardinality: injection ensures surjectivity, and is more efficient to check/propagate; example when using queens 1..n -->> 1..n for NQueens
4717		; check_relation_is_surjective(R,RangeType,WF)).
4718		% invert_relation_wf(R,IR,WF), total_relation_wf(IR,RangeType,DomType,WF).
4719
4720		surjection_has_to_be_total_injection(CardDom,CardRange) :- number(CardDom), CardDom=CardRange.
4721		% TO DO: determine the difference in size between Dom and Range and count how many times a range element can occur multiple times (would give better incremental checking)
4722
4723		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(7))],[int(1),int(2),int(3)],[int(7),int(6)],WF),WF)).
4724		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4725		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4726		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(7)),(int(3),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4727		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(7)),(int(3),int(8))],[int(1),int(2),int(3)],[int(7),int(6)],WF),WF)).
4728		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_total_surjection_wf([(int(2),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4729
4730		:- block not_total_surjection_wf(-,?,?,?), not_total_surjection_wf(?,-,-,?).
4731		not_total_surjection_wf(R,DomType,RangeType,WF) :-
4732		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4733		not_total_surjection2(PredRes,R,DomType,RangeType,WF).
4734		:- block not_total_surjection2(-,?,?,?,?).
4735		not_total_surjection2(pred_false,_R,_DomType,_RangeType,_WF).
4736		not_total_surjection2(pred_true,R,_DomType,RangeType,WF) :-
4737		range_wf(R,RelRange,WF),
4738		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
4739		[RangeType,b_operator(range,[R])],unknown),WF2),
4740		not_subset_of_wf(RangeType,RelRange,WF2).
4741		%not_surjective_relation_wf(R,DomType,RangeType,WF).
4742
4743		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:total_function_test_wf([(int(2),int(6)),(int(1),int(7)),(int(3),int(8))],[int(1),int(2),int(3)],[int(7),int(6)],pred_false,WF),WF)).
4744		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:total_function_test_wf([(int(2),int(6)),(int(1),int(7)),(int(3),int(6))],[int(1),int(2),int(3)],[int(7),int(6)],pred_true,WF),WF)). % used to be wfdet (see in_domain_wf above)
4745		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:total_function_test_wf([(int(2),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],pred_false,WF),WF)).
4746
4747		% reified total function check:
4748		total_function_test_wf(R,DomType,RangeType,PredRes,WF) :-
4749		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4750		(IsPF==pred_false -> PredRes=pred_false
4751		; domain_wf(R,RelDom,WF),
4752	?	conjoin_test(IsPF,IsTotal,PredRes,WF),
4753		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
4754		[DomType,b_operator(domain,[R])],unknown),WF2),
4755	?	subset_test(DomType,RelDom,IsTotal,WF2)
4756		).
4757
4758		/*******************************************/
4759		/* partial_injection(R,DomType,RangeType) */
4760		/* R : DomType >+> RangeType */
4761		/*******************************************/
4762
4763		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_injection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4764		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_injection_wf([(int(1),int(6)),(int(4),int(7)),(int(2),int(8))],[int(1),int(2),int(3),int(4)],[int(7),int(6),int(8),int(9)],WF),WF)).
4765		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:partial_injection_wf([(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4766		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1)],[int(7)]),
4767		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4768		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4769		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4770		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7)]),
4771		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4772		:- assert_must_succeed((bsets_clp:partial_injection([],[int(1)],[int(7)]))).
4773		:- assert_must_fail((bsets_clp:partial_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4774		:- assert_must_fail((bsets_clp:partial_injection([(int(1),int(7)), (int(2),int(1))],
4775		[int(1),int(2)],[int(7)]))).
4776		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4777		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4778
4779
4780		partial_injection(R,Domain,Range) :- init_wait_flags(WF),
4781		partial_injection_wf(R,Domain,Range,WF),
4782	?	ground_wait_flags(WF).
4783
4784		:- block partial_injection_wf(-,-,?,?).
4785		partial_injection_wf(FF,Domain,Range,WF) :- nonvar(FF),
4786		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
4787		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
4788		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
4789		partial_injection_wf(R,DomType,RangeType,WF) :-
4790		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF), % should we use very_large?
4791		partial_function_wf(ER,DomType,RangeType,WF),
4792		injective(ER,WF).
4793		% invert_relation_wf(R,IR,WF),
4794		% partial_function_wf(IR,RangeType,DomType,WF).
4795
4796		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective([(int(1),int(6)),(int(4),int(7)),(int(2),int(8))],WF),WF)).
4797		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:injective([(int(1),int(6)),(int(4),int(7)),(int(2),int(7))],WF),WF)).
4798
4799		:- block injective(-,?).
4800		injective(FF,WF) :-
4801		custom_explicit_sets:dom_range_for_specific_closure(FF,_FFDomain,_FFRange,function(bijection),WF),!.
4802		injective(avl_set(AVL),_WF) :- !,
4803		is_injective_avl_relation(AVL,_Range). % seems slightly faster than injective/3 code below
4804		injective(closure(P,T,B),WF) :- !,
4805		symbolic_injectivity_check(closure(P,T,B),WF).
4806		injective(Rel,WF) :- expand_custom_set_to_list_wf(Rel,ERel,_,injective,WF),
4807		injective(ERel,[],WF).
4808
4809		%:- use_module(library(lists),[maplist/3]).
4810		% for FD-sets we could setup all_different constraint
4811		:- block injective(-,?,?).
4812		injective([],_SoFar,_).
4813		% (maplist(get_fd_val,SoFar,FDL) -> clpfd:all_distinct(FDL) ; true). %clpfd_interface:clpfd_alldifferent(FDL) ; true).
4814		%get_fd_val(int(H),H).
4815		injective([(_From,To)|T],SoFar,WF) :-
4816		not_element_of_wf(To,SoFar,WF), /* check that it is injective */
4817		add_new_element_wf(To,SoFar,SoFar2,WF), %SoFar2=[To|SoFar], could also work and be faster ?
4818		injective(T,SoFar2,WF).
4819		% no case for global_set: it cannot be a relation; two cases below not required because of expand_custom_set_to_list
4820		%injective(avl_set(S),SoFar,WF) :- expand_custom_set_wf(avl_set(S),ES,inj,WF), injective(ES,SoFar,WF).
4821		%injective(closure(P,T,B),SoFar,WF) :- expand_custom_set_wf(closure(P,T,B),ES,inj,WF), injective(ES,SoFar,WF).
4822
4823
4824
4825		/* /: Dom >+> R */
4826
4827		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_injection([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4828		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_injection([(int(1),int(6)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4829		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_injection([(int(1),int(6)),(int(2),int(8))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4830		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_partial_injection([(int(1),int(6)),(int(3),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4831
4832		:- block not_partial_injection(-,?,?,?).
4833		not_partial_injection(R,DomType,RangeType,WF) :-
4834		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4835		not_partial_injection2(IsPF,R,DomType,RangeType,WF).
4836
4837		:- block not_partial_injection2(-,?,?,?,?).
4838		not_partial_injection2(pred_false,_R,_DomType,_RType,_WF).
4839		not_partial_injection2(pred_true,R,DomType,RType,WF) :-
4840		not_injection_wf(R,DomType,RType,WF).
4841
4842		not_injection_wf(R,DomType,RType,WF) :-
4843		invert_relation_wf(R,IR,WF),
4844		not_partial_function(IR,RType,DomType,WF).
4845
4846		/*****************************************/
4847		/* total_injection(R,DomType,RangeType) */
4848		/* R : DomType >-> RangeType */
4849		/*****************************************/
4850
4851		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:total_injection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4852		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:total_injection_wf([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4853		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_total_injection([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4854		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_total_injection([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4855		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1)],[int(7)]),
4856		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4857		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4858		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4859		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7)]),
4860		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4861		:- assert_must_fail((bsets_clp:total_injection([],[int(1)],[int(7)]))).
4862		:- assert_must_fail((bsets_clp:total_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4863		:- assert_must_fail((bsets_clp:total_injection([(int(1),int(7)), (int(2),int(1))],
4864		[int(1),int(2)],[int(7)]))).
4865		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4866		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4867
4868
4869		total_injection(R,Domain,Range) :- init_wait_flags(WF),
4870		total_injection_wf(R,Domain,Range,WF),
4871	?	ground_wait_flags(WF).
4872
4873		:- block total_injection_wf(-,-,?,?). % with just ?,-,?,? we may wait too long to start injective check
4874		% Note: no need to check: dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection)),
4875		total_injection_wf(R,DomType,RangeType,WF) :-
4876		check_card_greater_equal(RangeType,geq,DomType,_,_), % there must be more Range elements than domain elements; pigeonhole principle
4877	?	total_injection_wf2(R,DomType,RangeType,WF).
4878		total_injection_wf2(R,DomType,RangeType,WF) :-
4879		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
4880	?	total_function_wf(ER,DomType,RangeType,WF),
4881		injective(ER,WF).
4882
4883
4884		:- block not_total_injection(-,?,?,?), not_total_injection(?,-,-,?).
4885		not_total_injection(R,DomType,RangeType,WF) :-
4886		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4887		not_total_injection2(PredRes,R,DomType,RangeType,WF).
4888
4889		:- block not_total_injection2(-,?,?,?,?).
4890		not_total_injection2(pred_false,_R,_Dom,_Ran,_WF).
4891		not_total_injection2(pred_true,R,DomType,RangeType,WF) :-
4892		% TO DO: replace DomType and RangeType by full Type
4893		not_injection_wf(R,DomType,RangeType,WF).
4894
4895		/***********************************/
4896		/* partial_bijection(R,DomType,RangeType) */
4897		/* R : DomType >+>> RangeType */
4898		/***********************************/
4899
4900		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:partial_bijection_wf([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)). % used to be wfdet (see in_domain_wf above)
4901		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:partial_bijection_wf([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4902		:- assert_must_succeed((partial_bijection(X,[int(1),int(2)],[int(7),int(6)]),
4903		kernel_objects:equal_object(X,[(int(1),int(6)),(int(2),int(7))]))).
4904		:- assert_must_succeed((partial_bijection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4905		X = [(int(2),int(7)),(int(3),int(6))])).
4906		:- assert_must_fail((partial_bijection(X,[int(1),int(2)],[int(7),int(6),int(5)]),
4907		X = [(int(2),int(7)),(int(1),int(6))])).
4908
4909		partial_bijection(R,Domain,Range) :- init_wait_flags(WF),
4910		partial_bijection_wf(R,Domain,Range,WF),
4911	?	ground_wait_flags(WF).
4912
4913		partial_bijection_wf(R,DomType,RangeType,WF) :-
4914		partial_injection_wf(R,DomType,RangeType,WF),
4915	?	partial_surjection_wf(R,DomType,RangeType,WF).
4916
4917		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_bijection([(int(1),int(6)),(int(2),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4918		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_partial_bijection([(int(1),int(6)),(int(2),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4919
4920		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_bijection([(int(2),int(7)),(int(1),int(6))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
4921
4922		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_bijection([(int(2),int(7)),(int(3),int(6))],[int(1),int(2),int(3),int(4)],[int(7),int(6)],WF),WF)).
4923		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_partial_bijection([(int(2),int(7)),(int(1),int(6))],[int(1),int(2)],[int(7),int(6),int(5)],WF),WF)).
4924
4925
4926		:- block not_partial_bijection(-,?,?,?), not_partial_bijection(?,-,-,?).
4927		not_partial_bijection(R,DomType,RangeType,WF) :-
4928		% >+>> = +->> + injective
4929		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF),
4930		not_partial_bijection2(PredRes,R,DomType,RangeType,WF).
4931
4932		:- block not_partial_bijection2(-,?,?,?,?).
4933		not_partial_bijection2(pred_false,_R,_DomType,_RangeType,_WF).
4934		not_partial_bijection2(pred_true,R,DomType,RangeType,WF) :-
4935		not_injection_wf(R,DomType,RangeType,WF).
4936
4937
4938
4939		/* The transitive (not reflexive) closure of a relation (closure1) */
4940
4941		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:relational_trans_closure([(int(1),int(2)),(int(2),int(6))],[(int(1),int(2)),(int(1),int(6)),(int(2),int(6))]))).
4942		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:relational_trans_closure([(int(1),int(2)),(int(2),int(6)),(int(1),int(3))],[(int(1),int(2)),(int(1),int(3)),(int(1),int(6)),(int(2),int(6))]))).
4943		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:relational_trans_closure([(int(6),int(7)),(int(1),int(2)),(int(2),int(6)),(int(1),int(3))],[(int(1),int(2)),(int(1),int(3)),(int(1),int(6)),(int(2),int(6)),(int(1),int(7)),(int(2),int(7)),(int(6),int(7))]))).
4944		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4))],X),
4945		kernel_objects:equal_object(X,[(int(1),int(4))]))).
4946		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2))],X),
4947		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),
4948		(int(1),int(2))]))).
4949		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2)),(int(2),int(3))],X),
4950		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),(int(2),int(3)),
4951		(int(4),int(3)),(int(1),int(2)),(int(1),int(3))]))).
4952		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(A),int(2)),(int(2),int(6))],
4953		[(int(1),int(2)),(int(1),int(6)),(int(2),int(6))]),A=1)).
4954
4955		relational_trans_closure(Rel,Res) :- relational_trans_closure_wf(Rel,Res,no_wf_available).
4956
4957		% transitive closure for relations (closure1)
4958		:- block relational_trans_closure_wf(-,?,?).
4959		relational_trans_closure_wf(Relation,Result,WF) :-
4960		try_expand_and_convert_to_avl_with_check(Relation,ARelation,relational_trans_closure_wf),
4961		relational_trans_closure2(ARelation,Result,WF).
4962		:- block relational_trans_closure2(-,?,?).
4963		relational_trans_closure2(ARelation,Result,WF) :-
4964		(closure1_for_explicit_set(ARelation,Res)
4965		-> kernel_objects:equal_object_wf(Res,Result,relational_trans_closure_wf,WF)
4966		; expand_custom_set_to_list_wf(ARelation,ERelation,_,relational_trans_closure2,WF),
4967		is_full_relation(ERelation,WaitVar), % still required??
4968		% we could do a check_subset_of_wf(ERelation,Resul,WF) if Result is nonvar and ERelation not ground
4969		compute_trans_closure(ERelation,Result,WaitVar,WF)
4970		).
4971
4972		:- block compute_trans_closure(?,?,-,?).
4973		compute_trans_closure(Relation,Result,_,WF) :-
4974	?	compute_trans_closure2(Relation,1,Result,WF).
4975
4976		compute_trans_closure2(Relation,Cnt,Result,WF) :-
4977		one_closure_iteration(Relation,Relation,Relation,Result1,Added,Done,WF),
4978	?	compute_trans_closure3(Relation,Cnt,Result1,Added,Done,Result,WF).
4979
4980		:- block compute_trans_closure3(?,?,?,?,-,?,?).
4981		compute_trans_closure3(Relation,Cnt,Result1,Added,_Done,Result,WF) :-
4982		( equal_object_wf(Result1,Relation,relational_trans_closure_wf,WF), % should we do equality_objects here?
4983		equal_object_optimized_wf(Result,Result1,compute_trans_closure,WF)
4984		;
4985		Added==possibly_added,
4986		not_equal_object_wf(Result1,Relation,WF), % not a fixpoint; continue
4987		IterCnt is Cnt+1,
4988	?	compute_trans_closure2(Result1,IterCnt,Result,WF)
4989		).
4990
4991		:- block one_closure_iteration(?,?,-,?,?,?,?).
4992		one_closure_iteration([],_,IterRes,OutRel,Added,Done,WF) :-
4993		equal_object_wf(IterRes,OutRel,one_closure_iteration,WF),
4994		(var(Added) -> Added=not_added ; true),
4995		Done=done.
4996		one_closure_iteration([(X,Y)|T],ExpandedPreviousRel,PreviousRel,OutRel,Added,Done,WF) :-
4997		add_tuples(ExpandedPreviousRel,X,Y,PreviousRel,IntRel,Added,DoneTuples,WF),
4998	?	one_closure_iteration_block(DoneTuples,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
4999
5000		:- block one_closure_iteration_block(-,?,?,?,?,?,?,?).
5001		one_closure_iteration_block(_,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF) :-
5002	?	one_closure_iteration(T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
5003
5004		add_tuples([],_,_,OutRel,OutRel,_Added,done,_).
5005		add_tuples([(X,Y)|T],OX,OY,InRel,OutRel,Added,Done,WF) :-
5006		% add tuple (X,OY) if we have Y=OX
5007		equality_objects_wf(Y,OX,EqRes,WF),
5008	?	add_tuples_aux(EqRes,X,T,OX,OY,InRel,OutRel,Added,Done,WF).
5009
5010		:- block add_tuples_aux(-,?,?,?,?,?,?,?,?,?).
5011		add_tuples_aux(pred_true,X,T,OX,OY,InRel,OutRel,possibly_added,Done,WF) :-
5012		add_element_wf((X,OY),InRel,IntRel,WF), % add transitive couple X -> OY
5013	?	add_tuples(T,OX,OY,IntRel,OutRel,_,Done,WF).
5014		add_tuples_aux(pred_false,_X,T,OX,OY,InRel,OutRel,Added,Done,WF) :- % no transitive couple needed
5015	?	add_tuples(T,OX,OY,InRel,OutRel,Added,Done,WF).
5016
5017
5018		:- assert_must_succeed((is_full_relation(X,R),var(R),X=[],R==true)).
5019		:- assert_must_succeed((is_full_relation(X,R),var(R),X=[(A,B)|T],var(R),A=int(1),var(R),B=A,var(R),T=[],R==true)).
5020		:- block is_full_relation(-,?).
5021		is_full_relation([],R) :- !,R=true.
5022	?	is_full_relation([H|T],W) :- !, is_full_relation_aux(H,T,W).
5023		is_full_relation(X,R) :-
5024		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation(X,R)),fail.
5025
5026		:- block is_full_relation_aux(-,?,?).
5027	?	is_full_relation_aux((X,Y),T,W) :- !, is_full_relation_aux2(X,Y,T,W).
5028		is_full_relation_aux(X,T,W) :-
5029		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation_aux(X,T,W)),fail.
5030		:- block is_full_relation_aux2(-,?,?,?), is_full_relation_aux2(?,-,?,?).
5031	?	is_full_relation_aux2(_X,_Y,T,W) :- is_full_relation(T,W).
5032
5033		/* ------------------ */
5034
5035		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_closure1_wf((int(1),int(3)),[(int(1),int(2)),(int(2),int(1)),(int(2),int(3))],WF),WF)). % used to be wfdet (see in_domain_wf above)
5036
5037		in_closure1_wf(Pair,Relation,WF) :- %Pair = (_A,B),
5038		%in_domain_wf_lazy(A,Relation,WF), % done below
5039		%check_element_of_wf((_,B),Relation,WF), % multiple solutions for _, see test 634, 637
5040	?	in_closure1_membership_test_wf(Pair,Relation,pred_true,WF).
5041
5042
5043		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_closure1_wf((int(1),int(3)),[(int(1),int(2)),(int(2),int(1)),(int(3),int(3))],WF),WF)).
5044
5045		not_in_closure1_wf(Pair,Relation,WF) :-
5046	?	in_closure1_membership_test_wf(Pair,Relation,pred_false,WF).
5047
5048		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[],Res,_WF),Res==pred_false)).
5049		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(2))],Res,_WF),Res==pred_true)).
5050		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(3))],Res,_WF),Res==pred_false)).
5051		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(3)),(int(3),int(2))],Res,_WF),Res==pred_true)).
5052		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(3)),(int(3),int(2))],pred_true,WF),WF)). % used to be wfdet (see in_domain_wf above)
5053		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_closure1_membership_test_wf((int(11),int(3)),[(int(11),int(3))],pred_true,WF),WF)).
5054		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_closure1_membership_test_wf((int(11),int(3)),[(int(11),int(33))],pred_false,WF),WF)).
5055		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_closure1_membership_test_wf((int(1),int(3)),[(int(11),int(3))],pred_false,WF),WF)).
5056		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_closure1_membership_test_wf((int(11),int(22)),[(int(11),int(3)),(int(33),int(2)),(int(3),int(22)),(int(11),int(3))],pred_true,WF),WF)). % used to be wfdet (see in_domain_wf above)
5057		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_closure1_membership_test_wf((int(11),int(11)),[(int(11),int(3))],pred_false,WF),WF)).
5058
5059		:- block force_in_domain(-,?,?,?).
5060		force_in_domain(pred_false,_A,_Relation,_WF).
5061		force_in_domain(pred_true,A,Relation,WF) :- % force A to be in domain, avoid enumeration warnings,...
5062		% maybe only for non-ground A
5063		in_domain_wf_lazy(A,Relation,WF). % slowdown Loop.mch (tests 634, 637) if we use in_domain_wf ?
5064
5065		% (x,y) : closure1(Rel)
5066		:- block in_closure1_membership_test_wf(?,-,?,?).
5067		in_closure1_membership_test_wf((A,B),CSRelation,MemRes,WF) :-
5068		is_custom_explicit_set(CSRelation,in_closure1),
5069		!,
5070	?	image_for_closure1_wf(CSRelation,[A],Image,WF),
5071		force_in_domain(MemRes,A,CSRelation,WF),
5072		membership_test_wf(Image,B,MemRes,WF).
5073		in_closure1_membership_test_wf((X,Y),Relation,MemRes,WF) :-
5074		expand_custom_set_to_list_wf(Relation,ERelation,_,in_closure1_membership_test_wf,WF),
5075		Discarded = [], % pairs discarded in current iteration
5076		force_in_domain(MemRes,X,Relation,WF),
5077		in_closure1_membership_test_wf2(ERelation,X,Y,Discarded,MemRes,WF).
5078
5079		:- block in_closure1_membership_test_wf2(-,?,?,?,?,?).
5080		in_closure1_membership_test_wf2([],_X,_Y,_,MemRes,_WF) :- MemRes=pred_false.
5081		in_closure1_membership_test_wf2([(V,W)|Rest],X,Y,Discarded,MemRes,WF) :- % TO DO: Rest==[] -->
5082		equality_objects_wf(V,X,VXResult,WF),
5083		in_closure1_membership_test_wf3(VXResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5084
5085		:- block in_closure1_membership_test_wf3(-,?,?,?,?,?,?,?,?).
5086		in_closure1_membership_test_wf3(pred_false,V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5087		in_closure1_membership_test_wf2(Rest,X,Y,[(V,W)|Discarded],MemRes,WF).
5088		in_closure1_membership_test_wf3(pred_true,V,W,Rest,X,Y,Discarded,MemRes,WF) :- % V=X
5089		propagate_false(MemRes,WYResult),
5090		% TODO: Res=[],Discarded=[] -> MemRes=WYResult
5091		equality_objects_wf(W,Y,WYResult,WF), % MemRes = pred_false => WYResult = pred_false
5092		in_closure1_membership_test_wf4(WYResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5093
5094		:- block in_closure1_membership_test_wf4(-,?,?,?,?,?,?,?,?).
5095		in_closure1_membership_test_wf4(pred_false,_V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5096		append(Discarded,Rest,Restart),
5097		in_closure1_membership_test_wf2(Restart,W,Y,[],MemRes1,WF),
5098		propagate_false(MemRes,MemRes1), % MemRes = pred_false -> MemRes1=pred_false
5099		when(nonvar(MemRes1),
5100		(MemRes1=pred_true -> MemRes=pred_true
5101		; in_closure1_membership_test_wf2(Rest,X,Y,Discarded,MemRes,WF) % (V,W) not in Discarded: was not useful
5102		)).
5103		in_closure1_membership_test_wf4(pred_true,_V,_W,_Rest,_X,_Y,_Discarded,MemRes,_WF) :- % W=Y
5104		MemRes = pred_true.
5105		/* ------------------ */
5106
5107		:- block propagate_false(-,?).
5108		propagate_false(pred_false,pred_false).
5109		propagate_false(pred_true,_).
5110