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		kernel_tools: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		kernel_tools: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		kernel_tools: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),!,
1842		(preferences:preference(use_smt_mode,true) -> S=[(El,_)|_]
1843		; get_enumeration_starting_wait_flag(not_empty_domain_wf,WF,LWF),
1844		in_domain_lwf(El,S,LWF,WF)).
1845	?	in_domain_wf(El,Rel,WF) :- in_domain_wf_lazy(El,Rel,WF).
1846		:- block in_domain_lwf(-,-,-,?).
1847		% 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
1848		%:- block in_domain_lwf(-,-,?,?),in_domain_lwf(?,-,-,?). % this annotation fails test 1703
1849		in_domain_lwf(El,Rel,LWF,WF) :- % tools_printing:print_term_summary(in_domain_lwf(El,Rel,LWF)),
1850		(var(Rel) -> ground_value_check(El,GrVal),
1851		in_domain_lwf2(El,Rel,LWF,GrVal,WF) % we could also wait at least until WF0 is fully grounded?
1852		; not_empty_set_unless_closure_wf(Rel,WF),
1853		in_domain_wf_lazy(El,Rel,WF)).
1854
1855		:- block in_domain_lwf2(?,-,-,-,?).
1856		in_domain_lwf2(El,Rel,_LWF,_Grval,WF) :- % tools_printing:print_term_summary(in_domain_lwf2(El,Rel,_LWF,_Grval)),
1857		(var(Rel) -> Rel = [(El,_)|_]
1858		% can create a choice point when unifying with large avl_set:, see rule_Rule_DB_PSR_0003_C
1859		% maybe we should delay even further
1860		; not_empty_set_unless_closure_wf(Rel,WF),
1861		in_domain_wf_lazy(El,Rel,WF)).
1862
1863		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
1864		not_empty_set_unless_closure_wf(Rel,WF) :- not_empty_set_wf(Rel,WF).
1865
1866		% does not instantiate to [(El,_)|_]
1867		:- block in_domain_wf_lazy(?,-,?).
1868		in_domain_wf_lazy(_DomainElement,[],_WF) :- !,fail.
1869		in_domain_wf_lazy(DomainElement,avl_set(A),_WF) :-
1870		kernel_tools:ground_value(DomainElement), !,
1871		check_in_domain_of_avlset(DomainElement,A).
1872		% TO DO: check for infinite closures
1873		in_domain_wf_lazy(DomainElement,ES,WF) :-
1874		is_custom_explicit_set(ES,in_domain_wf_lazy),
1875		domain_of_explicit_set_wf(ES,Dom,WF),!,
1876	?	check_element_of_wf(DomainElement,Dom,WF).
1877		in_domain_wf_lazy(El,Rel,WF) :-
1878		expand_custom_set_to_list_wf(Rel,Relation,Done,in_domain_wf_lazy,WF),
1879		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
1880		% if Done == true -> we can use maybe clpfd_inlist or clpfd:element or quick_propagate
1881		quick_propagation_domain_element_list(Done,Relation,El,WF),
1882		in_domain2(El,Relation,WF,LWF).
1883
1884		% a custom implementation of quick_propagation_element_information for checking domain elements and lists only
1885		:- use_module(clpfd_lists,[try_in_fd_value_list_check/4]).
1886		:- block quick_propagation_domain_element_list(-,?,?,?).
1887		quick_propagation_domain_element_list(_,_,_,_) :- preferences:preference(use_clpfd_solver,false),!.
1888		quick_propagation_domain_element_list(_,_,El,_) :- ground(El),!.
1889		quick_propagation_domain_element_list(_,RelList,El,WF) :-
1890		try_in_fd_value_list_check(RelList,(El,_),couple_left(_),WF). % use couple_left to ignore range values
1891
1892
1893		:- block in_domain2(?,-,?,?).
1894		in_domain2(El,[(X,_Y)|T],WF,LWF) :-
1895		(T==[]
1896		-> equal_object_wf(El,X,in_domain2,WF)
1897		; kernel_objects:equality_objects_lwf(El,X,EqRes,LWF,WF),
1898		in_domain3(EqRes,El,T,WF,LWF)
1899		).
1900
1901		:- block in_domain3(-,?,?,?,?).
1902		in_domain3(pred_true,_El,_T,_WF,_LWF).
1903		in_domain3(pred_false,El,T,WF,LWF) :-
1904		get_new_subsidiary_wait_flag(LWF,in_domain2(El,T),WF,NewLWF), % not necessary if T only has single element
1905		in_domain2(El,T,WF,NewLWF).
1906
1907
1908		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[],WF),WF)).
1909		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(2),int(7))],WF),WF)).
1910		:- 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)).
1911		:- 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)).
1912		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(1),[(int(1),int(2))],_))).
1913		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2)),(int(3),int(4))],_))).
1914		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[],_))).
1915		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2))],_))).
1916		:- 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)).
1917		/* a more efficient version than using not_element_of and computing domain */
1918
1919
1920		:- block not_in_domain_wf(?,-,?).
1921		not_in_domain_wf(DomainElement,ES,WF) :- is_custom_explicit_set(ES,not_in_domain),
1922		domain_of_explicit_set_wf(ES,Dom,WF),!,
1923		not_element_of_wf(DomainElement,Dom,WF).
1924		not_in_domain_wf(El,Rel,WF) :-
1925		expand_custom_set_to_list_wf(Rel,Relation,_,not_in_domain,WF),
1926		not_in_domain2(Relation,El,WF).
1927		:- block not_in_domain2(-,?,?).
1928		not_in_domain2([],_,_WF).
1929		not_in_domain2([(X,_Y)|T],E,WF) :- not_equal_object_wf(E,X,WF), not_in_domain2(T,E,WF).
1930
1931
1932
1933
1934		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([],[],WF),WF)).
1935		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([(int(1),int(3))],[int(3)],WF),WF)).
1936		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf(
1937		[(int(0),int(55)),(int(2),int(3)),(int(1),int(3))],[int(3),int(55)],WF),WF)).
1938		:- assert_must_succeed((bsets_clp:range_wf([],Res,_WF),Res=[])).
1939		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2))],Res,_WF),
1940		kernel_objects:equal_object(Res,[int(2)]))).
1941		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(1)),(int(2),int(1))],Res,_WF),
1942		kernel_objects:equal_object(Res,[int(1)]))).
1943		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1))],Res,_WF),
1944		kernel_objects:equal_object(Res,[int(1),int(2)]))).
1945		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],Res,_WF),
1946		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1947		:- assert_must_succeed((bsets_clp:range_wf(X,Res,_WF),
1948		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],
1949		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1950		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1951		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(2))])).
1952		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1953		X = [(int(2),int(1)),(int(1),int(2)),(int(2),int(2))])).
1954		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1955		X = [])).
1956		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1957		([int(0),int(1)],[int(0),int(1)]),([int(0),int(2)],[int(0),int(2)]),
1958		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1959		([int(1),int(2)],[int(1),int(2)]),([int(1),int(3)],[int(1),int(3)]),
1960		([int(1),int(4)],[int(1),int(4)]),([int(2)],[int(2)]),([int(2),int(3)],[int(2),int(3)]),
1961		([int(2),int(4)],[int(2),int(4)]),([int(3)],[int(3)]),([int(3),int(4)],
1962		[int(3),int(4)]),([int(4)],[int(4)])],_Res,_WF))).
1963		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1964		([int(0),int(1)],[int(0),int(1)]),
1965		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1966		([int(1),int(2)],[int(1),int(2)])],_Res,_WF))).
1967
1968
1969		:- block range_wf(-,-,?).
1970		range_wf(Rel,Res,WF) :- Res ==[],!, empty_set_wf(Rel,WF).
1971		range_wf(Rel,Res,WF) :- Rel ==[],!, empty_set_wf(Res,WF).
1972		range_wf(Rel,Res,WF) :- range_wf1(Rel,Res,WF),
1973		propagate_result_to_input(Res,Rel,range,WF).
1974
1975		:- block range_wf1(-,?,?).
1976		range_wf1(Rel,Res,WF) :-
1977		is_custom_explicit_set(Rel,range_wf1),
1978		range_of_explicit_set_wf(Rel,Range,WF), !,
1979	?	equal_object_wf(Range,Res,range_wf1,WF).
1980		range_wf1(Rel,Res,WF) :-
1981		% TO DO : propagate information that card of Res <= card of Rel; similar thing for domain
1982		expand_custom_set_to_list_wf(Rel,Relation,_,range_wf1,WF),
1983		newrange2(Relation,[],Res,WF),
1984		quick_propagate_range(Relation,Res,WF).
1985
1986
1987		:- block quick_propagate_range(-,?,?).
1988		quick_propagate_range([],_,_WF).
1989		quick_propagate_range([(_,Y)|T],FullRes,WF) :-
1990		quick_propagation_element_information(FullRes,Y,WF,FullRes1), % should we use a stronger check ?
1991		quick_propagate_range(T,FullRes1,WF).
1992
1993		:- block newrange2(-,?,?,?).
1994		newrange2([],_SoFar,Res,WF) :-
1995	?	empty_set_wf(Res,WF).
1996		newrange2([(X,Y)|T],SoFar,Res,WF) :-
1997		(Res==[]
1998		-> MemRes=pred_true, check_element_of_wf(Y,SoFar,WF)
1999		; membership_test_wf(SoFar,Y,MemRes,WF),
2000		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
2001		(var(MemRes) -> prop_empty_pred_true(Res,MemRes) %,print(delay_range(Y,T)),nl
2002		% TO DO: we could look further in T if we can decide membership for other elements in T ?
2003		; true)
2004		),
2005		newrange3(MemRes,Y,T,SoFar,Res,WF).
2006
2007		:- block prop_empty_pred_true(-,?).
2008		prop_empty_pred_true([],R) :- !, R=pred_true.
2009		prop_empty_pred_true(_,_).
2010
2011		% 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
2012		% checking is stopped if EqFlag becomes nonvar
2013		% tested by testcase 1061
2014		:- block card_greater_equal_check(-,?,-), card_greater_equal_check(?,-,-).
2015		card_greater_equal_check(_,_,Flag) :- nonvar(Flag),!. % no longer required; even though we could prune failure !? done later in newrange2/newdomain2 ??!!
2016		card_greater_equal_check([],Set2,Flag) :- !,empty_set(Set2),
2017		Flag=pred_false. % Flag set indicates that both sets have same size
2018		card_greater_equal_check(_,[],_) :- !.
2019		card_greater_equal_check([_|T],[_|R],Flag) :- !, card_greater_equal_check(T,R,Flag).
2020		% To do: deal with AVL args as Result + also use efficient_card_for_set for closures
2021		%card_greater_equal_check([_|T],Set,Flag) :- efficient_card_for_set(B,CardB,CodeB),!,
2022		% f: 1..7 -->> 1..n & n>=7 & n<10 still does not work well
2023		% TO DO: can we merge code with check_card_greater_equal
2024		card_greater_equal_check(_,_,_).
2025
2026
2027		:- block newrange3(-,?,?,?,?,?).
2028	?	newrange3(pred_true,_Y,T,SoFar,Res,WF) :- newrange2(T,SoFar,Res,WF).
2029		newrange3(pred_false,Y,T,SoFar,Res,WF) :-
2030		kernel_objects:mark_as_non_free(Y,range), % Y is linked to a particular X -> it is not free
2031		add_element_wf(Y,SoFar,SoFar2,WF),
2032		equal_cons_wf(Res,Y,Res2,WF),
2033	?	newrange2(T,SoFar2,Res2,WF).
2034
2035
2036		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([],Res,_WF),Res=[])).
2037		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([int(1),int(2)],Res,_WF),
2038		Res=[(int(1),int(1)),(int(2),int(2))])).
2039		:- 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)).
2040		:- 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)).
2041		:- assert_must_fail((bsets_clp:identity_relation_over_wf([int(1)|_],_,_WF),fail)). /* check: no loop */
2042
2043		:- block identity_relation_over_wf(-,?,?).
2044		identity_relation_over_wf(Set1,IDRel,WF) :-
2045		expand_custom_set_to_list_wf(Set1,ESet1,_,identity_relation_over_wf,WF),
2046		identity_relation_over2(ESet1,IDRel,WF).
2047
2048		:- block identity_relation_over2(-,?,?).
2049		identity_relation_over2([],Res,WF) :- empty_set_wf(Res,WF).
2050		identity_relation_over2([X|T1],Res,WF) :- equal_cons_wf(Res,(X,X),T2,WF), % equal_object([(X,X)|T2],Res),
2051		identity_relation_over2(T1,T2,WF).
2052
2053
2054
2055		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_identity((int(1),int(1)),[int(1),int(2)],WF),WF)).
2056		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[int(1),int(2)],_WF))).
2057		:- assert_must_fail((bsets_clp:in_identity((int(3),int(3)),[int(1),int(2)],_WF))).
2058		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[],_WF))).
2059		in_identity((X,Y),Domain,WF) :-
2060	?	equal_object_wf(X,Y,in_identity,WF), check_element_of_wf(X,Domain,WF).
2061
2062		:- assert_must_fail((bsets_clp:not_in_identity((int(1),int(1)),[int(1),int(2)],_WF))).
2063		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(1),int(2)),[int(1),int(2)],WF),WF)).
2064		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(3),int(3)),[int(1),int(2)],WF),WF)).
2065		:- assert_must_succeed((bsets_clp:not_in_identity((int(1),int(2)),[],_WF))).
2066		not_in_identity((X,Y),Domain,WF) :-
2067		equality_objects_wf(X,Y,Eq,WF),
2068		not_in_id2(Eq,X,Domain,WF).
2069
2070		:- block not_in_id2(-,?,?,?).
2071		not_in_id2(pred_true,X,Domain,WF) :- not_element_of_wf(X,Domain,WF).
2072		not_in_id2(pred_false,_,_,_).
2073
2074
2075		:- 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)).
2076		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([(int(1),int(2))], [(int(2),int(1))],WF),WF)).
2077		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([], [],WF),WF)).
2078		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [])).
2079		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [(int(2),int(2))])).
2080		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,[(int(1),int(2)),(int(7),int(6))],_WF),
2081		X = [(int(2),int(1)),(int(6),int(7))])).
2082		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],X,_WF),
2083		X = [(int(2),int(1)),(int(6),int(7))])).
2084		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],
2085		[(int(6),int(7)),(int(2),int(1))],_WF))).
2086		:- assert_must_succeed((bsets_clp:invert_relation_wf(closure([a,b],[string,boolean],b(truth,pred,[])),
2087		closure([b,a],[boolean,string],b(truth,pred,[])),_WF))).
2088
2089		:- block invert_relation_wf(-,-,?).
2090		invert_relation_wf(R,IR,WF) :-
2091		% (nonvar(R) -> invert_relation2(R,IR) ; invert_relation2(IR,R)).
2092		invert_relation2(R,IR,WF). % , print_term_summary(invert_relation(R,IR)).
2093		/* Optimization for some types of closures: Instead of expanding the closures, we just
2094		swap the parameters. This does not work with closures wich have only one parameter
2095		wich is a pair */
2096		invert_relation2(CS,R,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2097		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_1,WF).
2098		invert_relation2(R,CS,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2099		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_2,WF).
2100		%invert_relation2(closure([P1,P2],[T1,T2],Clo),closure([P2,P1],[T2,T1],Clo)) :- !.
2101		invert_relation2(R,IR,WF) :- %try_expand_custom_set_wf(R,ER,invert,WF),
2102		% (nonvar(R) -> invert_relation3(R,IR)
2103		% ; invert_relation3(IR,R),(ground(IR)-> true ; invert_relation3(R,IR))).
2104		invert_relation3(R,IR,WF,1), invert_relation3(IR,R,WF,1).
2105
2106		:- block invert_relation3(-,?,?,?).
2107		invert_relation3(closure(P,T,B),Res,WF,_) :- invert_explicit_set(closure(P,T,B),ICS),
2108		equal_object_wf(Res,ICS,invert_relation3_1,WF).
2109		invert_relation3(avl_set(S),Res,WF,_) :- invert_explicit_set(avl_set(S),ICS),
2110		equal_object_wf(Res,ICS,invert_relation3_2,WF).
2111	?	invert_relation3([],Res,WF,_) :- empty_set_wf(Res,WF).
2112		invert_relation3([(X,Y)|T],Res,WF,Depth) :-
2113		D1 is Depth+1, get_wait_flag(D1,invert_relation3,WF,LWF),
2114		equal_cons_lwf(Res,(Y,X),IT,LWF,WF),
2115		invert_relation3(T,IT,WF,D1).
2116
2117
2118
2119
2120		tuple_of(X,Y,R) :- check_element_of((X,Y),R).
2121		%tuple_of_wf(X,Y,R,WF) :- check_element_of_wf((X,Y),R,WF).
2122
2123
2124		% RELATIONAL COMPOSITION (;)
2125
2126		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2127		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2128		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2129		[(int(11),int(12)),(int(11),int(33))],
2130		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2131		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(12)),
2132		[(int(11),int(12)),(int(11),int(33))],
2133		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2134		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2135		[(int(11),[int(33),int(32)])],
2136		[([int(32),int(33)],int(22))],WF),WF)).
2137		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:in_composition_wf((int(11),int(33)),
2138		[(int(11),int(12)),(int(11),int(33))],
2139		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2140		% check if (X,Y) element of (F ; G)
2141		in_composition_wf((X,Y),F,G,WF) :-
2142		check_element_of_wf((X,Z1),F,WF), % no need to enumerate Z (TODO: check)
2143		equal_object_wf(Z1,Z2,check_element_of_wf,WF),
2144		check_element_of_wf((Z2,Y),G,WF).
2145
2146		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(11),int(33)),
2147		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2148		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(33),int(22)),
2149		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2150
2151		% just evaluates arguments; TODO: improve or at least pass Type (for symbolic composition)
2152		not_in_composition_wf(Couple,F,G,WF) :-
2153		rel_composition_wf(F,G,Comp,_UnknownType,WF),
2154		not_element_of_wf(Couple,Comp,WF).
2155
2156		:- 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))],
2157		[(int(1),int(1)),(int(5),int(7)),(int(3),int(33))]))).
2158		:- 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))],[]))).
2159		:- 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))],[],[]))).
2160		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([],[],[]))).
2161		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2162		[(int(1),int(11))],X),X = [])).
2163		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],[],X),X = [])).
2164		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2165		[(int(2),int(11))],X),
2166		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2167		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(2))],[(int(2),int(11))],X),
2168		ground(X), bsets_clp:equal_object(X,[(int(1),int(11)),(int(7),int(11))]))).
2169		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(5))],
2170		[(int(2),int(11)),(int(2),int(4))],X),
2171		kernel_objects:equal_object(X,[(int(1),int(11)),(int(1),int(4))]))).
2172		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(1),int(5))],
2173		[(int(2),int(11)),(int(5),int(11))],X),
2174		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2175		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),[int(1)]),(int(1),[int(2),int(5)])],
2176		[([int(1),int(2)],int(13)),([int(5),int(2)],int(12))],X),
2177		kernel_objects:equal_object(X,[(int(1),int(12))]))).
2178
2179		rel_composition(Rel1,Rel2,Comp) :- % only used in unit_tests above
2180		init_wait_flags(WF,[rel_composition]),
2181		rel_composition_wf(Rel1,Rel2,Comp,_UnknownType,WF),
2182		ground_wait_flags(WF).
2183
2184		:- block rel_composition_wf(-,-,?,?,?).
2185		rel_composition_wf(Rel1,Rel2,Comp,_,WF) :-
2186		(Rel1==[] ; Rel2==[]),
2187		!,
2188		empty_set_wf(Comp,WF).
2189		rel_composition_wf(Rel1,Rel2,Comp,Type,WF) :- rel_composition1(Rel1,Rel2,Comp,Type,WF).
2190
2191		:- block rel_composition1(-,?,?,?,?),rel_composition1(?,-,?,?,?).
2192		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2193		(Rel1==[] ; Rel2==[]),!, empty_set_wf(Comp,WF).
2194	?	rel_composition1(Rel1,Rel2,Comp,Type,WF) :- keep_symbolic(Rel1),
2195		(Rel2 = avl_set(_) -> SYMBOLIC=false ; SYMBOLIC=symbolic),
2196		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3),
2197		!,
2198		equal_object_wf(Comp,Rel3,rel_composition1_0,WF).
2199		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2200		rel_composition_for_explicit_set(Rel1,Rel2,Res),!, % treats finite Rel1 and avl_set for Rel2
2201		equal_object_wf(Res,Comp,rel_composition1_1,WF).
2202		rel_composition1(Rel1,Rel2,Comp,Type,WF) :- Rel2=closure(_,_,_),
2203	?	keep_symbolic(Rel2),
2204		(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
2205		-> !,
2206		expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1,WF),
2207		rel_compose_with_inf_fun(Relation1,Domain,Rel2,Comp,WF) % this is like map Rel2 over Rel1 in functional programmming
2208		; symbolic_composition(Rel1,Rel2,false,Type,Rel3),
2209		!,
2210		expand_custom_set_wf(Rel3,CRes,rel_composition,WF),% do we need to expand ?
2211		equal_object_optimized(CRes,Comp,rel_composition1_3)
2212		).
2213		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2214		expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1_2,WF),
2215		expand_custom_set_to_list_wf(Rel2,Relation2,_,rel_composition1_3,WF),
2216		rel_compose2(Relation1,Relation2,Comp,WF).
2217
2218		:- use_module(btypechecker, [l_unify_types_strict/2]).
2219		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3) :-
2220		get_set_type(Type,couple(TX,TZ)),
2221		get_relation_types(Rel1,TX1,TY1),
2222		get_relation_types(Rel2,TY2,TZ2),
2223		(l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2]) -> true
2224		; add_internal_error('Could not unify range/domain types: ',l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2])),
2225		fail
2226		),
2227		ground((TX1,TY1,TZ)), % avoid creating a closure with non-ground type list
2228		rel_comp_closure(Rel1,Rel2,TX1,TY1,TZ,SYMBOLIC,Rel3).
2229		% generate a closure for {xx,zz | #(yy).(xx|->yy : Rel1 & yy|->zz : Rel2)}
2230		% TO DO: maybe detect special cases: Rel1 is a function/cartesian product, e.g., (((0 .. 76) * (0 .. 76)) * {FALSE}) ; {(FALSE|->0),(TRUE|->1)}
2231		:- use_module(bsyntaxtree, [safe_create_texpr/4]).
2232		rel_comp_closure(Rel1,Rel2,TX,TY,TZ,SYMBOLIC,closure(Args,Types,CBody)) :-
2233		Args = ['_rel_comp1','_rel_comp2'], Types = [TX,TZ],
2234		couple_member_pred('_rel_comp1',TX,'_zzzz_unary',TY,Rel1, Pred1),
2235		couple_member_pred('_zzzz_unary',TY,'_rel_comp2',TZ,Rel2, Pred2),
2236		UsedIds = ['_rel_comp1','_rel_comp2','_zzzz_unary'], % avoid having to call find_identifier_uses
2237		%conjunct_predicates([Pred1,Pred2],P12a), bsyntaxtree:check_computed_used_ids(P12a,UsedIds),
2238		safe_create_texpr(conjunct(Pred1,Pred2),pred,[used_ids(UsedIds)],P12),
2239		%b_interpreter_components:create_unsimplified_exists([b(identifier('_zzzz_unary'),TY,[])],P12,Body),
2240		bsyntaxtree:create_exists_opt_liftable([b(identifier('_zzzz_unary'),TY,[])],P12,Body), % cf Thales_All/rule_zcpa2 test 2287
2241		(SYMBOLIC==symbolic
2242		-> mark_bexpr_as_symbolic(Body,CBody)
2243		; CBody=Body).
2244
2245		% generate predicate for X|->Y : Rel
2246		couple_member_pred(X,TX,Y,TY,Rel, Pred) :-
2247		Pred = b(member(b(couple(b(identifier(X),TX,[]),
2248		b(identifier(Y),TY,[])),couple(TX,TY),[]),
2249		b(value(Rel),set(couple(TX,TY)),[])),pred,[]).
2250
2251
2252
2253		:- block rel_compose2(-,?,?,?).
2254		rel_compose2([],_,Out,WF) :- empty_set_wf(Out,WF).
2255		rel_compose2([(X,Y)|T],Rel2,Out,WF) :-
2256		rel_extract(Rel2,X,Y,OutXY,[],WF),
2257		% rel_extract(Rel2,X,Y,Out,OutRem),
2258		rel_compose2(T,Rel2,OutRem,WF),
2259		union_wf(OutRem,OutXY,Out,WF). % used to call union wihout wf; makes test 1394 fail
2260
2261		:- block rel_extract(-,?,?,?,?,?).
2262		rel_extract([],_,_,Rem,Rem,_WF). % should we use equal_object here ?????
2263		rel_extract([(Y1,Z)|T],X,Y,Res,Rem,WF) :-
2264		rel_extract(T,X,Y,CT,Rem,WF),
2265		equality_objects_wf(Y1,Y,EqRes,WF),
2266		rel_extract2(EqRes,Z,X,CT,Res).
2267
2268		:- block rel_extract2(-,?,?,?,?).
2269		rel_extract2(pred_true, Z, X,CT,Res) :- add_element((X,Z),CT,Res).
2270		rel_extract2(pred_false,_Z,_X,CT,Res) :- Res = CT.
2271
2272
2273		% relational composition of a finite relation with an infinite or symbolic function
2274		rel_compose_with_inf_fun(R,Dom,Fun,CompRes,WF) :- !,
2275		rel_compose_with_inf_fun_acc(R,Dom,Fun,[],CompRes,WF).
2276		:- block rel_compose_with_inf_fun_acc(-,?,?,?,?,?).
2277		rel_compose_with_inf_fun_acc([],_Dom,_Rel2,Acc,Comp,WF) :-
2278		equal_object_wf(Comp,Acc,rel_compose_with_inf_fun_acc,WF).
2279		rel_compose_with_inf_fun_acc([(X,Y)|T],Dom,Fun,Acc,CompRes,WF) :-
2280		membership_test_wf(Dom,Y,MemRes,WF), % check if Y is in the domain of the symbolic relation
2281		rel_compose_with_inf_fun_acc_aux(MemRes,X,Y,T,Dom,Fun,Acc,CompRes,WF).
2282
2283		:- block rel_compose_with_inf_fun_acc_aux(-,?,?,?, ?,?,?,?, ?).
2284		rel_compose_with_inf_fun_acc_aux(pred_true,X,Y,T,Dom,Fun,Acc,CompRes,WF) :-
2285		apply_to(Fun,Y,FY,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
2286		add_element_wf((X,FY),Acc,NewAcc,WF),
2287		rel_compose_with_inf_fun_acc(T,Dom,Fun,NewAcc,CompRes,WF).
2288		rel_compose_with_inf_fun_acc_aux(pred_false,_X,_Y,T,Dom,Fun,Acc,Comp,WF) :-
2289		rel_compose_with_inf_fun_acc(T,Dom,Fun,Acc,Comp,WF).
2290
2291		% TO DO: if we obtain a list such as [(int(1),X),...] in Acc rather than an avl_set,
2292		% we may still be able to sort and avoid quadratic comparisons if e.g.
2293		% first component is a data-type where equality can be decided by unification (integer, bool, global(GS), ...)
2294		% we could put the optimisation into add_element_wf ?
2295		% TO DO: special version for avl_set as relation?
2296
2297		/*
2298		Note: old version; has performance problem, 2021/02_Feb/CDS
2299		the add_element_wf calls below can only construct/instantiate result when empty_set_wf reached
2300		and a lot of pending co-routines pile up for long relation lists
2301
2302		:- block rel_compose_with_inf_fun(-,?,?,?,?).
2303		rel_compose_with_inf_fun([],_Dom,_Rel2,Comp,WF) :- empty_set_wf(Comp,WF).
2304		rel_compose_with_inf_fun([(X,Y)|T],Dom,Fun,CompRes,WF) :-
2305		membership_test_wf(Dom,Y,MemRes,WF), rel_compose_with_inf_fun_aux(MemRes,X,Y,T,Dom,Fun,CompRes,WF).
2306
2307		:- block rel_compose_with_inf_fun_aux(-,?,?,?, ?,?,?,?).
2308		rel_compose_with_inf_fun_aux(pred_true,X,Y,T,Dom,Fun,CompRes,WF) :-
2309		apply_to(Fun,Y,FY,WF),
2310		add_element_wf((X,FY),CT,CompRes,WF),
2311		rel_compose_with_inf_fun(T,Dom,Fun,CT,WF).
2312		rel_compose_with_inf_fun_aux(pred_false,_X,_Y,T,Dom,Fun,Comp,WF) :-
2313		rel_compose_with_inf_fun(T,Dom,Fun,Comp,WF).
2314		*/
2315
2316		:- assert_must_abort_wf(bsets_clp:rel_iterate_wf([],int(-1),_R,set(couple(integer,integer)),WF),WF).
2317		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([], int(2),[],set(couple(integer,integer)),_WF))).
2318		:- 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))).
2319		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([(pred_true,pred_true)], int(0),
2320		[(pred_true,pred_true),(pred_false,pred_false)],set(couple(boolean,boolean)),_WF))).
2321		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:rel_iterate_wf([(int(1),int(2)),
2322		(int(2),int(4)),(int(4),int(6))], int(2),[(int(1),int(4)),(int(2),int(6))],
2323		set(couple(integer,integer)),WF),WF)).
2324		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF), R=[],
2325		bsets_clp:equal_object(X,R))).
2326		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF),
2327		R=[(int(1),int(2)),(int(2),int(3))],
2328		bsets_clp:equal_object(X,R))).
2329		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(2),X,set(couple(integer,integer)),_WF),
2330		R=[(int(1),int(2)),(int(2),int(3))],
2331		bsets_clp:equal_object(X,[(int(1),int(3))]))).
2332		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2333		R=[(int(1),int(2)),(int(2),int(3))],
2334		bsets_clp:equal_object(X,[]))).
2335		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2336		R=[(int(1),int(2)),(int(2),int(3)),(int(1),int(1))],
2337		bsets_clp:equal_object(X,[(int(1),int(1)),(int(1),int(2)),(int(1),int(3))]))).
2338
2339		rel_iterate_wf(Rel,int(Nr),Res,Type,WF) :-
2340		opt_push_wait_flag_call_stack_info(WF,b_operator_call(iterate,
2341		[Nr,Rel],unknown),WF2),
2342		rel_iterate1(Nr,Rel,Res,Type,WF2).
2343
2344		:- block rel_iterate1(-,?,?,?,?).
2345		rel_iterate1(X,Rel,Res,Type,WF) :-
2346		%kernel_tools:value_variables(Rel,GrV),
2347		rel_iterate2(X,Rel,Res,Type,WF).
2348
2349		rel_iterate2(X,Rel,Res,Type,WF) :-
2350		( X=1 -> equal_object_wf(Res,Rel,rel_iterate2,WF)
2351		; X>1 -> X1 is X-1,
2352		rel_iterate2(X1,Rel,R1,Type,WF),
2353		rel_composition_wf(Rel,R1,Res,Type,WF)
2354		; X=0 -> rel_iterate0(Rel,Type,Res,WF)
2355		; add_wd_error('negative index in iterate',X,WF)
2356		).
2357
2358		:- use_module(bsyntaxtree,[get_set_type/2]).
2359		:- block rel_iterate0(?,-,?,?).
2360		rel_iterate0(_Rel,EType,Res,WF) :-
2361		get_set_type(EType,couple(Type,Type)),
2362		event_b_identity_for_type(Type,Res,WF).
2363
2364		:- use_module(typing_tools,[is_infinite_type/1]).
2365		event_b_identity_for_type(Type,Res,WF) :-
2366		create_texpr(identifier('_zzzz_unary'),Type,[],TIdentifier1), % was [generated]
2367		create_texpr(identifier('_zzzz_binary'),Type,[],TIdentifier2), % was [generated]
2368		(is_infinite_type(Type) -> Info = [prob_annotation('SYMBOLIC')] ; Info =[]),
2369		create_texpr(equal(TIdentifier1,TIdentifier2),pred,Info,TPred),
2370		construct_closure(['_zzzz_unary','_zzzz_binary'],[Type,Type],TPred,CRes),
2371		% for small types we could do: all_objects_of_type(Type,All), identity_relation_over_wf(All,CRes,WF)
2372		%, print(constructed_eventb_identity(Res)),nl
2373		equal_object_wf(Res,CRes,WF).
2374
2375
2376		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([],[(int(1),int(11))],[],_WF))).
2377		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2378		[(int(1),int(11))],[(int(1),(int(2),int(11)))],_WF))).
2379		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2380		[(int(2),int(11))],[],_WF))).
2381		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2382		[(int(2),int(11))],X,_WF),
2383		X = [])).
2384		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2385		[(int(1),int(11))],X,_WF),
2386		kernel_objects:equal_object(X,[(int(1),(int(2),int(11)))]))).
2387		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(1),int(6))],
2388		[(int(1),int(11))],X,_WF),
2389		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(6),int(11)))]))).
2390		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2391		[(int(1),int(11)),(int(1),int(12))],X,_WF),
2392		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))]))).
2393		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2394		[(int(1),int(11)),(int(1),int(12))],
2395		[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))],_WF))).
2396		:- 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))),
2397		avl_set(node((fd(1,'Name'),fd(2,'Name')),true,1,empty,node((fd(2,'Name'),fd(3,'Name')),true,0,empty,empty))),
2398		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)))
2399		,_WF))).
2400
2401		:- block direct_product_wf(-,?,?,?),direct_product_wf(?,-,?,?).
2402		direct_product_wf(Rel1,Rel2,Prod,WF) :-
2403		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),
2404		try_expand_and_convert_to_avl_with_check(Rel2,E2,direct_product),
2405		direct_product_wf1(E1,E2,Prod,WF).
2406
2407		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2408		direct_product_explicit_set(Rel1,Rel2,Res),!,
2409		equal_object_wf(Prod,Res,direct_product_wf1,WF).
2410		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2411		expand_custom_set_to_list_wf(Rel1,Relation1,_,direct_product_wf1_1,WF),
2412		expand_custom_set_to_list_wf(Rel2,Relation2,_,direct_product_wf1_2,WF),
2413		direct_product2(Relation1,Relation2,Prod,WF),
2414		direct_product_backwards(Relation1,Relation2,Prod,WF).
2415
2416		:- block direct_product2(-,?,?,?).
2417		direct_product2([],_,Out,WF) :- equal_object_wf(Out,[],direct_product2,WF).
2418		direct_product2([(X,Y)|T],Rel2,Out,WF) :-
2419		direct_product_tuple(Rel2,X,Y,Out,OutRem,WF),
2420		direct_product2(T,Rel2,OutRem,WF).
2421
2422		:- block direct_product_tuple(-,?,?,?,?,?).
2423		direct_product_tuple([],_,_,Res,Rem,WF) :- equal_object_optimized_wf(Res,Rem,direct_product_tuple,WF).
2424		direct_product_tuple([(X2,Z)|T],X,Y,Res,Rem,WF) :-
2425		direct_product_tuple(T,X,Y,CT,Rem,WF),
2426		equality_objects_wf(X2,X,EqRes,WF),
2427		direct_product_tuple3(EqRes,X,Y,Z,CT,Res,WF).
2428
2429		:- block direct_product_tuple3(-,?,?,?,?,?,?).
2430		direct_product_tuple3(pred_true,X,Y,Z,CT,Res,WF) :-
2431		equal_cons_wf(Res,(X,(Y,Z)),CT,WF). /* no need for add_element as output uniquely determines X,Y,Z !?*/
2432		direct_product_tuple3(pred_false,_X,_Y,_Z,CT,Res,WF) :- equal_object_optimized_wf(Res,CT,direct_product_tuple3,WF).
2433
2434		:- block direct_product_backwards(?,?,-,?).
2435		% Propagate information backwards from result to arguments
2436		direct_product_backwards(R1,R2,Prod,WF) :-
2437		((kernel_tools:ground_value(R1);kernel_tools:ground_value(R2)) -> true
2438		; expand_custom_set_to_list_wf(Prod,ProdList,_,direct_product_backwards,WF),
2439		direct_product_propagate_back(ProdList,R1,R2,WF)
2440		).
2441
2442		:- block direct_product_propagate_back(-,?,?,?).
2443		direct_product_propagate_back([],_,_,_WF).
2444		direct_product_propagate_back([(X,(Y,Z))|T],R1,R2,WF) :-
2445		check_element_of_wf((X,Y),R1,WF), check_element_of_wf((X,Z),R2,WF),
2446		direct_product_propagate_back(T,R1,R2,WF).
2447
2448		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([],[(int(3),int(4))],[]))).
2449		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([(int(1),int(2))],
2450		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2451		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2452		[(int(3),int(4))],X), ground(X),
2453		equal_object(X,[((int(1),int(3)),(int(2),int(4)))]))).
2454		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2455		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2456		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))], [],X),X == [])).
2457		:- assert_must_succeed((bsets_clp:parallel_product([], [(int(3),int(4))],X),X == [])).
2458
2459		parallel_product(Rel1,Rel2,Prod) :- parallel_product_wf(Rel1,Rel2,Prod,no_wf_available).
2460
2461		:- block parallel_product_wf(-,?,?,?),parallel_product_wf(?,-,?,?).
2462		% NOTE: we now have in_parallel_product; as such parallel products are kept symbolic
2463		%parallel_product_wf(Rel1,Rel2,Prod,WF) :- (keep_symbolic(Rel1) -> true ; keep_symbolic(Rel2)),
2464		% print_term_summary(parallel_product(Rel1,Rel2,Prod)),nl,
2465		%% % TO DO: generate closure
2466		% %{xy,mn|#(x,y,m,n).(xy=(x,y) & mn=(m,n) & (x,m):S & (y,n):R)}
2467		% fail.
2468		parallel_product_wf(Rel1,Rel2,Prod,WF) :-
2469		expand_custom_set_to_list_wf(Rel1,Relation1,_,parallel_product_1,WF),
2470		expand_custom_set_to_list_wf(Rel2,Relation2,_,parallel_product_2,WF),
2471		parallel_product2(Relation1,Relation2,ProdRes,WF),
2472		equal_object_optimized_wf(ProdRes,Prod,parallel_product,WF).
2473
2474		:- use_module(kernel_equality,[conjoin_test/4]).
2475		%(Rel1||Rel2) = {(x,y),(m,n)| (x,m):Rel1 & (y,n):Rel2}
2476
2477		% TO DO: use this in b_interpreter_check:
2478		in_parallel_product_test(((X,Y),(M,N)),Rel1,Rel2,Result,WF) :-
2479		conjoin_test(MemRes1,MemRes2,Result,WF),
2480		membership_test_wf(Rel1,(X,M),MemRes1,WF),
2481		membership_test_wf(Rel2,(Y,N),MemRes2,WF).
2482
2483		:- 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)).
2484
2485		in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2486		in_parallel_product_test(El,Rel1,Rel2,pred_true,WF).
2487
2488
2489		:- 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))).
2490
2491		not_in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2492		in_parallel_product_test(El,Rel1,Rel2,pred_false,WF).
2493
2494
2495		:- block parallel_product2(-,?,?,?).
2496		parallel_product2([],_,Out,WF) :- empty_set_wf(Out,WF).
2497		parallel_product2([(X,Y)|T],Rel2,Out,WF) :-
2498		parallel_product_tuple(Rel2,X,Y,Out,Tail,WF),
2499		parallel_product2(T,Rel2,Tail,WF).
2500
2501		:- block parallel_product_tuple(-,?,?,?,?,?).
2502		parallel_product_tuple([],_,_,Tail1,Tail2,WF) :- equal_object_wf(Tail1,Tail2,parallel_product_tuple,WF).
2503		parallel_product_tuple([(X2,Y2)|T],X,Y,Rel2,Tail,WF) :-
2504		equal_object_wf(Rel2,[((X,X2),(Y,Y2))|RT],parallel_product_tuple,WF),
2505		parallel_product_tuple(T,X,Y,RT,Tail,WF).
2506
2507
2508		% -------------------------------------------------
2509
2510		:- 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
2511		:- 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)).
2512		:- 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)).
2513		:- 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)).
2514		:- 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)).
2515		:- 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)).
2516		:- assert_must_fail((bsets_clp:not_partial_function([],[int(1)],[int(7)],_WF))).
2517		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[int(7)],_WF),
2518		X = [(int(1),int(7))])).
2519		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7)],_WF),
2520		X = [(int(2),int(7)),(int(1),int(7))])).
2521		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2522		[int(7),int(6)],_WF),
2523		X = [([(int(1),int(2))],int(7)),
2524		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2525		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2526		[int(7),int(6)],_WF),
2527		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2528		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2529		[int(7),int(6)],_WF),
2530		X = [([(int(1),int(2))],int(7)),
2531		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2532		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[[int(7),int(6)]],_WF),
2533		X = [(int(1),[int(6),int(7)])])).
2534		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2535		X = [(int(2),int(7)),(int(1),int(7))])).
2536		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2537		X = [(int(2),int(7)),(int(2),int(6))])).
2538		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2539		X = [(int(2),int(7)),(int(1),int(2))])).
2540		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2541		X = [(int(2),int(7)),(int(3),int(6))])).
2542		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2543		X = [(int(2),int(7)),(int(2),int(5))])).
2544		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2545		X = [(int(1),int(7)),(int(2),int(6)),(int(2),int(7))])).
2546		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2547		X = [(int(1),int(7)),(int(5),int(75))])).
2548		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2549		X = [(int(1),int(7)),(int(0),int(7))])).
2550		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2551		X = [(int(1),int(7)),(int(-1),int(7))])).
2552		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2553		X = [(int(1),int(7)),(int(0),int(7))])).
2554		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('Name'),global_set('Code'),_WF),
2555		X = [(fd(1,'Name'),fd(1,'Code'))])).
2556		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2557		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2558		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2559		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2560		:- assert_must_succeed((bsets_clp:not_partial_function([(fd(1,'Code'),int(1)),(fd(1,'Code'),int(2))],
2561		global_set('Code'),global_set('NAT1'),_WF) )).
2562
2563		:- block not_partial_function(-,?,?,?).
2564		not_partial_function([],_Domain,_Range,_WF) :- !,fail.
2565		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
2566		% we do not need the Range; this means we can match more closures (e.g., lambda)
2567		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2568		not_subset_of_wf(FFDomain,Domain,WF).
2569		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),
2570		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2571		not_both_subset_of(FFDomain,FFRange,Domain,Range,WF).
2572		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2573		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
2574		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2575		get_range_id_expression(P,T,TRangeID),!,
2576		subset_test(FFDomain,Domain,SubRes,WF),
2577		when(nonvar(SubRes),
2578		(SubRes=pred_false -> true % not a subset -> it is not a partial function over the domain
2579		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
2580		not_partial_function(R,Domain,Range,WF) :-
2581		expand_and_convert_to_avl_set_warn(R,AER,not_partial_function,'ARG /: ? +-> ?',WF),!,
2582		% TO DO: expand_and_convert_to_avl_set_catch and provide symbolic treatment similar to partial_function
2583		% e.g., to support f = NATURAL1 * {22,33} & not(f: NATURAL1 +-> NATURAL)
2584		is_not_avl_partial_function(AER,Domain,Range,WF).
2585		not_partial_function(R,Domain,Range,WF) :-
2586		expand_custom_set_to_list_wf(R,ER,_,not_partial_function,WF),
2587		not_pf(ER,[],Domain,Range,WF).
2588
2589		is_not_avl_partial_function(AER,Domain,Range,WF) :-
2590		(is_avl_partial_function(AER)
2591		-> is_not_avl_relation_over_domain_range(AER,Domain,Range,WF)
2592		; true
2593		).
2594
2595		:- block not_pf(-,?,?,?,?).
2596		not_pf([],_,_,_,_) :- fail.
2597		not_pf([(X,Y)|T],SoFar,Dom,Ran,WF) :-
2598		membership_test_wf_with_force(SoFar,X,MemRes,WF),
2599		not_pf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2600
2601		:- block not_pf2(-,?,?,?,?,?,?,?).
2602		not_pf2(pred_true,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* then not a function */
2603		not_pf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
2604		membership_test_wf_with_force(Dom,X,MemRes,WF), % creates a choice point in SMT mode
2605		not_pf2a(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2606
2607		:- block not_pf2a(-,?,?,?,?,?,?,?).
2608		not_pf2a(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* function, but domain wrong */
2609		not_pf2a(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
2610		remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,_LWF,Done), %% provide _LWF ??
2611		not_pf2b(Done,X,Y,T,SoFar,Dom2,Ran,WF).
2612
2613		:- block not_pf2b(-, ?,?,?, ?,?,?, ?).
2614		not_pf2b(_Done, X,Y,T, SoFar,Dom2,Ran, WF) :-
2615		add_element_wf(X,SoFar,SoFar2,WF),
2616		(T==[] -> not_element_of_wf(Y,Ran,WF)
2617		; membership_test_wf_with_force(Ran,Y,MemRes,WF),
2618		prop_empty_pred_false(T,MemRes), % if T=[] -> Y must not be in Ran
2619		not_pf3(MemRes,T,SoFar2,Dom2,Ran,WF)).
2620
2621		:- block prop_empty_pred_false(-,?).
2622		prop_empty_pred_false([],R) :- !, R=pred_false.
2623		prop_empty_pred_false(_,_).
2624
2625		:- block not_pf3(-,?,?,?,?,?).
2626		not_pf3(pred_false,_T,_SoFar,_Dom2,_Ran,_WF). /* illegal range */
2627		not_pf3(pred_true,T,SoFar,Dom2,Ran,WF) :-
2628		not_pf(T,SoFar,Dom2,Ran,WF).
2629
2630		:- 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)).
2631		:- 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)).
2632		:- 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)).
2633		:- 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)).
2634		:- assert_must_succeed((bsets_clp:partial_function([],[int(1)],[int(7)]))).
2635		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1)],[int(7)]),
2636		X = [(int(1),int(7))])).
2637		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),
2638		equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2639		:- assert_must_succeed((findall(X,bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),L),
2640		length(L,Len), Len >= 4,
2641		(preferences:get_preference(convert_comprehension_sets_into_closures,true) -> true ; Len=4) )).
2642		:- assert_must_succeed((bsets_clp:partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2643		[int(7),int(6)]),
2644		equal_object(X,[([(int(1),int(2))],int(7)),
2645		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
2646		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2647		[int(7),int(6)],_WF),
2648		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2649		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2650		[int(7),int(6)],_WF),
2651		X = [([(int(1),int(2))],int(7)),
2652		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2653		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[int(1)],[[int(7),int(6)]],_WF),
2654		X = [(int(1),[int(6),int(7)])])).
2655		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2656		X = [(int(1),int(7)),(int(5),int(75))])).
2657		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2658		X = [(int(1),int(7)),(int(0),int(7))])).
2659		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2660		X = [(int(1),int(7)),(int(-1),int(7))])).
2661		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2662		X = [(int(1),int(7)),(int(0),int(7))])).
2663		:- assert_must_fail((bsets_clp:partial_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
2664		X = [(int(2),int(7)),(int(2),int(6))])).
2665		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('Name'),global_set('Code'),_WF),
2666		X = [(fd(1,'Name'),fd(1,'Code'))])).
2667		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2668		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2669		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2670		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2671
2672		partial_function(R,Domain,Range) :- init_wait_flags(WF,[partial_function]),
2673		partial_function_wf(R,Domain,Range,WF),
2674		ground_wait_flags(WF).
2675
2676		:- use_module(kernel_equality,[get_cardinality_powset_wait_flag/5]).
2677		:- use_module(closures,[is_lambda_value_domain_closure/5]).
2678		:- block partial_function_wf(-,-,?,?).
2679		partial_function_wf(R,_Domain,_Range,_WF) :- R==[], !.
2680		partial_function_wf(R,Domain,Range,WF) :- (Domain==[] ; Range==[]), !, empty_set_wf(R,WF).
2681		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2682		custom_explicit_sets:is_definitely_maximal_set(Range),
2683		% we do not need the Range; this means we can match more closures (e.g., lambda)
2684		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2685		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF).
2686		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2687		% TODO: this will fail if is_definitely_maximal_set was true above !
2688		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2689		% same as for total_function_wf check
2690		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2691		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
2692		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2693		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2694		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2695		get_range_id_expression(P,T,TRangeID),
2696		!,
2697		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2698		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2699		[b_operator(range,[FF]),Range],unknown),WF3),
2700		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF3). % we could use symbolic_range_subset_check
2701		partial_function_wf(R,Domain,Range,WF) :-
2702		expand_and_convert_to_avl_set_catch(R,AER,partial_function_wf,'ARG : ? +-> ?',ResultStatus,WF),!,
2703		(ResultStatus=avl_set
2704		-> is_avl_partial_function_over(AER,Domain,Range,WF)
2705		; % keep symbolic
2706		(debug_mode(off) -> true ; print('SYMBOLIC +-> check : '),translate:print_bvalue(R),nl),
2707		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
2708		symbolic_domain_subset_check(R,Domain,WF),
2709		symbolic_range_subset_check(R,Range,WF),
2710		symbolic_functionality_check(R,WF)
2711		).
2712		partial_function_wf(R,Domain,Range,WF) :-
2713		get_cardinality_powset_wait_flag(Domain,partial_function_wf,WF,Card,CWF),
2714		% probably we should compute real cardinality of set of partial functions over Domain +-> Range ?
2715		% the powset waitflag uses 2^Card as priority; is the number of partial functions when Range contains just a single element
2716		% slows down test 1088: TO DO investigate
2717		% get_cardinality_partial_function_wait_flag(Domain,Range,partial_function_wf,WF,Card,_,CWF),
2718		%% Maybe we should only enumerate partial functions for domain variables ; e.g., not f <+ {x |-> y} : T +-> S
2719		%% print_bt_message(pf_dom_card(Card)),nl, %%%
2720		% probably we should use a special version when R is var
2721		propagate_empty_set_wf(Domain,dom_pf,R,WF),
2722		propagate_empty_set_wf(Range,ran_pf,R,WF),
2723		(var(R) -> pf_var_r(R,var,Domain,Range,Card,WF,CWF) ; pf_var_r(R,nonvar,Domain,Range,Card,WF,CWF)).
2724
2725		% symbolic dom(R) <: Domain check for closures
2726		symbolic_domain_subset_check(R,Domain,WF) :-
2727		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2728		[b_operator(domain,[R]),Domain],unknown),WF2),
2729		domain_subtraction_wf(Domain,R,Res,WF2), % works symbolically
2730		(debug_mode(off) -> true ; print('Domain Violations: '),translate:print_bvalue(Res),nl),
2731		empty_set_wf(Res,WF2). % empty_set does a symbolic treatment calling gen_typed_ids and b_not_test_exists:
2732		% symbolic ran(R) <: Range check for closures
2733		symbolic_range_subset_check(R,Range,WF) :-
2734		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2735		[b_operator(range,[R]),Range],unknown),WF2),
2736		range_subtraction_wf(R,Range,Res,WF2), % works symbolically
2737		(debug_mode(off) -> true ; print('Range Violations: '),translate:print_bvalue(Res),nl),
2738		empty_set_wf(Res,WF2). % works symbolically
2739		symbolic_functionality_check(Closure,WF) :-
2740		custom_explicit_sets:symbolic_functionality_check_closure(Closure,ViolationsClosure),!,
2741		(debug_mode(off) -> true ; print('FUNCTIONALITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2742		empty_set_wf(ViolationsClosure,WF). % works symbolically
2743		symbolic_functionality_check(R,WF) :-
2744		add_error_wf(symbolic_functionality_check,'Could not check functionality of:',R,R,WF).
2745
2746		symbolic_injectivity_check(Closure,WF) :-
2747		custom_explicit_sets:symbolic_injectivity_check_closure(Closure,ViolationsClosure),!,
2748		(debug_mode(off) -> true ; print('INJECTIVITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2749		empty_set_wf(ViolationsClosure,WF). % works symbolically
2750		symbolic_injectivity_check(R,WF) :-
2751		add_error_wf(symbolic_functionality_check,'Could not check injectivity of:',R,R,WF).
2752
2753
2754		is_avl_partial_function_over(AER,Domain,Range,WF) :-
2755		is_avl_partial_function(AER),
2756		is_avl_relation_over_domain(AER,Domain,WF),
2757		is_avl_relation_over_range(AER,Range,WF).
2758
2759		% symbolically check that the range of lambda closure is a subset of a given Range
2760		% TRangeID is obtained by calling get_range_id_expression(P,T,TRangeID)
2761		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2762		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2763		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2764		% CHECK not(#P.(Pred & TRangeID /: Range))
2765		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2766		is_empty_closure_wf(P,T,Pred2,WF2). % do we need to rename _lambda_result_ using rename_lambda_result_id ?
2767		% now the negation thereof:
2768		check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2769		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
2770		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2771		% CHECK (#P.(Pred & TRangeID /: Range))
2772		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2773		is_non_empty_closure_wf(P,T,Pred2,WF2).
2774		test_lambda_closure_range(P,T,Pred,TRangeID,Range,Res,WF) :-
2775		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset, % it is actually a reify check
2776		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2777		% reify not(#P.(Pred & TRangeID /: Range))
2778		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2779		test_empty_closure_wf(P,T,Pred2,Res,WF2).
2780
2781		get_not_in_range_pred_aux(Pred,TRangeID,Range,NewPred) :- % construct (Pred & TRangeID /: Range)
2782		ExpectedRange = b(value(Range),set(RanT),[]),
2783		get_texpr_type(TRangeID,RanT),
2784		safe_create_texpr(not_member(TRangeID,ExpectedRange),pred,NotMemCheck),
2785		conjunct_predicates([Pred,NotMemCheck],NewPred).
2786
2787
2788		% if first argument is empty, second argument must also be empty
2789		:- block propagate_empty_set_wf(-,?,?,?).
2790		propagate_empty_set_wf([],_PP,A,WF) :- !, %print(prop_empty(_PP,A)),nl,
2791		kernel_objects:empty_set_wf(A,WF). % TO DO: add WF
2792		propagate_empty_set_wf(_,_,_,_).
2793
2794		:- block pf_var_r(-,?,?,?,?,?,-).
2795		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
2796		expand_and_convert_to_avl_set_warn(R,AER,pf_var_r,'ARG : ? +-> ?',WF),!,
2797		is_avl_partial_function_over(AER,Domain,Range,WF).
2798		pf_var_r(R,_,Domain,Range,Card,WF,CWF) :-
2799		expand_custom_set_to_list_wf(R,ER,_,partial_function_wf,WF),
2800		%get_last_wait_flag(partial_fun(Domain),WF,LWF),
2801	?	pf_w(ER,[],Domain,Range,Card,_Large,WF,CWF).
2802
2803		pf_w(T,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2804		(Card==0 -> T=[]
2805	?	; pf(T,SoFar,Dom,Ran,Card,Large,WF,LWF)).
2806
2807		:- block pf(-,?,?,?,?,?,?,-).
2808		pf(LIST,_,_,_,_,_WF,_,_LWF) :- LIST==[],!. % avoid leaving choicepoint
2809		pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF) :- nonvar(AVL),AVL=avl_set(_A),
2810		add_internal_error('AVL arg: ',pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF)),fail.
2811		pf([],_,_,_,_,_WF,_,_LWF).
2812		pf(LIST,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2813		(var(LIST) -> ListWasVar = true ; ListWasVar = false), % is ListWasVar = true we are doing the enumeration driven by LWF being ground
2814		LIST = [(X,Y)|T],
2815	?	dec_card(Card,NC),/* Card ensures we do not build too big lists */
2816		Dom \== [],
2817	?	remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done),
2818	?	check_element_of_wf(Y,Ran,WF),
2819	?	pf1(Done, X,Y,T,SoFar,Dom2,Ran,NC,Large,WF,LWF).
2820
2821		:- block dec_card(-,?).
2822		dec_card(inf,NewC) :- !, NewC=inf.
2823		dec_card(C,NewC) :- C>0, NewC is C-1.
2824
2825		:- block pf1(-, ?,?,?,?,?,?,?,?,?,?).
2826		pf1(_Done, X,_Y,T,SoFar,Dom2,Ran,Card,Large,WF,LWF) :-
2827		not_element_of_wf(X,SoFar,WF), /* check that it is a function */
2828		%% check_element_of_wf(Y,Ran,WF), % this check is now done above in pf
2829		add_new_element_wf(X,SoFar,SoFar2,WF),
2830	?	pf_w(T,SoFar2,Dom2,Ran,Card,Large,WF,LWF).
2831
2832		remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done) :- compute_large(Dom,Large),
2833		((ListWasVar==true,var(X),var(Y),Large==false,
2834		preference(convert_comprehension_sets_into_closures,false), % not in symbolic mode
2835		kernel_tools:ground_value(Dom))
2836		-> %% (X, Y are free and we drive the enumeration: we can influence which element is taken from Dom
2837		remove_a_minimal_element(X,Dom,Dom2,WF,Done) %%%%%%%%%% added Jul 15 2008
2838	?	; remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,LWF,Done)
2839		).
2840		compute_large(Dom,Large) :- % check if the domain is large; ensure that we compute this only once
2841		(nonvar(Large) -> true
2842		; var(Dom) -> true
2843		; dont_expand_this_explicit_set(Dom) -> Large=large
2844		; Large=false).
2845
2846		:- assert_must_succeed(( bsets_clp:remove_a_minimal_element(X,[int(1)],R,_WF,Done),
2847		X==int(1), Done==true, R=[] )).
2848		:- 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),
2849		X==int(2), Done==true, R=[int(3)] )).
2850		:- 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),
2851		X==int(1), R=[int(2),int(3)], Done==true )).
2852		:- 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),
2853		X==int(3), R=[], Done==true )).
2854		:- assert_must_succeed(( init_wait_flags(WF), CL=closure(['_zzzz_binary'],[integer],b(member( b(identifier('_zzzz_binary'),integer,[]),
2855		b(interval(b(value(int(1)),integer,[]),b(value(int(10)),integer,[])),set(integer),[])),pred,[])),
2856		bsets_clp:remove_a_minimal_element(X,CL,R,WF,Done), ground_wait_flags(WF),
2857		X=int(9), Done==true, kernel_objects:equal_object(R,[int(10)]) )).
2858
2859		/* usage: restrict number of possible choices if element to remove is free */
2860		/* select one element; and disallow all elements appearing before it in the list */
2861		remove_a_minimal_element(X,Set,Res,WF,Done) :-
2862		expand_custom_set_to_list_wf(Set,ESet,EDone,remove_a_minimal_element,WF),
2863		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done).
2864
2865		:- use_module(kernel_equality,[get_cardinality_wait_flag/4]).
2866		:- block remove_a_minimal_element2(?,?,-,?,?,?).
2867		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done) :- var(ESet),
2868		% should not happen as we wait for EDone
2869		add_internal_error('Illegal call: ',remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done)),
2870		fail.
2871		remove_a_minimal_element2(X,ESet,_EDone,Res,WF,Done) :-
2872		ESet \= [],
2873		(ESet = [El]
2874		-> X=El, empty_set_wf(Res,WF), Done=true % only one choice
2875		; get_cardinality_wait_flag(ESet,remove_a_minimal_element2,WF,CWF),
2876		remove_a_minimal_element3(X,ESet,Res,WF,Done,CWF)
2877		).
2878
2879		:- block remove_a_minimal_element3(?,?,?,?,?,-).
2880		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- var(Res), !,
2881		append(_,[X|TRes],ESet), % WHAT IF Res has been instantiated in the meantime ???
2882		equal_object_wf(Res,TRes,remove_a_minimal_element2_2,WF),Done=true.
2883		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- %print(remove_min_nonvar_res(Res)),nl,
2884		equal_cons_wf(ESet,X,Res,WF), Done=true.
2885
2886
2887		% reified version of partial function test partial_function_wf:
2888		:- 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)).
2889		:- 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)).
2890		:- 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)).
2891		:- 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)).
2892		:- 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)).
2893		:- 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)).
2894
2895		:- use_module(kernel_equality,[subset_test/4]).
2896		:- block partial_function_test_wf(-,?,?,-,?), partial_function_test_wf(?,-,-,-,?).
2897		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_true,!,
2898		partial_function_wf(FF,Domain,Range,WF).
2899		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_false,!,
2900		not_partial_function(FF,Domain,Range,WF). % TO DO: remove not_partial_function to use check_is_partial_function?
2901		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2902		custom_explicit_sets:is_definitely_maximal_set(Range),
2903		% we do not need the Range; this means we can match more closures (e.g., lambda)
2904		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2905		subset_test(FFDomain,Domain,Res,WF).
2906		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2907		% TODO: this will fail if is_definitely_maximal_set was true above !
2908		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2909		% same as for total_function_wf check
2910		subset_test(FFDomain,Domain,DomainOk,WF),
2911		(DomainOk==pred_false -> Res = pred_false
2912		; conjoin_test(DomainOk,RangeOk,Res,WF),
2913		subset_test(FFRange,Range,RangeOk,WF)).
2914		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2915		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2916		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2917		get_range_id_expression(P,T,TRangeID),
2918		!,
2919		subset_test(FFDomain,Domain,DomainOk,WF),
2920		(DomainOk == pred_false -> Res=pred_false
2921		; conjoin_test(DomainOk,RangeOk,Res,WF),
2922		test_lambda_closure_range(P,T,Pred,TRangeID,Range,RangeOk,WF)
2923		).
2924		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2925		expand_and_convert_to_avl_set_warn(R,AER,partial_function_test_wf,'ARG : ? +-> ?',WF),!,
2926		% TO DO: use expand_and_convert_to_avl_set_catch
2927		(is_avl_partial_function(AER)
2928		-> % TO DO: we could do something similar to this instead: is_not_avl_relation_over_domain_range
2929		domain_of_explicit_set_wf(avl_set(AER),FFDomain,WF),
2930		subset_test(FFDomain,Domain,DomainOk,WF),
2931		(DomainOk == pred_false -> Res=pred_false
2932		; range_of_explicit_set_wf(avl_set(AER),FFRange,WF),
2933		conjoin_test(DomainOk,RangeOk,Res,WF),
2934		subset_test(FFRange,Range,RangeOk,WF)
2935		)
2936		; Res=pred_false).
2937		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2938		expand_custom_set_to_list_wf(R,ER,_,partial_function_test_wf,WF),
2939		check_is_partial_function_acc_wf(ER,[],Domain,Range,Res,WF).
2940
2941		:- block check_is_partial_function_acc_wf(-,?,?,?,?,?).
2942		check_is_partial_function_acc_wf([],_,_,_,Res,_WF) :- !, Res=pred_true.
2943		check_is_partial_function_acc_wf([(A,FA)|T],Acc,Dom,Ran,Res,WF) :- !,
2944		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF),
2945		(MemResDomRan==pred_false
2946		-> Res = pred_false
2947		; membership_test_wf(Acc,A,MemResNotFunc,WF),
2948		negate(MemResNotFunc,MemResFunctionality),
2949		conjoin_test(MemResDomRan,MemResFunctionality,PF_Head,WF),
2950		(PF_Head == pred_false -> Res = pred_false
2951		; T==[] -> Res=PF_Head
2952		; add_element_wf(A,Acc,NewAcc,WF),
2953	?	conjoin_test(PF_Head,PF_Tail,Res,WF),
2954		check_is_partial_function_acc_wf(T,NewAcc,Dom,Ran,PF_Tail,WF))
2955		).
2956
2957		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF) :-
2958		membership_test_wf(Dom, A,MemResDom,WF), % use membership_test_wf_with_force for SMT mode ??
2959		(MemResDom == pred_false -> MemResDomRan = pred_false
2960		; membership_test_wf(Ran,FA,MemResRan,WF),
2961		conjoin_test(MemResDom,MemResRan,MemResDomRan,WF)).
2962
2963		:- 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)).
2964		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[int(7)]),
2965		X = [(int(1),int(7))])).
2966		:- assert_must_succeed((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
2967		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2968		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3))]],[int(7),int(6)]),
2969		kernel_objects:equal_object(X,[([(int(1),int(3))],int(7)),([(int(1),int(2))],int(7))]))).
2970		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2971		[int(7),int(6)]),
2972		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
2973		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
2974		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[[int(7),int(6)]]),
2975		kernel_objects:equal_object(X,[(int(1),[int(6),int(7)])]))).
2976		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2977		[[int(7),int(6)]]),
2978		kernel_objects:equal_object(X,[([(int(1),int(2))],[int(6),int(7)]),
2979		([(int(2),int(3)),(int(1),int(3))],[int(6),int(7)])]))).
2980		:- assert_must_succeed((bsets_clp:total_function(X,[ [(int(1),int(3)),(int(2),int(3))]],
2981		[int(6)]),
2982		kernel_objects:equal_object(X,[ ([(int(2),int(3)),(int(1),int(3))], int(6)) ]))).
2983		:- assert_must_succeed((bsets_clp:total_function(X,global_set('Name'),
2984		[[],[fd(1,'Code'),fd(2,'Code')],[fd(1,'Code')],[fd(2,'Code')]]),
2985		kernel_objects:enumerate_basic_type(X,set(couple(global('Name'),set(global('Code'))))),
2986		kernel_objects:equal_object(X,[(fd(3,'Name'),[fd(2,'Code')]),(fd(1,'Name'),[fd(2,'Code')]),(fd(2,'Name'),[])]))).
2987
2988		%:- assert_must_succeed(( kernel_waitflags:init_wait_flags(WF),bsets_clp:total_function_wf(TF,global_set('Code'),
2989		% closure([zzzz],[set(set(couple(integer,boolean)))],
2990		% member(identifier(zzzz),
2991		% pow_subset(value(closure([zzzz],[set(couple(integer,boolean))],
2992		% member('ListExpression'(['Identifier'(zzzz)]),
2993		% 'Seq'(value([pred_true /* bool_true */,pred_false /* bool_false */])))))))),WF),
2994		% 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 */)]]),
2995		% (fd(2,'Code'), [[],[(int(1),pred_true /* bool_true */)],[(int(1),pred_true /* bool_true */),(int(2),pred_true /* bool_true */)]]) ]),
2996		% kernel_waitflags:ground_wait_flags(WF) )).
2997
2998		:- assert_must_succeed((bsets_clp:total_function([],[],[int(7)]))).
2999
3000		:- assert_must_fail((bsets_clp:total_function([],[int(1)],[int(7)]))).
3001		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3002		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
3003		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3004		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(5))]))).
3005		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3006		kernel_objects:equal_object(X,[(int(2),int(7))]))).
3007		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3008		[int(7),int(6)]),
3009		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
3010		([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3011		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3012		[int(7),int(6)]),
3013		kernel_objects:equal_object(X,[([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3014
3015		total_function(R,Domain,Range) :- init_wait_flags(WF,[total_function]),
3016		total_function_wf(R,Domain,Range,WF),
3017		ground_wait_flags(WF).
3018
3019
3020		:- assert_must_succeed((bsets_clp:total_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
3021		nonvar(X),X=[(A,B),(C,D)],A==int(1),C==int(2),\+ ground(B),\+ ground(D), B=int(7),D=int(7) )).
3022
3023		:- block total_function_wf(-,-,-,?).
3024		total_function_wf(FF,Domain,_Range,WF) :- FF == [],!,
3025		empty_set_wf(Domain,WF).
3026		total_function_wf(FF,Domain,Range,WF) :-
3027		Range == [],!,
3028		empty_set_wf(FF,WF), empty_set_wf(Domain,WF).
3029		total_function_wf(FF,Domain,Range,WF) :-
3030		% TO DO: if FF or Domain nonvar but \= [] -> check if other variable becomes []
3031	?	total_function_wf1(FF,Domain,Range,WF).
3032
3033		:- block total_function_wf1(?,-,?,?).
3034		total_function_wf1(FF,Domain,_Range,WF) :-
3035		FF==[],!,
3036		empty_set_wf(Domain,WF).
3037		total_function_wf1(FF,Domain,Range,WF) :-
3038		custom_explicit_sets:is_definitely_maximal_set(Range),
3039		% we do not need the Range; this means we can match more closures (e.g., lambda)
3040		(nonvar(FF),
3041		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF)
3042		-> !,
3043		equal_object_wf(FFDomain,Domain,total_function_wf1_1,WF)
3044		; var(FF),
3045		get_wait_flag1(WF,WF1), var(WF1),
3046		\+ (custom_explicit_sets:get_card_for_specific_custom_set(Domain,Card), number(Card)),
3047		% we have a total_function over a possibly infinite domain,
3048		% better wait: maybe a recursive of other closure will be produced for FF
3049		!,
3050		when( (nonvar(FF) ; nonvar(WF1)), total_function_wf1(FF,Domain,Range,WF))
3051		).
3052		total_function_wf1(FF,Domain,Range,WF) :- nonvar(FF),
3053		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3054		equal_object_wf(FFDomain,Domain,total_function_wf1_2,WF),
3055		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
3056		total_function_wf1(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3057		total_function_avl_set(AEF,Domain,Range,WF).
3058		total_function_wf1(FF,Domain,Range,WF) :-
3059		% want to replace FF by closure: needs to be a variable!
3060		var(FF),
3061		% if the total function can not be build up explicitly (i.e. infinite domain)
3062		% TODO: can / should this be relaxed?
3063		custom_explicit_sets:is_infinite_explicit_set(Domain), % get_card_for_specific_custom_set or is_infinite_or_symbolic_closure
3064		% TO DO: delay if Domain infinite or closure and not yet known and range is type
3065		kernel_objects:infer_value_type(Domain,set(DomT)),
3066		kernel_objects:infer_value_type(Range,set(RanT)),
3067		!,
3068		% IDEA : TF = %x.(x:Domain|DEFAULT) <+ SFF, where SFF is partial function and DEFAULT is some default value
3069		% build up a partial function instead (fulfilling all constraints)
3070		% better? : %x.(x:Domain|IF x:dom(SFF) THEN SFF(x) ELSE DEFAULT)?
3071		partial_function_wf(SFF,Domain,Range,WF),
3072		% next, build up a total function mapping everything to a default value
3073		% this function will be overriden by the partial function to fulfilling
3074		% given constraints
3075		% 1. identifiers for closure
3076		create_texpr(identifier('__domid__'),DomT,[],TDomId),
3077		create_texpr(identifier('__ranid__'),RanT,[],TRanId),
3078		% 2. domain identifier might take all values of the domain
3079		create_texpr(member(TDomId,b(value(Domain),set(DomT),[])),pred,[],DomMember),
3080		% 3. pick a single value for the range identifier
3081		check_element_of_wf(RangeElement,Range,WF),
3082		%% external_functions:observe_value(RangeElement,"range"),external_functions:observe_value(SFF,"pf"),
3083		create_texpr(equal(TRanId,b(value(RangeElement),RanT,[])),pred,[],RanMember),
3084		% 4. conjunct and form closure (should be treated symbolically)
3085		conjunct_predicates([RanMember,DomMember],Pred),
3086		Default = closure(['__domid__','__ranid__'],[DomT,RanT],Pred),
3087		% 5. override default values where needed
3088		override_relation(Default,SFF,FF,WF),
3089		get_last_wait_flag(enum_symb_tf,WF,LastWF),
3090		when(nonvar(LastWF), % if we enum too early test 1619 fails; see also test 2022
3091		% as partial_function_wf does not fully enumerate the new variable SFF we may have to enumerate SFF; see test 2328
3092		(enumerate_basic_type_wf(RangeElement,RanT,WF),
3093		enumerate_basic_type_wf(SFF,set(couple(DomT,RanT)),WF)
3094		)).
3095		total_function_wf1(R,Domain,Range,WF) :-
3096		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
3097		% TO DO: maybe avoid converting intervals which are not fully instantiated ?
3098		% 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
3099		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
3100		propagate_empty_set_wf(Range,tf_range,ER,WF), % if the range of a total function is empty then the function must be empty
3101	?	total_function_wf2(ER,EDomain,Range,WF).
3102
3103		:- block total_function_wf2(?,-,?,?).
3104		total_function_wf2(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3105		total_function_avl_set(AEF,Domain,Range,WF).
3106		total_function_wf2(R,Domain,Range,WF) :-
3107		cardinality_as_int_wf(Domain,int(Card),WF),
3108	?	total_function_wf3(R,Card,Domain,Range,WF).
3109
3110		total_function_wf3(FF,Card,Domain,Range,WF) :-
3111		nonvar(FF),
3112		(number(Card) -> (Card >= 1000 -> true ; is_symbolic_closure(FF)) ; true),
3113		% note: we can have symbolic closures with a finite domain: /*@symbolic */ %p.(p:BOOL|(%t.(t:NATURAL|t+100)))
3114		custom_explicit_sets:dom_for_lambda_closure(FF,FFDomain),
3115		% we have a lambda closure where we cannot determine the range, otherwise dom_range_for_specific_closure would have succeeded
3116		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 --> NATURAL
3117		FF = closure(P,T,Pred),
3118		get_range_id_expression(P,T,TRangeID),
3119		!,
3120		equal_object_wf(FFDomain,Domain,total_function1_closure,WF),
3121		% CHECK not(#P.(Pred & P /: Range))
3122		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF).
3123		% custom_explicit_sets:ran_symbolic(closure(P,T,Pred),RanSymbolic), check_subset_of_wf(RanSymbolic,Range,WF).
3124		total_function_wf3(R,Card,Domain,Range,WF) :- Card==inf,!,
3125		when(nonvar(R), total_function_symbolic(R,Domain,Range,WF)).
3126		total_function_wf3(R,Card,Domain,Range,WF) :-
3127		card_convert_int_to_peano(Card,PeanoCard),
3128		((nonvar(R);ground(PeanoCard))
3129		-> true
3130		; get_last_wait_flag(total_fun(Domain),WF,WF1)),
3131	?	when((nonvar(R);ground(PeanoCard);
3132		(nonvar(PeanoCard),nonvar(WF1))), /* mal 12/5/04: changed , into ; 17/3/2008: added WF1 */
3133		/* reason for delaying nonvar(Card): Card grounded bit by bit by cardinality; avoid
3134		triggering too early and missing tf_var */
3135		total_function1(R,Card,PeanoCard,Domain,Range,WF
3136		)).
3137
3138		:- use_module(bsyntaxtree,[safe_create_texpr/3]).
3139		:- use_module(library(lists),[last/2]).
3140		% for a closure get the identifier or proj expression that represents range values
3141		get_range_id_expression([PairID],[Type],Res) :- !,
3142		Type = couple(_,TX),
3143		TP = b(identifier(PairID),Type,[]),
3144		safe_create_texpr(second_of_pair(TP),TX,Res). % prj2(PairID) ,
3145		%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
3146		% but currently lambda closure detection in dom_for_lambda_closure cannot handle such closures anyway
3147		get_range_id_expression(P,T,b(identifier(ID),Type,[])) :- last(P,ID), last(T,Type).
3148
3149		total_function_avl_set(AEF,Domain,Range,WF) :-
3150		(Domain = avl_set(Dom) -> is_avl_total_function_over_domain(AEF,Dom)
3151		; is_avl_partial_function(AEF),
3152		domain_of_explicit_set_wf(avl_set(AEF),AEF_Domain,WF),
3153		equal_object_wf(AEF_Domain,Domain,total_function_avl_set,WF)
3154		),
3155		is_avl_relation_over_range(AEF,Range,WF).
3156
3157		total_function_symbolic(FF,Domain,Range,WF) :-
3158		(debug_mode(off) -> true ; print('SYMBOLIC --> check : '),translate:print_bvalue(FF),nl),
3159		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
3160		domain_wf(FF,Domain,WF),
3161		symbolic_range_subset_check(FF,Range,WF),
3162		symbolic_functionality_check(FF,WF).
3163
3164		total_function1(FF,Card,PeanoCard,Domain,Range,WF) :- Card==inf,PeanoCard=inf,!,
3165		total_function_symbolic(FF,Domain,Range,WF).
3166		total_function1(FF,_,_,Domain,Range,WF) :-
3167		expand_and_convert_to_avl_set_catch(FF,AEF,total_function1,'ARG : ? --> ?',ResultStatus,WF),!,
3168		(ResultStatus=avl_set -> total_function_avl_set(AEF,Domain,Range,WF)
3169		; % keep symbolic
3170		% TO DO: ensure no pending co-routine infinite_peano in card_convert_int_to_peano
3171		total_function_symbolic(FF,Domain,Range,WF)
3172		).
3173		total_function1(R,_,Card,Domain,Range,WF) :-
3174		try_expand_custom_set_wf(R,ER,total_function1,WF),
3175	?	total_function2(ER,Card,Domain,Range,WF).
3176
3177		total_function2(ER,Card,Domain,Range,WF) :-
3178		var(ER),ground(Card),!,
3179		tf_var(TotalFunction,[],Card,Domain,Range,WF),
3180		ER=TotalFunction.
3181		total_function2(ER,Card,Domain,Range,WF) :-
3182		(ground(Card)
3183		-> get_wait_flag(0,tot_fun,WF,LWF) % we seem to know the domain exactly now; see e.g. test 1316
3184		; 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 ?
3185	?	tf(ER,[],Card,Domain,Range,WF,LWF).
3186
3187		:- block tf(-,?,-,?,?,?,?),tf(-,?,?,?,?,?,-).
3188		tf([],_,0,Dom,_,WF,_) :- empty_set_wf(Dom,WF).
3189		tf(FUN,SoFar,s(Card),Dom,Ran,WF,LWF) :- var(FUN),nonvar(Dom), % try setting up skeleton for total fun
3190		remove_exact_first_element(X,Dom,Dom2),not_element_of_wf(X,SoFar,WF),var(FUN),!,
3191		FUN = [(X,Y)|T], tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3192		tf([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF,LWF) :-
3193		not_element_of_wf(X,SoFar,WF),
3194		remove_element_wf(X,Dom,Dom2,WF), %mal: 17/3/08 changed to _wf version
3195	?	tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3196		tf(CS,SoFar,Card,Dom,Ran,WF,LWF) :- nonvar(CS), is_custom_explicit_set(CS),
3197		expand_custom_set_to_list_wf(CS,ER,_,tf,WF),
3198		tf(ER,SoFar,Card,Dom,Ran,WF,LWF).
3199		tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF) :-
3200		check_element_of_wf(Y,Ran,WF),
3201		%when((nonvar(T);nonvar(Card)), /* mal 12/5/04: changed , into ; */
3202		add_new_element_wf(X,SoFar,SoFar2,WF), %%% try_expand_and_convert_to_avl
3203	?	tf(T,SoFar2,Card,Dom2,Ran,WF,LWF).
3204
3205		:- block tf_var(-,?,-,?,?,?).
3206		tf_var(F,_,Card,Dom,_,WF) :- Card==0,!,F=[],empty_set_wf(Dom,WF). % avoid choice point
3207		tf_var([],_,0,Dom,_,WF) :- empty_set_wf(Dom,WF).
3208		tf_var([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF) :-
3209		/* supposes that X + Y are unbound */
3210		/* TO DO: rewrite like enumerate <-------------------------- */
3211		((var(X),var(Y)) -> true ; (print_message(warning,'Nonvar in tf_var: '),
3212		print_message(warning,((X,Y))))),
3213		remove_exact_first_element(X,Dom,Dom2),
3214		not_element_of_wf(X,SoFar,WF),
3215		check_element_of_wf(Y,Ran,WF),
3216		add_new_element_wf(X,SoFar,SoFar2,WF),
3217		tf_var(T,SoFar2,Card,Dom2,Ran,WF).
3218
3219
3220
3221		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1)],[int(7)]),
3222		X = [(int(1),int(7))])).
3223		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3224		kernel_objects:equal_object(X,[(int(2),int(8)),(int(1),int(7))]))).
3225		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1)],[int(7),int(3)]),
3226		X = [(int(1),int(7))])).
3227		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(3)]),
3228		X = [(int(1),int(3)),(int(2),int(3))])).
3229		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3230		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
3231		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3232		X = [(int(1),int(7)),(int(1),int(8))])).
3233
3234
3235
3236		total_bijection(R,Domain,Range) :- init_wait_flags(WF,[total_bijection]),
3237		total_bijection_wf(R,Domain,Range,WF),
3238		ground_wait_flags(WF).
3239
3240		:- block total_bijection_wf(?,-,?,?).
3241		total_bijection_wf(FF,Domain,Range,WF) :- nonvar(FF),
3242		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3243		equal_object_wf(FFDomain,Domain,total_bijection_wf_1,WF),
3244		equal_object_wf(FFRange,Range,total_bijection_wf_2,WF).
3245		%(R,Domain,Range,WF) :- Domain==Range,!, print(eq_domain_range),nl, total_injection_wf(R,Domain,Range,WF).
3246		total_bijection_wf(R,Domain,Range,WF) :-
3247		same_cardinality_wf(Domain,Range,WF),
3248		total_injection_wf2(R,Domain,Range,WF). % TO DO: use cardinality_as_int_wf ? makes test 1194 fail
3249
3250		%Note: we used to call custom code: total_bijection_wf2(R,Domain,Card,Range,WF).
3251		% total_injection_wf2 gives a considerable performance boost, e.g., for test 1222 ClearSy/alloc_large.mch or NQueens with >->>
3252
3253		:- 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)).
3254		:- 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)).
3255		:- 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)).
3256		:- 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)).
3257		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1)],[int(7)],_WF),
3258		X = [(int(1),int(7))])).
3259		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3260		X = [(int(2),int(7)),(int(1),int(7))])).
3261		:- assert_must_succeed((bsets_clp:not_total_function([],[int(1)],[int(7)],_WF))).
3262		:- assert_must_succeed((bsets_clp:not_total_function([],[global_set('NAT1')],[global_set('Name')],_WF))).
3263		:- assert_must_succeed((bsets_clp:not_total_function([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3264		:- assert_must_succeed((bsets_clp:not_total_function([(int(1),int(7)), (int(2),int(1))],
3265		[int(1),int(2)],[int(7)],_WF))).
3266		:- assert_must_succeed((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3267		X = [(int(2),int(7)),(int(2),int(6))])).
3268
3269		:- block not_total_function(-,?,?,?), not_total_function(?,-,?,?).
3270		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
3271		% we do not need the Range; this means we can match more closures (e.g., lambda)
3272		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
3273		not_equal_object_wf(FFDomain,Domain,WF).
3274		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),
3275		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3276		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
3277		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
3278		% this clause is unsound: it prevents finite partial functions as solutions to not_total_function over an infinite domain
3279		%not_total_function(FF,Domain,Range,WF) :-
3280		% custom_explicit_sets:get_card_for_specific_custom_set(Domain,Card),
3281		% Card == inf, !, % cardinality is too large for avl expansion to be viable
3282		% not_partial_function(FF,Domain,Range,WF). % so search for something that is not even a partial function
3283		not_total_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
3284		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
3285		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
3286		get_range_id_expression(P,T,TRangeID),!,
3287		equality_objects_wf(FFDomain,Domain,SubRes,WF), % compare: subset_test for not_partial_function
3288		when(nonvar(SubRes),
3289		(SubRes=pred_false -> true % not equal -> it is not a total function over the domain
3290		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
3291		not_total_function(R,Domain,Range,WF) :-
3292		try_expand_and_convert_to_avl_with_check(R,ER,not_total_function_range),
3293		try_expand_and_convert_to_avl_unless_large_wf(Range,ERange,WF),
3294		not_total_function2(ER,Domain,ERange,WF).
3295
3296		% repeat block, in case Domain or R is a closure
3297		:- block not_total_function2(-,?,?,?), not_total_function2(?,-,?,?).
3298		not_total_function2(R,Domain,Range,WF) :-
3299		expand_and_convert_to_avl_set_warn(R,AER,not_total_function2,'ARG /: ? --> ?',WF),
3300		!,
3301		not_total_function_avl(AER,Domain,Range,WF).
3302		not_total_function2(R,Domain,ERange,WF) :-
3303		expand_custom_set_to_list_wf(R,ER,_,not_total_function2,WF),
3304		try_expand_and_convert_to_avl_with_check(Domain,EDomain,keep_intervals(1000),not_total_function_domain),
3305		not_tf(ER,[],EDomain,ERange,WF).
3306
3307		not_total_function_avl(_AER,Domain,_Range,_WF) :- is_infinite_explicit_set(Domain),!,
3308		true. % a finite AVL set cannot be a total function over an infinite domain
3309		not_total_function_avl(AER,Domain,Range,WF) :-
3310		expand_and_convert_to_avl_set_warn(Domain,ADom,not_total_function2,'? /: ARG --> ?',WF),
3311		!,
3312		(is_avl_total_function_over_domain(AER,ADom)
3313		->
3314		is_not_avl_relation_over_range(AER,Range,WF)
3315		; true
3316		).
3317		not_total_function_avl(AER,EDomain,ERange,WF) :-
3318		expand_custom_set_to_list_wf(avl_set(AER),ER,_,not_total_function_avl,WF),
3319		not_tf(ER,[],EDomain,ERange,WF).
3320
3321
3322		:- use_module(kernel_equality,[membership_test_wf_with_force/4]).
3323
3324		:- block not_tf(-,?,?,?,?).
3325		not_tf([],_,Domain,_,WF) :- not_empty_set_wf(Domain,WF).
3326		not_tf([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf_with_force(SoFar,X,MemRes,WF),
3327		not_tf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3328
3329		:- block not_tf2(-,?,?,?, ?,?,?,?). %, not_tf2(?,?,?,?, -,?,?), not_tf2(?,?,?,?, ?,-,?).
3330		not_tf2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).% :- check_element_of_lazy(X,SoFar,WF).
3331		not_tf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3332		%not_element_of_wf(X,SoFar,WF),
3333		membership_test_wf_with_force(Dom,X,MemRes,WF),
3334		not_tf3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3335
3336		:- block not_tf3(-, ?,?,?,?, ?,?,?).
3337		not_tf3(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF).
3338		not_tf3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3339		remove_element_wf(X,Dom,Dom2,WF),
3340		membership_test_wf_with_force(Ran,Y,MemRes,WF),
3341		not_tf4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3342
3343		:- block not_tf4(-, ?,?,?,?, ?,?,?).
3344		not_tf4(pred_false,_X,_Y,_T,_SoFar,_Dom2,_Ran,_WF).
3345		not_tf4(pred_true,X,_Y,T,SoFar,Dom2,Ran,WF) :-
3346		%check_element_of_wf(Y,Ran,WF), %DO WE NEED THIS ????
3347		add_new_element_wf(X,SoFar,SoFar2,WF),
3348		not_tf(T,SoFar2,Dom2,Ran,WF).
3349
3350
3351
3352		:- 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)).
3353		:- 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)).
3354		:- 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)).
3355		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1)],[int(7)],_WF),
3356		X = [(int(1),int(7))])).
3357		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3358		X = [(int(2),int(7)),(int(1),int(6))])).
3359		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3360		X = [(int(1),int(7)),(int(2),int(6))])).
3361		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(3)],_WF),
3362		X = [(int(1),int(3)),(int(2),int(3))])).
3363		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3364		X = [(int(2),int(7)),(int(1),int(7))])).
3365		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1)],[int(7),int(8)],_WF),
3366		X = [(int(1),int(7))])).
3367		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7)],_WF),
3368		X = [(int(2),int(7))])).
3369		:- assert_must_succeed((bsets_clp:not_total_bijection([],[int(1)],[int(7)],_WF))).
3370		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3371		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(1),int(7)), (int(2),int(1))],
3372		[int(1),int(2)],[int(7)],_WF))).
3373		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3374		X = [(int(2),int(7)),(int(2),int(6))])).
3375
3376		:- block not_total_bijection(-,?,?,?), not_total_bijection(?,-,?,?).
3377		not_total_bijection(FF,Domain,Range,WF) :-
3378		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3379		not_equal_object_wf((FFDomain,FFRange),(Domain,Range),WF).
3380		not_total_bijection(avl_set(_),Domain,_Range,_WF) :-
3381	?	is_infinite_explicit_set(Domain),!.
3382		% a finite set cannot be a total bijection over an infinite domain, see test 1641
3383		not_total_bijection(R,Domain,Range,WF) :-
3384		try_expand_custom_set_wf(R,ER,not_total_bijection,WF),
3385		not_tot_bij(ER,[],Domain,Range,WF).
3386
3387		:- block not_tot_bij(-,?,?,?,?).
3388		not_tot_bij([],_,Domain,Range,WF) :- empty_not_tot_bij(Domain,Range,WF).
3389		not_tot_bij([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf(SoFar,X,MemRes,WF),
3390		not_tot_bij2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3391
3392		:- use_module(kernel_equality,[empty_set_test_wf/3]).
3393		:- block empty_not_tot_bij(-,?,?).
3394		empty_not_tot_bij(Domain,Range,WF) :-
3395		empty_set_test_wf(Domain,EqRes,WF),
3396		empty_not_tot_bij2(EqRes,Range,WF).
3397		:- block empty_not_tot_bij2(-,?,?).
3398		empty_not_tot_bij2(pred_false,_,_).
3399		empty_not_tot_bij2(pred_true,Range,WF) :- not_empty_set_wf(Range,WF).
3400
3401		:- block not_tot_bij2(-,?,?,?,?,?,?,?).
3402		not_tot_bij2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).
3403		not_tot_bij2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3404		membership_test_wf(Dom,X,MemRes,WF),
3405		not_tot_bij3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3406
3407		:- block not_tot_bij3(-,?,?,?,?,?,?,?).
3408		not_tot_bij3(pred_false,_X,_,_T,_SoFar,_Dom,_Ran,_WF). % X not a member of domain
3409		not_tot_bij3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3410		remove_element_wf(X,Dom,Dom2,WF),
3411		membership_test_wf(Ran,Y,MemRes,WF),
3412		not_tot_bij4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3413
3414		:- block not_tot_bij4(-,?,?,?,?,?,?,?).
3415		not_tot_bij4(pred_false,_X,_,_T,_SoFar,_Dom2,_Ran,_WF). % Y not a member of range
3416		not_tot_bij4(pred_true,X,Y,T,SoFar,Dom2,Ran,WF) :-
3417		remove_element_wf(Y,Ran,Ran2,WF),
3418		add_element_wf(X,SoFar,SoFar2,WF),
3419		not_tot_bij(T,SoFar2,Dom2,Ran2,WF).
3420
3421
3422
3423		:- 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)).
3424		:- 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)).
3425		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_restriction_wf([],[int(2),int(3)],[],WF),WF)).
3426		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[int(1)],[],_WF))).
3427		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[],[],_WF))).
3428		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(1)],[],_WF))).
3429		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(2)],[(int(1),int(2))],_WF))).
3430		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3431		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3432		kernel_objects:equal_object(X,R))).
3433		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,Y,R,_WF),
3434		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],Y=global_set('Name'),
3435		kernel_objects:equal_object(X,R))).
3436		:- assert_must_fail((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3437		X = [(int(1),fd(3,'Name')),(int(2),fd(1,'Name'))],
3438		kernel_objects:equal_object(X,R))).
3439
3440		:- block range_restriction_wf(-,?,?,?),range_restriction_wf(?,-,-,?).
3441
3442		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3443		ok_to_try_restriction_explicit_set(S,R,Res),
3444		range_restriction_explicit_set_wf(R,S,SR,WF),!,
3445		equal_object_wf(SR,Res,range_restriction,WF).
3446		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3447		expand_custom_set_to_list_wf(R,ER,_,range_restriction,WF),
3448		relation_restriction_wf(ER,S,Res,pred_true,range,WF).
3449
3450		% heuristic: should we try restriction_explicit_set or
3451		% is relation_restriction with its stronger constraint propagation better
3452		ok_to_try_restriction_explicit_set(S,R,Res) :-
3453		nonvar(S),
3454		(var(Res) -> true
3455		; S=avl_set(_),
3456		nonvar(R), R=avl_set(_) % otherwise constraint propagation from normal relation_restriction better
3457		).
3458
3459		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([],[int(2)],[],WF),WF)).
3460		:- 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)).
3461		:- 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)).
3462		:- 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)).
3463
3464		:- block range_subtraction_wf(-,?,?,?),range_subtraction_wf(?,-,-,?).
3465		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3466		S==[],!,
3467		equal_object_wf(R,Res,range_subtraction1,WF).
3468		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3469		ok_to_try_restriction_explicit_set(S,R,Res),
3470		range_subtraction_explicit_set_wf(R,S,SR,WF),!,
3471		equal_object_wf(SR,Res,range_subtraction2,WF).
3472		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3473		expand_custom_set_to_list_wf(R,ER,_,range_subtraction,WF),
3474		relation_restriction_wf(ER,S,Res,pred_false,range,WF).
3475
3476		:- 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)).
3477		:- 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)).
3478
3479		:- block in_range_restriction_wf(-,-,-,?).
3480		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3481		(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3482		; preference(convert_comprehension_sets_into_closures,true)),
3483		!,
3484		Rel \== [], % avoid setting up check_element_of for X then
3485		% x |-> y : Rel |>> Set <=> x|->y : Rel & y: Set
3486		check_element_of_wf(Pair,Rel,WF),
3487		Pair = (_,P2),
3488		check_element_of_wf(P2,Set,WF).
3489		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3490		range_restriction_wf(Rel,Set,Res,WF),
3491		check_element_of_wf(Pair,Res,WF).
3492
3493		:- 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)).
3494		:- 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)).
3495
3496		:- block not_in_range_restriction_wf(-,-,-,?).
3497		not_in_range_restriction_wf(Pair,Rel,Set,WF) :-
3498		range_restriction_wf(Rel,Set,Res,WF),
3499		not_element_of_wf(Pair,Res,WF).
3500
3501		:- 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)).
3502		:- 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)).
3503
3504		:- block in_range_subtraction_wf(-,-,-,?).
3505		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3506		(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3507		; preference(convert_comprehension_sets_into_closures,true)),
3508		!,
3509		Rel \== [], % avoid setting up check_element_of for X then
3510		% x |-> y : Rel |>> Set <=> x|->y : Rel & y/: Set
3511		check_element_of_wf(Pair,Rel,WF),
3512		Pair = (_,P2),
3513		not_element_of_wf(P2,Set,WF).
3514		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3515		range_subtraction_wf(Rel,Set,Res,WF),
3516		check_element_of_wf(Pair,Res,WF).
3517
3518		:- 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)).
3519		:- 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)).
3520
3521		:- block not_in_range_subtraction_wf(-,-,-,?).
3522		not_in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3523		range_subtraction_wf(Rel,Set,Res,WF),
3524		not_element_of_wf(Pair,Res,WF).
3525
3526
3527
3528		:- 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)).
3529		:- 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)).
3530
3531		:- block in_domain_restriction_wf(-,-,-,?).
3532		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3533		(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3534		; preference(convert_comprehension_sets_into_closures,true)),
3535		!,
3536		Rel \== [], % avoid setting up check_element_of for X then
3537		% x |-> y : Set <| Rel <=> x|->y : Rel & x: Set
3538		check_element_of_wf(Pair,Rel,WF),
3539		Pair = (P1,_),
3540		check_element_of_wf(P1,Set,WF).
3541		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3542		domain_restriction_wf(Set,Rel,Res,WF),
3543		check_element_of_wf(Pair,Res,WF).
3544
3545		:- 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)).
3546		:- 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)).
3547
3548		:- block not_in_domain_restriction_wf(-,-,-,?).
3549		not_in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3550		domain_restriction_wf(Set,Rel,Res,WF),
3551		not_element_of_wf(Pair,Res,WF).
3552
3553		:- 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)).
3554		:- 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)).
3555		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_restriction_wf([int(2),int(3)],[],[],WF),WF)).
3556		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],[],_WF))).
3557		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],R,_WF), R==[])).
3558		:- assert_must_fail((bsets_clp:domain_restriction_wf(_,[],R,_WF), R=[int(_)|_])).
3559		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2)],[(int(1),int(2))],[],_WF))).
3560		:- assert_must_succeed((bsets_clp:domain_restriction_wf([],[(int(1),int(2))],[],_WF))).
3561		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3562		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2)),(int(2),_)],_,_WF))).
3563		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2),int(1)],X,R,_WF),
3564		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3565		kernel_objects:equal_object(X,R))).
3566
3567
3568		:- block domain_restriction_wf(?,-,?,?),domain_restriction_wf(-,?,-,?).
3569		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3570		ok_to_try_restriction_explicit_set(S,R,Res),
3571		domain_restriction_explicit_set_wf(S,R,SR,WF),!,
3572		equal_object_wf(SR,Res,domain_restriction,WF).
3573		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3574		expand_custom_set_to_list_wf(R,ER,_,domain_restriction,WF),
3575		relation_restriction_wf(ER,S,Res,pred_true,domain,WF).
3576
3577		% a predicate to compute domain/range restriction/subtraction
3578		:- block relation_restriction_wf(?,-,- ,?,?,?),
3579		relation_restriction_wf(-,?,? ,?,?,?).
3580		relation_restriction_wf([],_S,Res,_AddWhen,_DomOrRange,WF) :-
3581		empty_set_wf(Res,WF).
3582		relation_restriction_wf([(X,Y)|T],S,Res,AddWhen,DomOrRange,WF) :-
3583		(DomOrRange=domain
3584		-> membership_test_wf(S,X,MemRes,WF) % TO DO: pass WF !
3585		; membership_test_wf(S,Y,MemRes,WF)),
3586		(nonvar(MemRes)
3587		%MemRes==AddWhen % MemRes already set; we will ensure that (X,Y) in Res below; this slows down Alstom Compilation Regle !
3588		% 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 ?
3589		-> true %,(MemRes==AddWhen -> true ; print_term_summary(relation_restriction([(X,Y)|T],S,Res,AddWhen,DomOrRange)),nl)
3590		; (AddWhen=pred_true -> InResult=MemRes
3591		; negate(InResult,MemRes)), % from bool_pred
3592		membership_test_wf(Res,(X,Y),InResult,WF)
3593		% TO DO: same for explicit version; gets called e.g. if S = 1..n (1..n <| [1,2,3] = [1,2])
3594		% can now solve e.g. {x|x <| [1,2,3] = [1,2] & card(x)=2} = {{1,2}}
3595		% or x <| s = [1,2,3] \/ {29|->29} & x <: 1..100 & s = %i.(i:1..50|i)
3596		),
3597		relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF).
3598		:- block relation_restriction_aux(-,?,?,?,?,?, ?,?,?).
3599		relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF) :-
3600		MemRes==AddWhen,!, % (X,Y) should be added to result
3601		% TO DO: collect result until we delay ? and then do equal_object ?
3602		equal_cons(Res,(X,Y),RT), % was : equal_object([(X,Y)|RT],Res),
3603		%equal_cons_wf(Res,(X,Y),RT,WF), % makes tests 982, 1302, 1303 fail; TO DO: investigate
3604		%when(nonvar(RT), % causes problem for test 982
3605		relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3606		relation_restriction_aux(_MemRes,_X,_,T,S,RT,AddWhen,DomOrRange,WF) :-
3607		% the couple is filtered out
3608		relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3609
3610
3611		:- 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)).
3612		:- 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)).
3613		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([int(1)],[],[],WF),WF)).
3614		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([],[(int(11),int(21))],[(int(11),int(21))],WF),WF)).
3615		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2))],[],_WF))).
3616		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(3)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3617		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2)),(int(2),int(X))],R,_WF),
3618		R=[(int(2),int(YY))], YY==X)).
3619		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(5),int(3)],X,R,_WF),
3620		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3621		kernel_objects:equal_object(X,R))).
3622		:- block domain_subtraction_wf(?,-,?,?),domain_subtraction_wf(-,?,-,?).
3623		domain_subtraction_wf(S,R,Res,WF) :- S==[],!,
3624		equal_object_wf(R,Res,domain_subtraction1,WF).
3625		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3626		ok_to_try_restriction_explicit_set(S,R,Res),
3627		domain_subtraction_explicit_set_wf(S,R,SR,WF),!,
3628		equal_object_wf(SR,Res,domain_subtraction2,WF).
3629		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3630		expand_custom_set_to_list_wf(R,ER,_,domain_subtraction,WF),
3631		try_expand_and_convert_to_avl_with_check(S,AS,keep_intervals(500),domain_subtraction),
3632		% (ground(ER) -> domain_subtraction_acc(ER,AS,[],Res) ;
3633		relation_restriction_wf(ER,AS,Res,pred_false,domain,WF)
3634		% )
3635		.
3636
3637
3638
3639		:- 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)).
3640		:- 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)).
3641
3642		:- block in_domain_subtraction_wf(-,-,-,?).
3643
3644		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3645		(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3646		; preference(convert_comprehension_sets_into_closures,true)),
3647		!,
3648		Rel \== [], % avoid setting up check_element_of for X then
3649		% x |-> y : Set <<| Rel <=> x|->y : Rel & x/: Set
3650		check_element_of_wf(Pair,Rel,WF),
3651		Pair = (P1,_),
3652		not_element_of_wf(P1,Set,WF).
3653		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3654		domain_subtraction_wf(Set,Rel,Res,WF),
3655		check_element_of_wf(Pair,Res,WF).
3656
3657
3658		:- 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)).
3659		:- 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)).
3660
3661		:- block not_in_domain_subtraction_wf(-,-,-,?).
3662		not_in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3663		domain_subtraction_wf(Set,Rel,Res,WF),
3664		not_element_of_wf(Pair,Res,WF).
3665
3666		% similar to kernel_objects, but adds case for [_|_]
3667		treat_arg_symbolically(X) :- var(X),!.
3668		treat_arg_symbolically([H|T]) :- \+ ground(H) ; treat_arg_symbolically(T).
3669		treat_arg_symbolically(global_set(_)).
3670		treat_arg_symbolically(freetype(_)).
3671		treat_arg_symbolically(closure(P,T,B)) :- \+ kernel_objects:small_interval(P,T,B).
3672
3673		:- 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)).
3674		:- 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)).
3675		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(1),int(3))],X,_WF),
3676		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(3))]))).
3677		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(3),int(6))],X,_WF),
3678		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(2)),(int(3),int(6))]))).
3679
3680		:- block override_relation(-,-,?,?). % overwrite AST node
3681		override_relation(R,S,Res,WF) :- R==[],!, equal_object_wf(S,Res,override_relation1,WF).
3682		override_relation(R,S,Res,WF) :- S==[],!, equal_object_wf(R,Res,override_relation2,WF).
3683		override_relation(R,S,Res,WF) :- Res==[],!, empty_set_wf(S,WF), empty_set_wf(R,WF).
3684		override_relation(R,S,Res,WF) :- /* R <+ S */
3685		override_custom_explicit_set_wf(R,S,ORes,WF),!,
3686		equal_object_wf(ORes,Res,override_relation3,WF).
3687		override_relation(R,S,Res,WF) :- /* R <+ S */
3688		domain_wf(S,DS,WF),
3689		domain_subtraction_wf(DS,R,DSR,WF),
3690		union_wf(DSR,S,Res,WF). % in principle we could call disjoint_union_wf, but fails 1112, 1751
3691
3692		:- 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)).
3693		:- 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)).
3694		:- 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)).
3695		:- 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)).
3696
3697		:- block in_override_relation_wf(-,-,-,?).
3698		in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3699		check_element_of_wf(Pair,Rel1,WF).
3700		in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3701		check_element_of_wf(Pair,S,WF).
3702		in_override_relation_wf((X,Y),Rel1,S,WF) :-
3703		(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3704		; preference(convert_comprehension_sets_into_closures,true)),
3705		!,
3706		domain_wf(S,DS,WF),
3707		membership_test_wf(DS,X,MemRes,WF),
3708		in_override_aux(MemRes,X,Y,Rel1,S,WF).
3709		in_override_relation_wf(Pair,Rel1,S,WF) :-
3710		override_relation(Rel1,S,Res,WF),
3711		check_element_of_wf(Pair,Res,WF).
3712
3713		:- block in_override_aux(-,?,?,?,?,?).
3714		in_override_aux(pred_true,X,Y,_R,S,WF) :-
3715		check_element_of_wf((X,Y),S,WF).
3716		in_override_aux(pred_false,X,Y,R,_S,WF) :-
3717		check_element_of_wf((X,Y),R,WF).
3718
3719		:- 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)).
3720		:- 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)).
3721
3722		:- block not_in_override_relation_wf(-,-,-,?).
3723		not_in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3724		not_element_of_wf(Pair,Rel1,WF).
3725		not_in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3726		not_element_of_wf(Pair,S,WF).
3727		not_in_override_relation_wf((X,Y),Rel1,S,WF) :-
3728		(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3729		; preference(convert_comprehension_sets_into_closures,true)),
3730		!,
3731		domain_wf(S,DS,WF),
3732		membership_test_wf(DS,X,MemRes,WF),
3733		not_in_override_aux(MemRes,X,Y,Rel1,S,WF).
3734		not_in_override_relation_wf(Pair,Rel1,S,WF) :-
3735		override_relation(Rel1,S,Res,WF),
3736		not_element_of_wf(Pair,Res,WF).
3737
3738		:- block not_in_override_aux(-,?,?,?,?,?).
3739		not_in_override_aux(pred_true,X,Y,_R,S,WF) :-
3740		not_element_of_wf((X,Y),S,WF).
3741		not_in_override_aux(pred_false,X,Y,R,_S,WF) :-
3742		not_element_of_wf((X,Y),R,WF).
3743
3744		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override([],int(1),int(3),[(int(1),int(3))],WF),WF)).
3745		:- 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)).
3746		:- 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)).
3747
3748		% override for a single pair
3749		:- block override(-,?,?,?,?), override(?,-,?,?,?),
3750		override(?,?,-,?,?). % also wait on Y; try to generate avl if possible; can only be used in substitution anyway
3751		/* R <+ {X |-> Y} as used by substitution R(X) := Y */
3752		override(R,X,Y,Res,WF) :-
3753		override_pair_explicit_set(R,X,Y,ORes),!,
3754		equal_object_wf(ORes,Res,override1,WF).
3755		override(R,X,Y,Res,WF) :-
3756		if(try_expand_custom_set_to_list(R,ER,_,override),
3757		(
3758		override2(ER,X,Y,[(X,Y)],ORes,WF),
3759		equal_object_wf(ORes,Res,override2,WF)),
3760		( %print_term_summary(exception(R)), % Virtual Timeout exception occured
3761		override_relation(R,[(X,Y)],Res,WF)
3762		)).
3763
3764		:- block override2(-,?,?,?,?,?).
3765		override2([],_X,_Y,Remainder,Res,WF) :- equal_object_optimized_wf(Remainder,Res,override2,WF). %equal_object(Remainder,Res).
3766		override2([(V,W)|T],X,Y,Remainder,Res,WF) :-
3767		equality_objects_wf(V,X,EqRes,WF),
3768		override2c(EqRes,V,W,T,X,Y,Remainder,Res,WF).
3769
3770		:- block override2c(-, ?,?,?, ?,?,?,?,?).
3771		override2c(pred_true,_V,_W,T,X,Y,_Remainder,Res,WF) :-
3772		equal_cons_wf(Res,(X,Y),T2,WF),
3773		override2(T,X,Y,[],T2,WF). /* set remainder to [], we have already added (X,Y) */
3774		override2c(pred_false,V,W,T,X,Y,Remainder,Res,WF) :-
3775		equal_cons_wf(Res,(V,W),T2,WF),
3776		override2(T,X,Y,Remainder,T2,WF).
3777
3778
3779
3780		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],WF),WF)).
3781		:- 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)).
3782		:- 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)).
3783		:- 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)).
3784		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],_WF)).
3785		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(2)],[],_WF)).
3786		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(3)],[],_WF)).
3787		:- assert_must_succeed((bsets_clp:image_wf([(int(1),int(2)),(int(1),int(3))],
3788		[int(X)],R,_WF), X=1, kernel_objects:equal_object(R,[int(2),int(3)]))).
3789		:- assert_must_succeed((bsets_clp:image_wf([([int(1),int(2)],int(6)),
3790		([int(1),int(2),int(3)],int(7)),
3791		([int(2),int(1)],int(8))],
3792		[[int(X),int(1)]],R,_WF), X=2,
3793		kernel_objects:equal_object(R,[int(6),int(8)]))).
3794		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2))],[int(1),int(2)],[int(2)],_WF)).
3795		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(1)],_WF)).
3796		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[],_WF)).
3797
3798
3799		:- block image_wf(-,?,?,?).
3800		image_wf(Rel,_,Res,WF) :- Rel==[],!,empty_set_wf(Res,WF).
3801		image_wf(Rel,S,Res,WF) :-
3802		image_for_id_closure(Rel,S,Img),!, % we don't require S to be known here
3803		equal_object_wf(Img,Res,image_wf_id_closure,WF).
3804		image_wf(Rel,S,Res,WF) :-
3805		image_wf0(Rel,S,Res,WF).
3806
3807		:- block image_wf0(?,-,?,?).
3808		image_wf0(Rel,S,Res,WF) :- /* Res = Rel[S] */
3809		(S==[] -> empty_set_wf(Res,WF)
3810		; opt_push_wait_flag_call_stack_info(WF,b_operator_call(image,[Rel,S],unknown),WF2),
3811		image1(Rel,S,Res,WF2) ).
3812
3813		keep_symbolic(R) :- var(R),!,fail.
3814		keep_symbolic(closure(_,_,_)) :- preferences:get_preference(convert_comprehension_sets_into_closures,true),!.
3815	?	keep_symbolic(R) :- dont_expand_this_explicit_set(R).
3816
3817		:- block image1(-,?,?,?).
3818		image1(Rel,S,Res,WF) :-
3819		image_for_explicit_set(Rel,S,Img,WF),!,
3820		equal_object_wf(Img,Res,image1_1,WF),
3821		quick_propagate_subset_range(Res,Rel,WF).
3822		%image1(Rel,S,Res,WF) :- expand_custom_set_to_list(S,ES),!, image_of_set(ES,Rel,Res,WF).
3823		image1(Rel,Set,Res,WF) :-
3824	?	keep_symbolic(Rel),
3825		(preferences:get_preference(convert_comprehension_sets_into_closures,true), % in this case keep_symbolic is always true
3826		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
3827		-> debug_println(9,infinite_for_image1(Set)),
3828		fail
3829		; true),
3830		( dom_for_specific_closure(Rel,Domain,function(_),WF)
3831		-> !, expand_custom_set_to_list_wf(Set,ESet,_,image1,WF), % TO DO: what if keep_symbolic(Set)
3832		image_for_inf_fun(ESet,Domain,Rel,[],Res,WF)
3833		; get_relation_types(Rel,DomType,RangeType),!,
3834		expand_custom_set_to_list_wf(Set,ESet,_,image1_2,WF),
3835		kernel_tools:ground_value_check(Rel,GRel),
3836		when(nonvar(GRel), image_for_large_relation(ESet,Rel,DomType,RangeType,[],Res,WF))
3837		/* ; get_relation_types(Rel,DomType,RangeType),!,
3838		% Rel should not be expanded, compute closure by calculating {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3839		print_term_summary(image_for_large_relation(DomType,RangeType,Set,Rel)),nl,
3840		image_closure(Set,Rel,DomType,RangeType,Closure ),
3841		translate:print_bvalue(Closure),nl,
3842		expand_custom_set(Closure,CRes), print(expanded),nl,
3843		equal_object(CRes,Res,image1_2) */
3844		).
3845		image1(Rel,S,Res,WF) :-
3846		expand_custom_set_to_list_wf(Rel,Relation,_,image1_2,WF), % bad if Rel is a big closure !
3847		% image_for_list_relation(Relation,S,Res).
3848		propagate_singleton_image(Relation,S,Res,WF),
3849		% TO DO: we could propagate cardinality constraints about Relation,S and Res
3850		% we could also try to infer all_different constraints in case card(S)=card(Res) and f is a function
3851		image_for_list_relation(Relation,S,[],Res,WF). %,print_term_summary(image2_res(Relation,S,Res)).
3852
3853		% propagate that f[{x}] = {r1,...,rk} => x|->ri : f (or {x}*{r1,...,rk} <: f); see test 1532
3854		propagate_singleton_image(R,S,Res,_) :-
3855		(var(S) ; var(Res) ; nonvar(R), is_custom_explicit_set(R,psi)), !.
3856		propagate_singleton_image(Relation,S,avl_set(Res),WF) :-
3857		custom_explicit_sets:singleton_set(S,El), % we have the image by a singleton set {El}
3858		expand_custom_set_to_list_wf(avl_set(Res),LR,_,prop_singleton,WF),
3859		!,
3860		l_check_element_of(LR, El, Relation, WF). % propagate x|->ri : f (will force membership)
3861		propagate_singleton_image(_,_,_,_).
3862
3863		l_check_element_of([],_,_,_).
3864		l_check_element_of([H|T],El,Relation,WF) :-
3865		check_element_of_wf((El,H),Relation,WF),
3866		l_check_element_of(T,El,Relation,WF).
3867
3868		% quick_propagate_in_range(Set, Relation,WF) : propagate that Set <: ran(Relation)
3869		:- block quick_propagate_subset_range(-,?,?).
3870		quick_propagate_subset_range(avl_set(_),_,_) :- !.
3871		quick_propagate_subset_range([],_,_) :- !.
3872		quick_propagate_subset_range([H|T],Relation,WF) :- is_custom_explicit_set(Relation,range_wf1),
3873		range_of_explicit_set_wf(Relation,Range,WF), !,
3874		quick_propagation_element_information(Range,H,WF,NewRange),
3875		quick_propagate_subset_range2(T,NewRange,WF).
3876		quick_propagate_subset_range(_,_,_).
3877
3878		:- block quick_propagate_subset_range2(-,?,?).
3879		quick_propagate_subset_range2([H|T],NewRange,WF) :- !,
3880		quick_propagation_element_information(NewRange,H,WF,NewRange1),
3881		quick_propagate_subset_range2(T,NewRange1,WF).
3882		quick_propagate_subset_range2(_,_,_).
3883
3884		:- use_module(btypechecker, [unify_types_strict/2]).
3885		get_relation_types(Value,Domain,Range) :-
3886		kernel_objects:infer_value_type(Value,VT),
3887		unify_types_strict(VT,set(couple(Domain,Range))). % deal also with seq types
3888		% VT=set(couple(Domain,Range)).
3889
3890		:- block image_for_large_relation(-,?,?,?,?,?,?), image_for_large_relation(?,?,?,?,-,?,?).
3891		image_for_large_relation([],_,_,_,Acc,Res,WF) :- equal_object_wf(Acc,Res,WF).
3892		image_for_large_relation([XX|T],Rel,DomType,RangeType,Acc,Res,WF) :-
3893		Body = b(member(b(couple(b(value(XX),DomType,[]),
3894		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3895		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]),
3896		% TO DO: simplify above if we have Rel = closure(P,T,B); which we usually will
3897		custom_explicit_sets:expand_normal_closure_direct([yy],[RangeType],Body,HRes,_Done,WF), % do not memoize this (many different values)
3898		union_wf(Acc,HRes,NewAcc,WF),
3899		(T == [] -> equal_object_wf(NewAcc,Res,WF)
3900		; image_for_large_relation(T,Rel,DomType,RangeType,NewAcc,Res,WF)).
3901
3902		/* no longer used
3903		% construct a closure for {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3904		image_closure(Set,Rel,DomType,RangeType,Closure ) :- custom_explicit_sets:singleton_set(Set,XX),!,
3905		% do not set up existential quantifier if Set is singleton set
3906		Closure = closure([yy],[RangeType],Body),
3907		Body = b(member(b(couple(b(value(XX),DomType,[]),
3908		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3909		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]).
3910		image_closure(Set,Rel,DomType,RangeType,Closure ) :-
3911		Closure = closure([yy],[RangeType],Body),
3912		couple_member_pred(xx,DomType,yy,RangeType,Rel, Predxxyy),
3913		Body = b(exists([b(identifier(xx),DomType,[])],
3914		b(conjunct(
3915		b(member(b(identifier(xx),DomType,[]),b(value(Set),set(DomType),[])),pred,[]), % TO DO : force evaluation !
3916		Predxxyy),
3917		pred,[])),pred,[used_ids([yy])]).
3918		*/
3919
3920		% very similar to rel_compose_with_inf_fun, indeed f[S] = ran((id(S);f))
3921		:- block image_for_inf_fun(-,?,?,?,?,?).
3922		image_for_inf_fun([],_Dom,_Rel2,Acc,Comp,WF) :- equal_object_wf(Acc,Comp,WF).
3923		image_for_inf_fun([X|T],Dom,Fun,Acc,CompRes,WF) :-
3924		membership_test_wf(Dom,X,MemRes,WF),
3925		image_for_inf_fun_aux(MemRes,X,T,Dom,Fun,Acc,CompRes,WF).
3926
3927		:- block image_for_inf_fun_aux(-,?,?, ?,?,?,?,?).
3928		image_for_inf_fun_aux(pred_true,X,T,Dom,Fun,Acc,CompRes,WF) :-
3929		apply_to(Fun,X,FX,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
3930		add_element_wf(FX,Acc,NewAcc,WF), % will block until Acc Known !!
3931		% TO DO USE: equal_cons_wf(CompRes,FX,CT,WF) + accumulator !,
3932		image_for_inf_fun(T,Dom,Fun,NewAcc,CompRes,WF).
3933		image_for_inf_fun_aux(pred_false,_X,T,Dom,Fun,Acc,Comp,WF) :-
3934		image_for_inf_fun(T,Dom,Fun,Acc,Comp,WF).
3935
3936
3937		/*
3938		:- block image_of_set(-,?,?,?,?), image_of_set(?,?,-,?,?).
3939		image_of_set([],Rel,ImageSoFar,Res,WF) :- equal_object(ImageSoFar,Res).
3940		image_of_set([H|T],Rel,ImageSoFar,Res,WF) :-
3941		image_of_element(Rel,H,ImageSoFar,SF2,WF),
3942		image_of_set(T,Rel,SF2,Res,WF).
3943
3944		image_of_element([],_,Acc,Res,WF) :- equal_object(Acc,Res).
3945		image_of_element([(A,B)|T],H,Acc,Res,WF) :- equality....
3946		image_of_element(avl_set(),H,Acc,Res,WF) :- ....
3947		image_of_element(closure(),....
3948		*/
3949
3950		% Computing the image of a relation which is stored as a list: traverse the relation
3951		:- block image_for_list_relation(-,?,?,?,?).
3952	?	image_for_list_relation([],_,_,Res,WF) :- empty_set_wf(Res,WF).
3953		image_for_list_relation([(X,Y)|T],S,ImageSoFar,Res,WF) :-
3954		((T==[], definitely_not_empty(Res))
3955		-> MemRes=pred_true, % we need at least one more element for Res
3956		check_element_of_wf(X,S,WF)
3957		; (Res==[],ImageSoFar==[]) -> MemRes=pred_false, not_element_of_wf(X,S,WF) % Result empty: X cannot be in S
3958		; membership_test_wf(S,X,MemRes,WF)
3959		),
3960	?	image4(MemRes,Y,T,S,ImageSoFar,Res,WF).
3961
3962		definitely_not_empty(Set) :- nonvar(Set), Set \== [], \+ functor(Set,closure,3). % Set \= closure(_,_,_).
3963
3964		:- block image4(-, ?,?,?, ?,?,?).
3965		image4(pred_true, Y,T,S, ImageSoFar,Res,WF) :-
3966		(Res==[]
3967		-> MemRes=pred_true, check_element_of_wf(Y,ImageSoFar,WF)
3968		; membership_test_wf(ImageSoFar,Y,MemRes,WF)
3969		),
3970	?	image5(MemRes,Y,T,S,ImageSoFar,Res,WF).
3971		image4(pred_false, _Y,T,S, ImageSoFar,Res,WF) :-
3972	?	image_for_list_relation(T,S,ImageSoFar,Res,WF).
3973
3974		:- block image5(-, ?,?,? ,?,?,?).
3975		image5(pred_true,_Y,T,S,ImageSoFar,Res,WF) :- /* we have already added Y to the image */
3976	?	image_for_list_relation(T,S,ImageSoFar,Res,WF).
3977		image5(pred_false,Y,T,S,ImageSoFar,Res,WF) :-
3978		add_element_wf(Y,ImageSoFar,ImageSoFar2,WF),
3979		kernel_objects:mark_as_non_free(Y,image), % Y has been added to image, no longer freely choosable
3980		equal_cons_wf(Res,Y,Res2,WF),
3981	?	image_for_list_relation(T,S,ImageSoFar2,Res2,WF).
3982
3983
3984
3985		:- 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)).
3986		:- 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)).
3987		:- 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)).
3988		% version for computing closure1(Rel)[S]
3989		:- block image_for_closure1_wf(-,?,?,?),image_for_closure1_wf(?,-,?,?).
3990		image_for_closure1_wf(Rel,S,Res,WF) :- (Rel==[] ; S==[]),!,empty_set_wf(Res,WF).
3991		image_for_closure1_wf(Rel,Set,Res,WF) :-
3992		try_expand_and_convert_to_avl_unless_large_wf(Set,ESet,WF),
3993	?	image_for_closure1_wf_aux(Rel,ESet,Res,WF).
3994
3995		:- use_module(library(avl),[avl_height/2]).
3996		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
3997		((nonvar(S),S=avl_set(_))
3998		-> closure1_for_explicit_set_from(Rel,S,Closure1Rel),!,
3999		% if S is known: start from S (currently only deals with Rel=avl_set(_)
4000		range_wf(Closure1Rel,Res,WF)
4001		; Rel=avl_set(AR), avl_height(AR,AR_Height),
4002		((set_smaller_than(S,4),AR_Height>4)
4003		-> !, % TO DO: we could do the same for small S if Rel is large
4004		when(ground(S), (expand_and_convert_to_avl_set(S,ES,image_for_closure1_wf_aux,'closure1(ARG)[?]') ->
4005		closure1_for_explicit_set_from(Rel,avl_set(ES),Closure1Rel),
4006		range_wf(Closure1Rel,Res,WF)
4007		; image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S))
4008		))
4009		; % Don't do this if avl_height too large; then it is probably better to compute the image for S only
4010		AR_Height < 13, % how big should we make this magic constant; or should we time-out ? 2^14=16384
4011		closure1_for_explicit_set(Rel,Closure1Rel),!, % we can compute it effiently; don't use code below
4012		image_wf(Closure1Rel,S,Res,WF)
4013		)
4014		).
4015		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
4016		propagate_result_in_range(Rel,S,Res,WF),
4017	?	image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S)).
4018
4019		% no need to treat avl_sets; already covered as special case above
4020		set_smaller_than([],_).
4021		set_smaller_than([_|T],N) :- N>1, nonvar(T), N1 is N-1, set_smaller_than(T,N1).
4022
4023		image_for_closure1_iterate(Rel,S,Acc,Res,WF,FIRST) :-
4024		image_wf0(Rel,S,Res1,WF),
4025		ground_value_check(Res1,RV),
4026	?	image_for_closure1_check_fix(RV,Rel,Acc,Res1,Res,WF,FIRST).
4027
4028		:- block image_for_closure1_check_fix(-,?,?,?,?,?,?).
4029		image_for_closure1_check_fix(_,Rel,Acc,Res1,Res,WF,FIRST) :-
4030		%try_expand_and_convert_to_avl_unless_large_wf(Res1,ERes1,WF),
4031		difference_set(Res1,Acc,New),
4032		try_expand_and_convert_to_avl(New,ENew), % we compute difference_set below; we most definitely will need an explicit finite representation
4033		(not_empty_set_wf(ENew,WF),
4034		union(ENew,Acc,Acc1), % Note: we do not call union_wf - should we do this
4035		% upon first iteration remove also S from New -> New2 and pass New2 to image_for_closure1_iterate
4036		% TO DO: investigate whether this also makes sense for further iterations; always remove S
4037		(FIRST=first_iteration(S) -> difference_set(ENew,S,New2) ; New2=ENew),
4038	?	image_for_closure1_iterate(Rel,New2,Acc1,Res,WF,not_first)
4039		;
4040		empty_set_wf(ENew,WF),equal_object_optimized_wf(Acc,Res,image_for_closure1_check_fix,WF)).
4041
4042		% propagate information that if closure1(Rel)[.] = Res => Res <: range(Rel)
4043		% x: 1..n --> 1..n & closure1(x)[{1}] = {} & n=100
4044		:- block propagate_result_in_range(?,?,-,?).
4045		propagate_result_in_range(Rel,_S,_Res,_WF) :-
4046		ground_value(Rel),!. % no propagation required
4047		propagate_result_in_range(Rel,S,[],WF) :- !,
4048		domain_wf(Rel,Domain,WF),
4049		not_subset_of_wf(S,Domain,WF).
4050		propagate_result_in_range(Rel,_,Res,WF) :-
4051		range_wf(Rel,Range,WF),
4052		check_subset_of_wf(Res,Range,WF).
4053
4054		:- use_module(probsrc(avl_tools),[avl_height_less_than/2]).
4055
4056		% version for computing iterate(K,Rel)[S]
4057		% iteration
4058		:- block image_for_iterate_wf(?,-,?,?,?,?), image_for_iterate_wf(?,?,-,?,?,?).
4059		image_for_iterate_wf(_Rel,_K,S,Res,_,WF) :- S==[],!,empty_set_wf(Res,WF).
4060		image_for_iterate_wf(Rel,int(K),S,Res,Type,WF) :-
4061		image_for_iterate_k(K,Rel,S,Res,Type,WF).
4062
4063		:- block image_for_iterate_k(-,?,?,?,?,?).
4064		image_for_iterate_k(K,Rel,S,Res,Type,WF) :-
4065		nonvar(Rel),
4066		Rel=avl_set(AVL),
4067		(var(S) -> avl_height_less_than(AVL,11) ; avl_height_less_than(AVL,3)),
4068		!, % compute the iteration once; possibly better constraint propagation and performance if S enumerated
4069		% e.g. x:{1,10,20} & iterate({1|->10,20|->1,10|->20},2)(x) = 20
4070		rel_iterate_wf(Rel,int(K),RelIterated,Type,WF),
4071		image_wf(RelIterated,S,Res,WF).
4072		image_for_iterate_k(K,Rel,S,Res,_,WF) :-
4073		image_for_iterate_k_loop(K,Rel,S,Res,WF).
4074
4075		:- block image_for_iterate_k_loop(?,?,-,?,?).
4076		image_for_iterate_k_loop(0,_Rel,Acc,Result,WF) :- !,
4077		equal_object_optimized_wf(Acc,Result,image_for_iterate_k,WF).
4078		image_for_iterate_k_loop(K,Rel,Acc,Result,WF) :-
4079		image_wf0(Rel,Acc,Acc1,WF), % we could try and detect fix point if K> some limit or time for iteration is measurable
4080		if((K>10, K mod 10 =:= 0, % check for fixpoint every 10 iterations
4081		nonvar(Acc1), Acc1=avl_set(_), quick_custom_explicit_set_approximate_size(Acc1,Size1),
4082		quick_custom_explicit_set_approximate_size(Acc,Size0),
4083		Size0=Size1, % only check for equality if approximate sizes match
4084		equal_explicit_sets_wf(Acc,Acc1,WF)),
4085		K1=0, % fixpoint found, no need to continue iterating
4086		K1 is K-1),
4087		image_for_iterate_k_loop(K1,Rel,Acc1,Result,WF).
4088
4089		special_operator_for_image(b(Rel,Type,_),Kind,Args) :- special_image_aux(Rel,Type,Kind,Args).
4090		special_image_aux(closure(Rel),_,closure,[Rel]). % we have closure1(Rel)[Set] -> avoid computing full closure
4091		special_image_aux(iteration(Rel,K),Type,iteration(Type),[Rel,K]).
4092		% TODO: reflexive closure, id_closure (this will probably be more natural as special case for a value)
4093
4094	?	image_for_special_operator(closure,[Rel],S,Res,WF) :- image_for_closure1_wf(Rel,S,Res,WF).
4095		image_for_special_operator(iteration(Type),[Rel,K],S,Res,WF) :-
4096		image_for_iterate_wf(Rel,K,S,Res,Type,WF).
4097
4098		:- use_module(kernel_objects,[singleton_set_element/4]).
4099		apply_fun_for_special_operator(Kind,EArgs,FunArg,Res,WF,Span) :-
4100		InitialSet = [FunArg], % TODO: try convert to AVL, note: closure1 not really useful in fun. application context
4101		image_for_special_operator(Kind,EArgs,InitialSet,SetRes,WF),
4102		singleton_set_element(SetRes,Res,Span,WF).
4103
4104		% iterate(%x.(x:NATURAL|x+2),2000)(20) much faster this way, 15 ms vs 4 seconds
4105		% iterate(%x.(x:NATURAL|x+2),2000)[{20}]: ditto
4106
4107
4108		% -----------------------------------
4109
4110		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:apply_to([(int(2),int(22))],int(2),int(22),WF),WF)).
4111		:- 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)
4112		:- 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)
4113		:- assert_must_succeed(bsets_clp:apply_to([(int(1),int(2))],int(1),int(2),_WF)).
4114		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),int(2),_WF),F=[(int(3),int(2)),(int(2),int(1))])).
4115		:- assert_must_succeed((bsets_clp:apply_to(F,X,int(1),_WF),F=[(int(3),int(2)),(int(2),int(1))],X=int(2))).
4116		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),_,_WF),F=[(int(3),[int(2),int(3)]),(int(2),[])])).
4117
4118		:- assert_must_fail(bsets_clp:apply_to([(int(1),int(2)),(int(1),int(3))],int(1),int(3),_WF)).
4119		/* input not a function */
4120		apply_to(R,X,Y,WF) :- apply_to(R,X,Y,unknown,unknown,WF).
4121		apply_to(R,X,Y,Span,WF) :- apply_to(R,X,Y,unknown,Span,WF).
4122
4123		% comment in to perform profiling at function call level; can lead to big slowdowns
4124		%:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4125		%:- use_module(source_profiler,[opt_add_source_location_hits/2]).
4126		%apply_to(_R,_X,_Y,_FunctionType,Span,_WF) :- opt_add_source_location_hits(Span,1),fail.
4127
4128		:- block apply_to(-,-,-,?,?,?).
4129		apply_to(R,X,Y,_FunctionType,Span,WF) :-
4130		% we could check if WD condition discharged in Span
4131		(\+ preferences:preference(find_abort_values,false) ; preference(data_validation_mode,true)),
4132		!,
4133		apply_to_var_block_abort(R,X,Y,R,Span,WF). % we have to know R before we can do anything
4134		apply_to(R,X,Y,FunctionType,Span,WF) :-
4135		(var(R),var(X) -> force_in_domain_wf(X,R,WF) ; true),
4136		apply_to1(R,X,Y,R,FunctionType,Span,WF).
4137
4138		:- use_module(kernel_non_empty_attr,[mark_var_set_as_non_empty/1]).
4139		force_in_domain_wf(El,S,WF) :-
4140		(preferences:preference(use_smt_mode,true) -> S=[(El,_)|_]
4141		; % TO DO: non-empty flag
4142		mark_var_set_as_non_empty(S),
4143		get_enumeration_starting_wait_flag(not_empty_domain_wf,WF,LWF), in_domain_lwf(El,S,LWF,WF)).
4144
4145		:- use_module(preferences,[preference/2]).
4146		:- use_module(clpfd_tables,[can_translate_function_to_element_constraint/2,check_apply_with_element_constraint/5]).
4147		:- block apply_to1(-,-,?,?,?,?,?).
4148		apply_to1(R,X,Y,InitialRel,FunctionType,Span,WF) :-
4149		(var(R) -> apply_to_var(R,X,Y,InitialRel,Span,WF)
4150		; R\=[], can_translate_function_to_element_constraint(R,FunctionType) ->
4151		check_apply_with_element_constraint(R,X,Y,FunctionType,WF)
4152		; apply_to_nonvar(R,X,Y,InitialRel,Span,WF),
4153		propagate_range_membership(R,Y)
4154		).
4155		:- block apply_to2(-,-,?,?,?,?).
4156		apply_to2(R,X,Y,InitialRel,Span,WF) :-
4157		(var(R)
4158		-> apply_to_var(R,X,Y,InitialRel,Span,WF)
4159		; apply_to_nonvar(R,X,Y,InitialRel,Span,WF)
4160		).
4161
4162		:- use_module(clpfd_lists,[get_finite_fdset_information/2,combine_fdset_information/3,
4163		assert_fdset_information/2,get_fdset_information/2]).
4164		% tested in test 1478; initially slows down NQueens
4165		%:- block propagate_range_membership(-,?). % not necessary
4166		propagate_range_membership([(_,RanEl)|T],X) :- nonvar(RanEl),
4167		preferences:preference(use_clpfd_solver,true),
4168		preferences:preference(find_abort_values,false),
4169		get_finite_fdset_information(RanEl,Info), % TO DO: try and detect if we can apply element/3 from clpfd
4170		\+ ground(X),
4171		get_fdset_information(X,InfoX),
4172		Info \= InfoX, % avoids NQueens slowdown; TO DO: check if more precise than InfoX; otherwise no use in collecting info
4173		!,
4174		propagate_range_membership(T,Info,X).
4175		propagate_range_membership(_,_).
4176		:- block propagate_range_membership(-,?,?).
4177		propagate_range_membership([],Info,El) :- !,
4178		% 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 ??
4179		assert_fdset_information(Info,El).
4180		propagate_range_membership([(_,RanEl)|T],Acc,X) :-
4181		nonvar(RanEl), % otherwise we have no info: we may just as well stop
4182		get_finite_fdset_information(RanEl,RInfo),
4183		combine_fdset_information(Acc,RInfo,NewAcc),
4184		NewAcc \= no_fdset_info,
4185		!,
4186		propagate_range_membership(T,NewAcc,X).
4187		propagate_range_membership(_,_,_).
4188
4189
4190		apply_to_var(R,X,Y,InitialRel,Span,WF) :-
4191		mark_var_set_as_non_empty(R),
4192		get_wait_flag(1.0,apply_to_var,WF,WF1), % see tests 1393, 1562??
4193		% was: get_wait_flag0(WF,WF1), but see test 1706 (in conjunction for improvement for test 2033)
4194		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 ??)
4195		(var(R) ->
4196		R=[(X,Y)|Tail],
4197		optional_functionality_check(Tail,X,WF)
4198		; apply_to_nonvar(R,X,Y,InitialRel,Span,WF))).
4199
4200		:- block apply_to_var_block_abort(-,?,?,?,?,?).
4201		apply_to_var_block_abort(R,X,Y,InitialRel,Span,WF) :-
4202		apply_to_nonvar(R,X,Y,InitialRel,Span,WF).
4203
4204		optional_functionality_check(Tail,X,WF) :-
4205		preferences:preference(disprover_mode,true),!,
4206		not_in_domain_wf(X,Tail,WF). % we assert that R is a function ; when disproving we can assume well-definedness
4207		% 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)
4208		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
4209
4210
4211		:- use_module(closures,[is_recursive_closure/3]).
4212		:- use_module(memoization,[is_memoization_closure/4,apply_to_memoize/8]).
4213		:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4214		:- if(\+ environ(no_wd_checking,true)).
4215		apply_to_nonvar([],X,_Y,InitialRel,Span,WF) :-
4216		\+ preferences:preference(find_abort_values,false),
4217		when(ground(X),add_wd_error_span('function applied outside of domain (#2): ', '@fun'(X,InitialRel),Span,WF)).
4218		:- endif.
4219		apply_to_nonvar([(X2,Y2)|T],X,Y,InitialRel,Span,WF) :-
4220		equality_objects_wf(X2,X,EqRes,WF),
4221		% this check on Y2 below is important if both Y and Y2 are instantiated but X,X2 not yet
4222		% example: aload_R07_cbc.mch (Savary) or cbc_sequence check for R08_ByteArray for aload_R07 event (test 1349)
4223		% however: slows down test 583 !
4224		(var(EqRes) -> equality_objects_wf(Y2,Y,EqResY,WF),
4225		prop_apply_eqxy(EqResY,EqRes) % propagate: if Y/=Y2 => X/=X2
4226		; EqResY=not_called),
4227		apply_to4(EqRes,EqResY,Y2,T,X,Y,InitialRel,Span,WF).
4228		apply_to_nonvar(avl_set(A),X,Y,_InitialRel,Span,WF) :-
4229		apply_to_avl_set(A,X,Y,Span,WF).
4230		apply_to_nonvar(closure(P,T,B),X,Y,_InitialRel,Span,WF) :-
4231		%is_custom_explicit_set(Closure,apply), % should also work for avl_set,...
4232	?	(is_memoization_closure(P,T,B,MemoID)
4233		% Function application with memoization; currently enabled by add /*@desc memo */ pragma to abstract constant
4234		-> apply_to_memoize(MemoID,P,T,B,X,Y,Span,WF)
4235		; 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 ??)
4236		-> % print_term_summary(apply_recursive_closure(X,P,T,B)),
4237		%hit_profiler:add_profile_hit(rec_apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4238		ground_value_check(X,XV), block_apply_closure_to_nonvar_groundx(XV,X,Y,P,T,B,Span,WF)
4239		; %hit_profiler:add_profile_hit(apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4240		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)).
4241
4242
4243		:- block block_apply_closure_to_nonvar_groundx(-,?,?, ?,?,?, ?,?).
4244		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).
4245
4246		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4247		kernel_tools:ground_bexpr(B),
4248		!, % then if the element of function succeeds there is no need to check WD
4249		if(check_element_of_function_closure(X,Y,P,T,B,WF),
4250		true, % No need to check for well-definedness; no pending choice points
4251		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
4252		).
4253		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4254		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF).
4255
4256		% if we first check preferences:preference(find_abort_values,false) to avoid a choice
4257		% point, we get a big slow-down on Alstom models; e.g., vesg_Mar12
4258		% WARNING: This choice point can be set up in WF0 !
4259		apply_closure_to_nonvar(X,Y,P,T,B,_,WF) :-
4260		(preferences:preference(find_abort_values,true) -> true ; !), % slow down ???!
4261		check_element_of_function_closure(X,Y,P,T,B,WF) .
4262		apply_closure_to_nonvar(X,_,P,T,B,Span,WF) :- % removing this clause doubles runtime of COMPUTE_GRADIENT_CHANGE
4263		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF).
4264
4265		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF) :-
4266		\+ preferences:preference(find_abort_values,false),
4267		not_in_domain_wf(X,closure(P,T,B),WF),
4268		when((ground(X),ground(closure(P,T,B))),
4269		add_wd_error_span('function applied outside of domain (#3): ', '@fun'(X,closure(P,T,B)),Span,WF)).
4270
4271
4272		% propagate equality_objects between range and domain elements for function application:
4273		:- block prop_apply_eqxy(-,-).
4274		prop_apply_eqxy(Eqy,Eqx) :- var(Eqy),!, (Eqx = pred_true -> Eqy = pred_true ; true).
4275		prop_apply_eqxy(pred_false,pred_false).
4276		prop_apply_eqxy(pred_true,_).
4277
4278		:- block apply_to4(-,?,?, -,?,?,?,?,?).
4279		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4280		var(EqResX),!, % Tail bound
4281		(Tail == []
4282		-> (preferences:preference(find_abort_values,false)
4283		-> EqResX = pred_true,
4284		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4285		; apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4286		)
4287		; 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,...)
4288		; Tail = closure(_,_,_) -> apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4289		; Tail \= [_|_] -> add_internal_error('Illegal Tail: ',apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)),fail
4290		; 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
4291		apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3)
4292		).
4293		apply_to4(pred_true,EqResY,Y2, Tail,X,Y,_InitialRel,_,WF) :-
4294		(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to4,WF) ; EqResY = pred_true),
4295		optional_functionality_check(Tail,X,WF).
4296		apply_to4(pred_false,_EqResY,_Y2,T,X,Y,InitialRel,Span,WF) :- apply_to2(T,X,Y,InitialRel,Span,WF).
4297
4298		% we delay setting up equality_objects until X3 is at least partially known, see test 1715 Alstom_essai2_boucle1
4299		% TO DO: we could check if X3==X above
4300		:- block apply_to4_call5(-,?,?, ?,?,?,?,?,?, -,?,?).
4301		apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, _X3,_Y3,_T3) :- nonvar(EqResX),!,
4302		apply_to4(EqResX,EqResY,Y2,Tail,X,Y,InitialRel,Span,WF).
4303		apply_to4_call5(EqResX,EqResY,Y2, _Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3) :- % X3 must now be bound
4304		equality_objects_wf(X3,X,EqRes3,WF),
4305		apply_to5(EqResX,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, InitialRel,Span,WF).
4306
4307		% version which wait suntil first argument known
4308		:- block apply_to4_block(-,?,?, ?,?,?,?,?,?).
4309		apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4310		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF).
4311
4312
4313		% apply_to5: implements a watched-literal style treatment of function application
4314		% we watch whether X unifies with two elements of the function, if only one element left we can force equality
4315		% TEST:
4316		% 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
4317		:- block apply_to5(-,?,-, ?,?,?,?, ?,?, ?,?,?),apply_to5(-,?,?, ?,?,?,-, ?,?, ?,?,?).
4318		apply_to5(EqRes,EqResY,EqRes3, Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4319		var(EqRes),!,
4320		% EqRes3 and T3 must be known; TO DO: improve predicate so that we have to wait on T3 only when EqRes3=pred_false
4321		(EqRes3 = pred_false -> % we cannot match next element, move tail one forward
4322		(T3 = [] -> EqRes=pred_true ; true),
4323		apply_to4(EqRes,EqResY,Y2,T3,X,Y,InitialRel,Span,WF)
4324		; /* EqRes3 = pred_true */
4325		% we match the next entry in the list; discard Y2 and jump to (X3,Y3) and return as solution
4326		equal_object_wf(Y3,Y,apply_to6,WF), optional_functionality_check(T3,X,WF),
4327		% TO DO: we could also do equality_objects if necessary between Y and Y3, as in apply_to4 for Y and Y2
4328		opt_force_false(EqRes)
4329		).
4330		apply_to5(pred_true,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, _InitialRel,_Span,WF) :-
4331		(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to5,WF) ; EqResY = pred_true),
4332		opt_force_false(EqRes3),
4333		optional_functionality_check([(X3,Y3)|T3],X,WF).
4334		apply_to5(pred_false,_EqResY,EqRes3, _Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4335		(var(EqRes3) -> % it can be that EqRes3 is about to be triggered
4336		equality_objects_wf(Y3,Y,EqResY3,WF),
4337		prop_apply_eqxy(EqResY3,EqRes3) % propagate: if Y/=Y3 => X/=X3
4338		; EqResY3=not_called),
4339		apply_to4(EqRes3,EqResY3,Y3, T3,X,Y,InitialRel,Span,WF).
4340
4341		opt_force_false(EqRes) :-
4342		(preference(find_abort_values,false) -> EqRes=pred_false
4343		; true). % TO DO: if EqRes becomes pred_true: raise abort_error as the relation was not a function
4344
4345
4346
4347		/********************************************/
4348		/* surjection_relation(R,Domain,Range) */
4349		/* R : Domain <->> Range */
4350		/********************************************/
4351		:- 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)
4352		:- 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)).
4353
4354		surjection_relation_wf(R,Domain,Range,WF) :-
4355		is_surjective(R,Range,WF),
4356		% TODO: is not optimal since ran(R)<:Range is already implied by is_surjective and
4357		% checked a second time by relation_over_wf/4
4358	?	relation_over_wf(R,Domain,Range,WF).
4359
4360		:- 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)).
4361
4362		not_surjection_relation_wf(R,Domain,Range,WF) :-
4363		expand_custom_set_to_list_wf(R,ER,Done,not_surjection_relation_wf,WF),
4364		not_tot_surj_rel(ER,Done,[],Domain,Range,Range,WF).
4365
4366		/*********************************************/
4367		/* total_surjection_relation(R,Domain,Range) */
4368		/* R : Domain <<->> Range */
4369		/*********************************************/
4370		:- 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)).
4371		:- 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)).
4372
4373
4374		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1)],[int(11),int(12)]),L),
4375		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
4376		length(SSL,1))).
4377		%:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11),int(12)]),L), length(L,7))).
4378		% the new domain predicate also instantiates from result; meaning that duplicate solutions are now generated
4379		:- 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))).
4380		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11)]),L),
4381		length(L,1))).
4382
4383		total_surjection_relation(R,Domain,Range) :- init_wait_flags(WF,[total_surjection_relation]),
4384		total_surjection_relation_wf(R,Domain,Range,WF), ground_wait_flags(WF).
4385
4386		total_surjection_relation_wf(R,Domain,Range,WF) :-
4387		relation_over_wf(R,Domain,Range,WF),
4388		check_relation_is_total(R,Domain,WF), % calls domain which now instantiates R if Domain known
4389		check_relation_is_surjective(R,Range,WF).
4390
4391
4392		:- 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)).
4393		:- 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)).
4394		:- 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)).
4395
4396		not_total_surjection_relation_wf(R,Domain,Range,WF) :-
4397		expand_custom_set_to_list_wf(R,ER,Done,not_total_surjection_relation_wf,WF),
4398	?	not_tot_surj_rel(ER,Done,Domain,Domain,Range,Range,WF).
4399
4400
4401		/********************************************/
4402		/* partial_surjection(R,DomType,RangeType) */
4403		/* R : DomType +->> RangeType */
4404		/********************************************/
4405
4406		:- 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)
4407		:- 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)).
4408		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4409		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4410		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],global_set('Name')),
4411		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4412		:- assert_must_succeed((bsets_clp:partial_surjection_wf(X,[int(1),int(2),int(3)],global_set('Name'),_WF),
4413		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4414		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4415		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4416		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4417		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))]))). /* mult. */
4418		:- assert_must_succeed((X=[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))],
4419		bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]))).
4420		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4421		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4422		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4423		X = [(int(2),int(7)),(int(1),int(7))])).
4424		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4425		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4426		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(_X,
4427		[int(1),int(2),int(3),int(4),int(5),int(6),int(7)],[int(2),int(3),int(4)]) )).
4428		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4429		X = [(int(2),int(7)),(int(2),int(6))])).
4430
4431		partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[partial_surjection]),
4432		partial_surjection_wf(R,Domain,Range,WF),
4433		ground_wait_flags(WF).
4434
4435		:- block partial_surjection_wf(-,-,?,?).
4436		partial_surjection_wf(R,Domain,Range,WF) :-
4437		check_card_greater_equal(Domain,geq,Range,CardDom,CardRange),
4438		(surjection_has_to_be_total_injection(CardDom,CardRange)
4439		% LAW: card(setX) = card(setY) => ff: setX +->> setY <=> ff: setX >-> setY
4440		-> total_function_wf(R,Domain,Range,WF),
4441		injective(R,WF)
4442		; is_surjective(R,Range,WF),
4443		partial_function_wf(R,Domain,Range,WF)
4444		).
4445
4446
4447		% check_card_greater_equal(A,B) : quick check that card(A) >= card(B); also works with infinite cardinality
4448		% TO DO: replace by a better constraint propagating predicate (also working for partially instantiated lists,...)
4449		% compared with computing card and setting up < constraint: will only compute card if it can be done efficiently + deals with inf
4450		% check_card_greater_equal(SetA,EQ,SetB) ; EQ=eq or geq
4451		:- block check_card_greater_equal(-,?,?,?,?).
4452		check_card_greater_equal([],_,R,0,0) :- !, empty_set(R).
4453		check_card_greater_equal(A,EQ,B,CA,CB) :- check_card_greater_equal2(A,EQ,B,CA,CB).
4454
4455		:- use_module(inf_arith,[block_inf_greater_equal/2]).
4456		:- block check_card_greater_equal2(?,?,-,?,?).
4457		check_card_greater_equal2(A,EQ,B,CardA,CardB) :-
4458		efficient_card_for_set(A,CardA,CodeA),
4459		efficient_card_for_set(B,CardB,CodeB),!,
4460		call(CodeA), call(CodeB),
4461		(EQ=eq -> CardA=CardB ; block_inf_greater_equal(CardA,CardB)).
4462		check_card_greater_equal2(_A,_,_B,'?','?').
4463
4464
4465		:- block is_surjective(-,-,?).
4466		is_surjective(R,Range,WF) :-
4467		(var(R) -> setup_surj_range(Range,R,WF)
4468		; range_wf(R,Range,WF)).
4469
4470		setup_surj_range(Range,R,WF) :-
4471		setup_range(Range,Res,DONE,WF),
4472		equal_when_done(Res,R,DONE).
4473		:- block equal_when_done(?,?,-).
4474	?	equal_when_done(Res,R,_DONE) :- equal_object(Res,R).
4475
4476
4477		:- block setup_range(-,?,?,?).
4478		setup_range(global_set(G),Res,DONE,WF) :-
4479		expand_custom_set_wf(global_set(G),ES,setup_range,WF),
4480		setup_range(ES,Res,DONE,WF).
4481		setup_range(freetype(ID),Res,DONE,WF) :-
4482		expand_custom_set_wf(freetype(ID),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4483		setup_range(avl_set(S),Res,DONE,WF) :-
4484		expand_custom_set_wf(avl_set(S),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4485		setup_range(closure(P,T,B),Res,DONE,WF) :-
4486		expand_custom_set_wf(closure(P,T,B),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4487		setup_range([],_,done,_WF).
4488		setup_range([H|T],[(_,H)|ST],DONE,WF) :- setup_range(T,ST,DONE,WF).
4489
4490
4491
4492		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],
4493		[int(1),int(2)],[int(7),int(6)],WF),WF)).
4494		:- 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)],
4495		[int(7),int(6),int(2)],WF),WF)).
4496		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4497		X = [(int(2),int(7)),(int(1),int(6))])).
4498		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4499		X = [(int(2),int(7)),(int(2),int(6))])).
4500		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4501		X = [(int(2),int(7)),(int(1),int(6))])).
4502		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4503		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4504		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4505		X = [(int(2),int(7)),(int(1),int(7))])).
4506		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4507		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4508
4509
4510
4511		/* /: Domain +->> Range */
4512		not_partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[not_partial_surjection]),
4513		not_partial_surjection_wf(R,Domain,Range,WF),
4514		ground_wait_flags(WF).
4515
4516		:- block not_partial_surjection_wf(-,?,?,?).
4517		not_partial_surjection_wf(R,DomType,RangeType,WF) :-
4518	?	partial_surjection_test_wf(R,DomType,RangeType,pred_false,WF).
4519
4520
4521		%not_surjective_relation_wf(R,DomType,RType,WF) :-
4522		% invert_relation_wf(R,IR,WF),
4523		% not_total_relation_wf(IR,RType,DomType,WF).
4524
4525
4526		:- 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)).
4527		:- 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)).
4528
4529		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF) :-
4530		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4531		(IsPF==pred_false -> PredRes=pred_false
4532		; range_wf(R,RelRan,WF),
4533	?	conjoin_test(IsPF,IsSurjective,PredRes,WF),
4534		subset_test(RangeType,RelRan,IsSurjective,WF)
4535		).
4536
4537
4538		:- 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)).
4539		:- 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)).
4540
4541		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4542		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4543		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4544		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4545		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4546		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(7))]))).
4547		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4548		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4549		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4550		kernel_objects:equal_object(X,[(int(2),int(7))]))).
4551		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4552		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(8))]))).
4553		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4554		kernel_objects:equal_object(X,[(int(2),int(7)),(int(3),int(6)),(int(1),int(7))]))).
4555		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4556		kernel_objects:equal_object(X,[]))).
4557
4558		/****************************************/
4559		/* total_relation_wf(R,Domain,Range,WF) */
4560		/* R : Domain <<-> Range */
4561		/****************************************/
4562
4563		total_relation_wf(R,Domain,Range,WF) :- relation_over_wf(R,Domain,Range,WF),
4564		check_relation_is_total(R,Domain,WF).
4565
4566		% this predicates assume that the relation's range and domain have already been checked
4567		check_relation_is_total(Relation,Domain,WF) :- domain_wf(Relation,Domain,WF).
4568		check_relation_is_surjective(Relation,Range,WF) :-
4569		range_wf(Relation,Range,WF). % we could also call is_surjective (which does setup_surj_range) ?
4570
4571		:- 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)).
4572		:- 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)).
4573		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4574		X = [(int(2),int(7)),(int(1),int(6))])).
4575		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4576		X = [(int(2),int(7)),(int(1),int(7))])).
4577		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4578		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4579		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4580		X = [(int(2),int(7)),(int(2),int(6))])).
4581		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4582		X = [(int(2),int(7))])).
4583		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4584		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(8))])).
4585		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4586		X = [(int(2),int(7)),(int(3),int(6)),(int(1),int(7))])).
4587
4588		:- block not_total_relation_wf(-,?,?,?).
4589		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
4590		% we do not need the Range; this means we can match more closures (e.g., lambda)
4591		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
4592		not_equal_object_wf(FFDomain,Domain,WF).
4593		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),
4594		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
4595		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
4596		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
4597		not_total_relation_wf(R,Domain,Range,WF) :-
4598		expand_custom_set_to_list_wf(R,ER,Done,not_total_relation_wf,WF),
4599		not_tot_surj_rel(ER,Done,Domain,Domain,[],Range,WF). % empty DelRange means we don't do surjective test
4600
4601		% can be used to check not total, not surj, not total surj relation
4602		:- block not_tot_surj_rel(-,?,?,?,?,?,?).
4603		not_tot_surj_rel([],_,DelDomain,_,DelRange,_,WF) :-
4604	?	at_least_one_set_not_empty(DelDomain,DelRange,WF).
4605		not_tot_surj_rel([_|_],Done,DelDom,Dom,_DelRan,_Ran,_WF) :- nonvar(Done),
4606		Done \= no_check_to_be_done,
4607		nonvar(DelDom),DelDom \= [],
4608		nonvar(Dom),is_infinite_explicit_set(Dom),
4609		!. % a finite expanded list can never be a total relation over an infinite domain
4610		not_tot_surj_rel([(X,Y)|T],_Done,DelDom,Dom,DelRan,Ran,WF) :-
4611		membership_test_wf(Dom,X,MemRes,WF),
4612		not_tr2(MemRes,X,Y,T,DelDom,Dom,DelRan,Ran,WF).
4613
4614		% check if one of the two sets is non-empty
4615		at_least_one_set_not_empty(Set1,Set2,_) :- (Set=Set1 ; Set=Set2),
4616		nonvar(Set),
4617		(Set=avl_set(_) ; Set=[_|_]), % we can avoid leaving choice point
4618		!.
4619		at_least_one_set_not_empty(Set1,_,WF) :- not_empty_set_wf(Set1,WF).
4620		at_least_one_set_not_empty(Set1,Set2,WF) :- empty_set_wf(Set1,WF),not_empty_set_wf(Set2,WF).
4621
4622		:- block not_tr2(-,?,?,?,?,?,?,?,?).
4623		not_tr2(pred_false,_X,_Y,_T,_DelDom,_Dom,_DelRan,_Ran,_WF).
4624		not_tr2(pred_true,X,Y,T,DelDom,Dom,DelRan,Ran,WF) :-
4625		delete_element_wf(X,DelDom,DelDom2,WF), % set DelDom initially to [] to avoid totality check
4626		membership_test_wf(Ran,Y,MemRes,WF),
4627		not_tr3(MemRes,Y,T,DelDom2,Dom,DelRan,Ran,WF).
4628
4629		:- block not_tr3(-,?,?,?,?,?,?,?).
4630		not_tr3(pred_false,_Y,_T,_DelDom2,_Dom,_DelRan,_Ran,_WF).
4631		not_tr3(pred_true,Y,T,DelDom2,Dom,DelRan,Ran,WF) :-
4632		delete_element_wf(Y,DelRan,DelRan2,WF), % set DelRan initially to [] to avoid surjection check
4633		not_tot_surj_rel(T,no_check_to_be_done,DelDom2,Dom,DelRan2,Ran,WF).
4634
4635		/******************************************/
4636		/* total_surjection(R,DomType,RangeType) */
4637		/* R : DomType -->> RangeType */
4638		/******************************************/
4639
4640		:- 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)
4641		:- 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
4642		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1)],[int(7)]),
4643		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4644		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4645		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4646		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7)]),
4647		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4648		:- assert_must_fail((bsets_clp:total_surjection([],[int(1)],[int(7)]))).
4649		:- assert_must_fail((bsets_clp:total_surjection([(int(7),int(7))],[int(1)],[int(7)]))).
4650		:- assert_must_fail((bsets_clp:total_surjection([(int(1),int(7)), (int(2),int(1))],
4651		[int(1),int(2)],[int(7)]))).
4652		:- assert_must_fail((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4653		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4654
4655
4656		total_surjection(R,Domain,Range) :- init_wait_flags(WF),
4657		total_surjection_wf(R,Domain,Range,WF),
4658		ground_wait_flags(WF).
4659
4660		:- block total_surjection_wf(-,-,?,?).
4661		total_surjection_wf(R,DomType,RangeType,WF) :-
4662		check_card_greater_equal(DomType,geq,RangeType,CardDom,CardRange),
4663		total_function_wf(R,DomType,RangeType,WF),
4664		% setup_surj_range(RangeType,R,WF).
4665		(surjection_has_to_be_total_injection(CardDom,CardRange)
4666		% LAW: card(setX) = card(setY) => ff: setX -->> setY <=> ff: setX >-> setY
4667		-> 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
4668		; check_relation_is_surjective(R,RangeType,WF)).
4669		% invert_relation_wf(R,IR,WF), total_relation_wf(IR,RangeType,DomType,WF).
4670
4671		surjection_has_to_be_total_injection(CardDom,CardRange) :- number(CardDom), CardDom=CardRange.
4672		% 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)
4673
4674		:- 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)).
4675		:- 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)).
4676		:- 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)).
4677		:- 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)).
4678		:- 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)).
4679		:- 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)).
4680
4681		:- block not_total_surjection_wf(-,?,?,?), not_total_surjection_wf(?,-,-,?).
4682		not_total_surjection_wf(R,DomType,RangeType,WF) :-
4683		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4684		not_total_surjection2(PredRes,R,DomType,RangeType,WF).
4685		:- block not_total_surjection2(-,?,?,?,?).
4686		not_total_surjection2(pred_false,_R,_DomType,_RangeType,_WF).
4687		not_total_surjection2(pred_true,R,_DomType,RangeType,WF) :-
4688		range_wf(R,RelRange,WF),
4689		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
4690		[RangeType,b_operator(range,[R])],unknown),WF2),
4691		not_subset_of_wf(RangeType,RelRange,WF2).
4692		%not_surjective_relation_wf(R,DomType,RangeType,WF).
4693
4694		:- 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)).
4695		:- 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)
4696		:- 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)).
4697
4698		% reified total function check:
4699		total_function_test_wf(R,DomType,RangeType,PredRes,WF) :-
4700		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4701		(IsPF==pred_false -> PredRes=pred_false
4702		; domain_wf(R,RelDom,WF),
4703		conjoin_test(IsPF,IsTotal,PredRes,WF),
4704		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
4705		[DomType,b_operator(domain,[R])],unknown),WF2),
4706		subset_test(DomType,RelDom,IsTotal,WF2)
4707		).
4708
4709		/*******************************************/
4710		/* partial_injection(R,DomType,RangeType) */
4711		/* R : DomType >+> RangeType */
4712		/*******************************************/
4713
4714		:- 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)).
4715		:- 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)).
4716		:- 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)).
4717		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1)],[int(7)]),
4718		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4719		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4720		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4721		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7)]),
4722		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4723		:- assert_must_succeed((bsets_clp:partial_injection([],[int(1)],[int(7)]))).
4724		:- assert_must_fail((bsets_clp:partial_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4725		:- assert_must_fail((bsets_clp:partial_injection([(int(1),int(7)), (int(2),int(1))],
4726		[int(1),int(2)],[int(7)]))).
4727		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4728		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4729
4730
4731		partial_injection(R,Domain,Range) :- init_wait_flags(WF),
4732		partial_injection_wf(R,Domain,Range,WF),
4733		ground_wait_flags(WF).
4734
4735		:- block partial_injection_wf(-,-,?,?).
4736		partial_injection_wf(FF,Domain,Range,WF) :- nonvar(FF),
4737		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
4738		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
4739		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
4740		partial_injection_wf(R,DomType,RangeType,WF) :-
4741		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF), % should we use very_large?
4742		partial_function_wf(ER,DomType,RangeType,WF),
4743		injective(ER,WF).
4744		% invert_relation_wf(R,IR,WF),
4745		% partial_function_wf(IR,RangeType,DomType,WF).
4746
4747		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective([(int(1),int(6)),(int(4),int(7)),(int(2),int(8))],WF),WF)).
4748		:- 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)).
4749
4750		:- block injective(-,?).
4751		injective(FF,WF) :-
4752		custom_explicit_sets:dom_range_for_specific_closure(FF,_FFDomain,_FFRange,function(bijection),WF),!.
4753		injective(avl_set(AVL),_WF) :- !,
4754		is_injective_avl_relation(AVL,_Range). % seems slightly faster than injective/3 code below
4755		injective(closure(P,T,B),WF) :- !,
4756		symbolic_injectivity_check(closure(P,T,B),WF).
4757		injective(Rel,WF) :- expand_custom_set_to_list_wf(Rel,ERel,_,injective,WF),
4758		injective(ERel,[],WF).
4759
4760		%:- use_module(library(lists),[maplist/3]).
4761		% for FD-sets we could setup all_different constraint
4762		:- block injective(-,?,?).
4763		injective([],_SoFar,_).
4764		% (maplist(get_fd_val,SoFar,FDL) -> clpfd:all_distinct(FDL) ; true). %clpfd_interface:clpfd_alldifferent(FDL) ; true).
4765		%get_fd_val(int(H),H).
4766		injective([(_From,To)|T],SoFar,WF) :-
4767		not_element_of_wf(To,SoFar,WF), /* check that it is injective */
4768		add_new_element_wf(To,SoFar,SoFar2,WF), %SoFar2=[To|SoFar], could also work and be faster ?
4769		injective(T,SoFar2,WF).
4770		% no case for global_set: it cannot be a relation; two cases below not required because of expand_custom_set_to_list
4771		%injective(avl_set(S),SoFar,WF) :- expand_custom_set_wf(avl_set(S),ES,inj,WF), injective(ES,SoFar,WF).
4772		%injective(closure(P,T,B),SoFar,WF) :- expand_custom_set_wf(closure(P,T,B),ES,inj,WF), injective(ES,SoFar,WF).
4773
4774
4775
4776		/* /: Dom >+> R */
4777
4778		:- 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)).
4779		:- 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)).
4780		:- 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)).
4781		:- 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)).
4782
4783		:- block not_partial_injection(-,?,?,?).
4784		not_partial_injection(R,DomType,RangeType,WF) :-
4785		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4786		not_partial_injection2(IsPF,R,DomType,RangeType,WF).
4787
4788		:- block not_partial_injection2(-,?,?,?,?).
4789		not_partial_injection2(pred_false,_R,_DomType,_RType,_WF).
4790		not_partial_injection2(pred_true,R,DomType,RType,WF) :-
4791		not_injection_wf(R,DomType,RType,WF).
4792
4793		not_injection_wf(R,DomType,RType,WF) :-
4794		invert_relation_wf(R,IR,WF),
4795		not_partial_function(IR,RType,DomType,WF).
4796
4797		/*****************************************/
4798		/* total_injection(R,DomType,RangeType) */
4799		/* R : DomType >-> RangeType */
4800		/*****************************************/
4801
4802		:- 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)).
4803		:- 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)).
4804		:- 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)).
4805		:- 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)).
4806		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1)],[int(7)]),
4807		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4808		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4809		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4810		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7)]),
4811		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4812		:- assert_must_fail((bsets_clp:total_injection([],[int(1)],[int(7)]))).
4813		:- assert_must_fail((bsets_clp:total_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4814		:- assert_must_fail((bsets_clp:total_injection([(int(1),int(7)), (int(2),int(1))],
4815		[int(1),int(2)],[int(7)]))).
4816		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4817		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4818
4819
4820		total_injection(R,Domain,Range) :- init_wait_flags(WF),
4821		total_injection_wf(R,Domain,Range,WF),
4822		ground_wait_flags(WF).
4823
4824		:- block total_injection_wf(-,-,?,?). % with just ?,-,?,? we may wait too long to start injective check
4825		% Note: no need to check: dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection)),
4826		total_injection_wf(R,DomType,RangeType,WF) :-
4827		check_card_greater_equal(RangeType,geq,DomType,_,_), % there must be more Range elements than domain elements; pigeonhole principle
4828		total_injection_wf2(R,DomType,RangeType,WF).
4829		total_injection_wf2(R,DomType,RangeType,WF) :-
4830		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
4831		total_function_wf(ER,DomType,RangeType,WF),
4832		injective(ER,WF).
4833
4834
4835		:- block not_total_injection(-,?,?,?), not_total_injection(?,-,-,?).
4836		not_total_injection(R,DomType,RangeType,WF) :-
4837		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4838		not_total_injection2(PredRes,R,DomType,RangeType,WF).
4839
4840		:- block not_total_injection2(-,?,?,?,?).
4841		not_total_injection2(pred_false,_R,_Dom,_Ran,_WF).
4842		not_total_injection2(pred_true,R,DomType,RangeType,WF) :-
4843		% TO DO: replace DomType and RangeType by full Type
4844		not_injection_wf(R,DomType,RangeType,WF).
4845
4846		/***********************************/
4847		/* partial_bijection(R,DomType,RangeType) */
4848		/* R : DomType >+>> RangeType */
4849		/***********************************/
4850
4851		:- 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)
4852		:- 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)).
4853		:- assert_must_succeed((partial_bijection(X,[int(1),int(2)],[int(7),int(6)]),
4854		kernel_objects:equal_object(X,[(int(1),int(6)),(int(2),int(7))]))).
4855		:- assert_must_succeed((partial_bijection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4856		X = [(int(2),int(7)),(int(3),int(6))])).
4857		:- assert_must_fail((partial_bijection(X,[int(1),int(2)],[int(7),int(6),int(5)]),
4858		X = [(int(2),int(7)),(int(1),int(6))])).
4859
4860		partial_bijection(R,Domain,Range) :- init_wait_flags(WF),
4861		partial_bijection_wf(R,Domain,Range,WF),
4862		ground_wait_flags(WF).
4863
4864		partial_bijection_wf(R,DomType,RangeType,WF) :-
4865		partial_injection_wf(R,DomType,RangeType,WF),
4866		partial_surjection_wf(R,DomType,RangeType,WF).
4867
4868		:- 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)).
4869		:- 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)).
4870
4871		:- 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)).
4872
4873		:- 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)).
4874		:- 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)).
4875
4876
4877		:- block not_partial_bijection(-,?,?,?), not_partial_bijection(?,-,-,?).
4878		not_partial_bijection(R,DomType,RangeType,WF) :-
4879		% >+>> = +->> + injective
4880		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF),
4881		not_partial_bijection2(PredRes,R,DomType,RangeType,WF).
4882
4883		:- block not_partial_bijection2(-,?,?,?,?).
4884		not_partial_bijection2(pred_false,_R,_DomType,_RangeType,_WF).
4885		not_partial_bijection2(pred_true,R,DomType,RangeType,WF) :-
4886		not_injection_wf(R,DomType,RangeType,WF).
4887
4888
4889
4890		/* The transitive (not reflexive) closure of a relation (closure1) */
4891
4892		:- 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))]))).
4893		:- 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))]))).
4894		:- 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))]))).
4895		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4))],X),
4896		kernel_objects:equal_object(X,[(int(1),int(4))]))).
4897		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2))],X),
4898		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),
4899		(int(1),int(2))]))).
4900		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2)),(int(2),int(3))],X),
4901		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),(int(2),int(3)),
4902		(int(4),int(3)),(int(1),int(2)),(int(1),int(3))]))).
4903		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(A),int(2)),(int(2),int(6))],
4904		[(int(1),int(2)),(int(1),int(6)),(int(2),int(6))]),A=1)).
4905
4906		relational_trans_closure(Rel,Res) :- relational_trans_closure_wf(Rel,Res,no_wf_available).
4907
4908		% transitive closure for relations (closure1)
4909		:- block relational_trans_closure_wf(-,?,?).
4910		relational_trans_closure_wf(Relation,Result,WF) :-
4911		try_expand_and_convert_to_avl_with_check(Relation,ARelation,relational_trans_closure_wf),
4912	?	relational_trans_closure2(ARelation,Result,WF).
4913		:- block relational_trans_closure2(-,?,?).
4914		relational_trans_closure2(ARelation,Result,WF) :-
4915		(closure1_for_explicit_set(ARelation,Res)
4916		-> kernel_objects:equal_object_wf(Res,Result,relational_trans_closure_wf,WF)
4917		; expand_custom_set_to_list_wf(ARelation,ERelation,_,relational_trans_closure2,WF),
4918		is_full_relation(ERelation,WaitVar), % still required??
4919		% we could do a check_subset_of_wf(ERelation,Resul,WF) if Result is nonvar and ERelation not ground
4920	?	compute_trans_closure(ERelation,Result,WaitVar,WF)
4921		).
4922
4923		:- block compute_trans_closure(?,?,-,?).
4924		compute_trans_closure(Relation,Result,_,WF) :-
4925	?	compute_trans_closure2(Relation,1,Result,WF).
4926
4927		compute_trans_closure2(Relation,Cnt,Result,WF) :-
4928		one_closure_iteration(Relation,Relation,Relation,Result1,Added,Done,WF),
4929	?	compute_trans_closure3(Relation,Cnt,Result1,Added,Done,Result,WF).
4930
4931		:- block compute_trans_closure3(?,?,?,?,-,?,?).
4932		compute_trans_closure3(Relation,Cnt,Result1,Added,_Done,Result,WF) :-
4933		( equal_object_wf(Result1,Relation,relational_trans_closure_wf,WF), % should we do equality_objects here?
4934		equal_object_optimized_wf(Result,Result1,compute_trans_closure,WF)
4935		;
4936		Added==possibly_added,
4937		not_equal_object_wf(Result1,Relation,WF), % not a fixpoint; continue
4938		IterCnt is Cnt+1,
4939	?	compute_trans_closure2(Result1,IterCnt,Result,WF)
4940		).
4941
4942		:- block one_closure_iteration(?,?,-,?,?,?,?).
4943		one_closure_iteration([],_,IterRes,OutRel,Added,Done,WF) :-
4944		equal_object_wf(IterRes,OutRel,one_closure_iteration,WF),
4945		(var(Added) -> Added=not_added ; true),
4946		Done=done.
4947		one_closure_iteration([(X,Y)|T],ExpandedPreviousRel,PreviousRel,OutRel,Added,Done,WF) :-
4948		add_tuples(ExpandedPreviousRel,X,Y,PreviousRel,IntRel,Added,DoneTuples,WF),
4949		one_closure_iteration_block(DoneTuples,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
4950
4951		:- block one_closure_iteration_block(-,?,?,?,?,?,?,?).
4952		one_closure_iteration_block(_,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF) :-
4953	?	one_closure_iteration(T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
4954
4955		add_tuples([],_,_,OutRel,OutRel,_Added,done,_).
4956		add_tuples([(X,Y)|T],OX,OY,InRel,OutRel,Added,Done,WF) :-
4957		% add tuple (X,OY) if we have Y=OX
4958		equality_objects_wf(Y,OX,EqRes,WF),
4959		add_tuples_aux(EqRes,X,T,OX,OY,InRel,OutRel,Added,Done,WF).
4960
4961		:- block add_tuples_aux(-,?,?,?,?,?,?,?,?,?).
4962		add_tuples_aux(pred_true,X,T,OX,OY,InRel,OutRel,possibly_added,Done,WF) :-
4963		add_element_wf((X,OY),InRel,IntRel,WF), % add transitive couple X -> OY
4964	?	add_tuples(T,OX,OY,IntRel,OutRel,_,Done,WF).
4965		add_tuples_aux(pred_false,_X,T,OX,OY,InRel,OutRel,Added,Done,WF) :- % no transitive couple needed
4966	?	add_tuples(T,OX,OY,InRel,OutRel,Added,Done,WF).
4967
4968
4969		:- assert_must_succeed((is_full_relation(X,R),var(R),X=[],R==true)).
4970		:- 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)).
4971		:- block is_full_relation(-,?).
4972		is_full_relation([],R) :- !,R=true.
4973		is_full_relation([H|T],W) :- !, is_full_relation_aux(H,T,W).
4974		is_full_relation(X,R) :-
4975		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation(X,R)),fail.
4976
4977		:- block is_full_relation_aux(-,?,?).
4978		is_full_relation_aux((X,Y),T,W) :- !, is_full_relation_aux2(X,Y,T,W).
4979		is_full_relation_aux(X,T,W) :-
4980		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation_aux(X,T,W)),fail.
4981		:- block is_full_relation_aux2(-,?,?,?), is_full_relation_aux2(?,-,?,?).
4982	?	is_full_relation_aux2(_X,_Y,T,W) :- is_full_relation(T,W).
4983
4984		/* ------------------ */
4985
4986		:- 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)
4987
4988		in_closure1_wf(Pair,Relation,WF) :- %Pair = (_A,B),
4989		%in_domain_wf_lazy(A,Relation,WF), % done below
4990		%check_element_of_wf((_,B),Relation,WF), % multiple solutions for _, see test 634, 637
4991		in_closure1_membership_test_wf(Pair,Relation,pred_true,WF).
4992
4993
4994		:- 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)).
4995
4996		not_in_closure1_wf(Pair,Relation,WF) :-
4997		in_closure1_membership_test_wf(Pair,Relation,pred_false,WF).
4998
4999		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[],Res,_WF),Res==pred_false)).
5000		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(2))],Res,_WF),Res==pred_true)).
5001		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(3))],Res,_WF),Res==pred_false)).
5002		:- 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)).
5003		:- 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)
5004		:- 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)).
5005		:- 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)).
5006		:- 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)).
5007		:- 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)
5008		:- 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)).
5009
5010		:- block force_in_domain(-,?,?,?).
5011		force_in_domain(pred_false,_A,_Relation,_WF).
5012		force_in_domain(pred_true,A,Relation,WF) :- % force A to be in domain, avoid enumeration warnings,...
5013		% maybe only for non-ground A
5014		in_domain_wf_lazy(A,Relation,WF). % slowdown Loop.mch (tests 634, 637) if we use in_domain_wf ?
5015
5016		% (x,y) : closure1(Rel)
5017		:- block in_closure1_membership_test_wf(?,-,?,?).
5018		in_closure1_membership_test_wf((A,B),CSRelation,MemRes,WF) :-
5019		is_custom_explicit_set(CSRelation,in_closure1),
5020		!,
5021		image_for_closure1_wf(CSRelation,[A],Image,WF),
5022		force_in_domain(MemRes,A,CSRelation,WF),
5023		membership_test_wf(Image,B,MemRes,WF).
5024		in_closure1_membership_test_wf((X,Y),Relation,MemRes,WF) :-
5025		expand_custom_set_to_list_wf(Relation,ERelation,_,in_closure1_membership_test_wf,WF),
5026		Discarded = [], % pairs discarded in current iteration
5027		force_in_domain(MemRes,X,Relation,WF),
5028		in_closure1_membership_test_wf2(ERelation,X,Y,Discarded,MemRes,WF).
5029
5030		:- block in_closure1_membership_test_wf2(-,?,?,?,?,?).
5031		in_closure1_membership_test_wf2([],_X,_Y,_,MemRes,_WF) :- MemRes=pred_false.
5032		in_closure1_membership_test_wf2([(V,W)|Rest],X,Y,Discarded,MemRes,WF) :- % TO DO: Rest==[] -->
5033		equality_objects_wf(V,X,VXResult,WF),
5034		in_closure1_membership_test_wf3(VXResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5035
5036		:- block in_closure1_membership_test_wf3(-,?,?,?,?,?,?,?,?).
5037		in_closure1_membership_test_wf3(pred_false,V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5038		in_closure1_membership_test_wf2(Rest,X,Y,[(V,W)|Discarded],MemRes,WF).
5039		in_closure1_membership_test_wf3(pred_true,V,W,Rest,X,Y,Discarded,MemRes,WF) :- % V=X
5040		propagate_false(MemRes,WYResult),
5041		% TODO: Res=[],Discarded=[] -> MemRes=WYResult
5042		equality_objects_wf(W,Y,WYResult,WF), % MemRes = pred_false => WYResult = pred_false
5043		in_closure1_membership_test_wf4(WYResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5044
5045		:- block in_closure1_membership_test_wf4(-,?,?,?,?,?,?,?,?).
5046		in_closure1_membership_test_wf4(pred_false,_V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5047		append(Discarded,Rest,Restart),
5048		in_closure1_membership_test_wf2(Restart,W,Y,[],MemRes1,WF),
5049		propagate_false(MemRes,MemRes1), % MemRes = pred_false -> MemRes1=pred_false
5050		when(nonvar(MemRes1),
5051		(MemRes1=pred_true -> MemRes=pred_true
5052		; in_closure1_membership_test_wf2(Rest,X,Y,Discarded,MemRes,WF) % (V,W) not in Discarded: was not useful
5053		)).
5054		in_closure1_membership_test_wf4(pred_true,_V,_W,_Rest,_X,_Y,_Discarded,MemRes,_WF) :- % W=Y
5055		MemRes = pred_true.
5056		/* ------------------ */
5057
5058		:- block propagate_false(-,?).
5059		propagate_false(pred_false,pred_false).
5060		propagate_false(pred_true,_).
5061