1		% (c) 2004-2024 Lehrstuhl fuer Softwaretechnik und Programmiersprachen,
2		% Heinrich Heine Universitaet Duesseldorf
3		% This software is licenced under EPL 1.0 (http://www.eclipse.org/org/documents/epl-v10.html)
4
5		:- module(bsets_clp,
6		[empty_sequence/1,
7		is_sequence/2, is_sequence_wf/3, not_is_sequence/2, not_is_sequence_wf/3,
8		not_is_non_empty_sequence_wf/3,
9		injective_sequence_wf/3,
10		not_injective_sequence/3,
11		not_non_empty_injective_sequence/3,
12		injective_non_empty_sequence/3,
13		finite_non_empty_sequence/3,
14		test_finite_non_empty_sequence/4,
15		permutation_sequence_wf/3,
16		not_permutation_sequence/3,
17		size_of_sequence/3,
18		prepend_sequence/4, append_sequence/4, prefix_sequence_wf/4,
19		suffix_sequence/4, concat_sequence/4,
20		disjoint_union_wf/4,
21		concatentation_of_sequences/3,
22		tail_sequence/4, first_sequence/4, front_sequence/4, last_sequence/4,
23		rev_sequence/3,
24
25
26		% maplet/3,
27		% relation/1,
28		relation_over/3, relation_over_wf/4,
29		not_relation_over/4,
30		domain_wf/3,
31
32		range_wf/3,
33		identity_relation_over_wf/3, in_identity/3, not_in_identity/3,
34		invert_relation_wf/3,
35		tuple_of/3,
36		in_composition_wf/4, not_in_composition_wf/4, rel_composition_wf/5,
37		direct_product_wf/4,
38		parallel_product_wf/4, in_parallel_product_wf/4, not_in_parallel_product_wf/4,
39		rel_iterate_wf/5,
40		event_b_identity_for_type/3,
41
42		not_partial_function/4,
43		partial_function/3, partial_function_wf/4, partial_function_test_wf/5,
44
45		total_function/3, total_function_wf/4, total_function_test_wf/5,
46
47		% enumerate_total_bijection/3,
48		total_bijection/3, total_bijection_wf/4,
49
50		not_total_function/4,
51		not_total_bijection/4,
52
53
54		range_restriction_wf/4, range_subtraction_wf/4,
55		in_range_restriction_wf/4, not_in_range_restriction_wf/4,
56		in_range_subtraction_wf/4, not_in_range_subtraction_wf/4,
57		domain_restriction_wf/4, domain_subtraction_wf/4,
58		in_domain_restriction_wf/4, not_in_domain_restriction_wf/4,
59		in_domain_subtraction_wf/4, not_in_domain_subtraction_wf/4,
60		override_relation/4,
61		in_override_relation_wf/4, not_in_override_relation_wf/4,
62		image_wf/4, image_for_closure1_wf/4,
63		special_operator_for_image/3, image_for_special_operator/5, apply_fun_for_special_operator/6,
64
65		in_domain_wf/3, not_in_domain_wf/3,
66		apply_to/4, apply_to/5, apply_to/6,
67		override/5,
68
69		%sum_over_range/2, mul_over_range/2,
70
71		disjoint_union_generalized_wf/3,
72
73		partial_surjection/3, not_partial_surjection_wf/4,
74		partial_surjection_test_wf/5,
75
76		total_relation_wf/4,
77		not_total_relation_wf/4,
78
79		surjection_relation_wf/4, total_surjection_relation_wf/4,
80		not_surjection_relation_wf/4, not_total_surjection_relation_wf/4,
81
82		total_surjection/3, total_surjection_wf/4,
83		not_total_surjection_wf/4,
84
85		partial_injection/3, partial_injection_wf/4,
86		not_partial_injection/4,
87
88		total_injection/3, total_injection_wf/4,
89		not_total_injection/4,
90
91		partial_bijection/3, partial_bijection_wf/4,
92		not_partial_bijection/4,
93
94		relational_trans_closure_wf/3, %relational_reflexive_closure/2,
95		in_closure1_wf/3, not_in_closure1_wf/3
96		]).
97
98
99		:- use_module(library(terms)).
100		:- use_module(self_check).
101
102		:- use_module(debug).
103		:- use_module(tools).
104
105		:- use_module(module_information,[module_info/2]).
106		:- module_info(group,kernel).
107		:- module_info(description,'This module provides more advanced operations for the basic datatypes of ProB (mainly for relations, functions, sequences).').
108
109		:- use_module(tools_printing).
110
111		:- use_module(delay).
112
113		:- use_module(typechecker).
114		:- use_module(error_manager).
115
116		:- use_module(kernel_objects).
117		:- use_module(kernel_records).
118		:- use_module(kernel_tools).
119
120		:- use_module(kernel_waitflags).
121		:- use_module(kernel_equality,[equality_objects_wf/4]).
122
123		:- use_module(custom_explicit_sets).
124		:- use_module(avl_tools,[avl_fetch_pair/3]).
125		:- use_module(bool_pred,[negate/2]).
126		:- use_module(closures,[is_symbolic_closure/1]).
127		:- use_module(bsyntaxtree, [conjunct_predicates/2,
128		mark_bexpr_as_symbolic/2,
129		create_texpr/4,
130		safe_create_texpr/3,
131		get_texpr_type/2
132		]).
133
134		/* --------- */
135		/* SEQUENCES */
136		/* ------- - */
137
138		:- assert_must_succeed((bsets_clp:empty_sequence([]))).
139		:- assert_must_fail((bsets_clp:empty_sequence([int(1)]))).
140	?	empty_sequence(X) :- empty_set(X). % TO DO: add WF
141
142		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_empty_sequence([(int(2),int(33)),(int(1),int(22))]))).
143		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_empty_sequence([(int(1),int(33))]))).
144		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_empty_sequence([]))).
145
146		not_empty_sequence(X) :- var(X),!,
147		X = [(int(1),_)|_].
148		not_empty_sequence(X) :- is_custom_explicit_set_nonvar(X),!,
149		is_non_empty_explicit_set(X).
150		not_empty_sequence([(int(_),_)|_]). % clousure, avl_set dealt with clause above
151
152		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_empty_sequence_wf([(int(1),int(33))],WF),WF)).
153		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_empty_sequence_wf([(int(1),pred_true),(int(2),pred_false)],WF),WF)).
154		not_empty_sequence_wf(X,_WF) :- nonvar(X),!, not_empty_sequence(X).
155		not_empty_sequence_wf(X,WF) :-
156		(preferences:preference(use_smt_mode,true) -> not_empty_sequence(X)
157		; get_enumeration_starting_wait_flag(not_empty_sequence_wf,WF,LWF),
158		not_empty_sequence_lwf(X,LWF)).
159
160		:- block not_empty_sequence_lwf(-,-).
161		not_empty_sequence_lwf(S,_) :- nonvar(S),!,not_empty_sequence(S).
162		not_empty_sequence_lwf([(int(1),_)|_],_).
163
164		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:is_sequence([(int(1),int(22))],[int(22)]))).
165		:- assert_must_succeed(bsets_clp:is_sequence(closure(['_zzzz_unit_tests'],[couple(integer,integer)],b(member(b(identifier('_zzzz_unit_tests'),couple(integer,integer),[generated]),b(value([(int(1),int(22))]),set(couple(integer,integer)),[])),pred,[])),[int(22)])).
166
167		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:is_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
168		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)]))).
169		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(1),int(33)),(int(0),int(22))],[int(22),int(33),int(44)]))).
170		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:is_sequence([(int(3),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
171		:- assert_must_succeed((is_sequence(R,global_set('Name')),R = [])).
172		:- assert_must_succeed((is_sequence(R,global_set('Name')),
173		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
174		:- assert_must_succeed((is_sequence(R,global_set('Name')),
175		R = [(int(1),fd(2,'Name'))] )).
176		:- assert_must_succeed((is_sequence(R,global_set('Name')),
177		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
178		:- assert_must_succeed((is_sequence(R,global_set('Name')),
179		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
180		:- assert_must_succeed((is_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
181		global_set('Name')) )).
182		:- assert_must_succeed((is_sequence(R,[int(1),int(2)]),
183		R = [(int(2),int(2)),(int(1),int(1))] )).
184		:- assert_must_fail((is_sequence(R,[int(1),int(2)]),
185		R = [(int(2),int(2)),(int(3),int(1))] )).
186		:- assert_must_fail((is_sequence(R,[int(1),int(2)]),
187		R = [(int(2),int(2)),(int(1),int(3))] )).
188		:- assert_must_fail((is_sequence(R,global_set('Name')),
189		R = [(int(0),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
190		:- assert_must_succeed((is_sequence(X,global_set('Name')),
191		(preferences:get_preference(randomise_enumeration_order,true) -> true
192		; kernel_objects:enumerate_basic_type(X,seq(global('Name')))),
193		X = [(int(1),fd(1,'Name'))])). % can take a long time with RANDOMISE_ENUMERATION_ORDER
194
195		is_sequence(X,Type) :- init_wait_flags(WF,[is_sequence]),
196		is_sequence_wf(X,Type,WF),
197	?	ground_wait_flags(WF).
198
199		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([int(1),int(2),int(3)],WF),WF)).
200		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([int(1)],WF),WF)).
201		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_domain([],WF),WF)).
202		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:is_sequence_domain([int(0)],WF),WF)).
203		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:is_sequence_domain([int(2),int(3)],WF),WF)).
204		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_is_sequence_domain([int(2),int(3)],WF),WF)).
205		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_is_sequence_domain([int(0)],WF),WF)).
206		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_is_sequence_domain([int(1)],WF),WF)).
207		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_is_sequence_domain([],WF),WF)).
208
209		% check if a set is the domain of a sequence, i.e., an interval 1..n with n>=0
210		:- use_module(custom_explicit_sets,[construct_interval_closure/3]).
211		:- use_module(kernel_cardinality_attr,[finite_cardinality_as_int_wf/3]).
212		:- block is_sequence_domain(-,?).
213		is_sequence_domain(Domain,WF) :-
214		finite_cardinality_as_int_wf(Domain,int(Max),WF),
215		construct_interval_closure(1,Max,Interval), equal_object_wf(Domain,Interval,is_sequence_domain,WF).
216		:- block not_is_sequence_domain(-,?).
217		not_is_sequence_domain(Domain,WF) :-
218		finite_cardinality_as_int_wf(Domain,int(Max),WF),
219		construct_interval_closure(1,Max,Interval), not_equal_object_wf(Domain,Interval,WF).
220
221		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_wf([(int(1),pred_true)],
222		[pred_true,pred_false],WF),WF)).
223		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:is_sequence_wf([(int(1),pred_true),(int(2),pred_false),(int(3),pred_true)],
224		[pred_true,pred_false],WF),WF)).
225		:- assert_must_succeed((bsets_clp:is_sequence_wf([(int(X),R)],[pred_true],_WF),X==1, R==pred_true)).
226		:- assert_must_succeed((bsets_clp:is_sequence_wf([(int(X),R),(int(Y),R)],[pred_true],_WF),X=2,
227		(preferences:preference(use_clpfd_solver,true) -> Y==1 ; Y=1), R==pred_true)).
228
229		is_sequence_wf(Seq,Range,WF) :- is_sequence_wf_ex(Seq,Range,WF,_).
230		% is_sequence_wf_ex also returns expansion; if it was done
231		:- block is_sequence_wf_ex(-,?,?,?).
232		is_sequence_wf_ex(FF,Range,WF,FF) :-
233		nonvar(FF), FF = closure(_,_,_),
234		custom_explicit_sets:is_definitely_maximal_set(Range),
235		% we do not need the Range; this means we can match more closures (e.g., lambda)
236		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
237		is_sequence_domain(FFDomain,WF).
238		is_sequence_wf_ex(Seq,Range,WF,Res) :-
239		expand_and_convert_to_avl_set_warn(Seq,AER,is_sequence_wf_ex,'ARG : seq(?)',WF),!,
240		is_avl_sequence(AER),
241		is_avl_relation_over_range(AER,Range,WF),
242		custom_explicit_sets:construct_avl_set(AER,Res).
243		is_sequence_wf_ex(X,Type,WF,EX) :-
244		% try_ensure_seq_numbering(X,1),
245		expand_custom_set_to_list_wf(X,EX,_,is_sequence_wf_ex,WF),
246		is_sequence2(EX,[],Type,0,_MinSize,WF).
247
248		% will make this much faster x:seq(STRING) & card(x)=400 & 401:dom(x) (40 ms rather than > 2 secs)
249		% but this does not work -eval_file /Users/leuschel/git_root/prob_examples/examples/Setlog/prob-ttf/plavis-TransData_SP_21_simplified.prob
250		%:- block try_ensure_seq_numbering(-,?).
251		%try_ensure_seq_numbering([H|T],NextNr) :- var(H),!, print(nr(NextNr)),nl,
252		% H = (int(NextNr),_), N1 is NextNr+1,
253		% try_ensure_seq_numbering(T,N1).
254		%try_ensure_seq_numbering(_,_).
255
256		:- block is_sequence2(-,?,?,?,?,?).
257		is_sequence2([],IndexesSoFar,_Type,Size,MinSize,_WF) :- MinSize = Size,
258		contiguous_set_of_indexes(IndexesSoFar,Size).
259		/* not very good to do the checking at the end; can we move part of the checking earlier ? */
260		is_sequence2([(int(Idx),X)|Tail],IndexesSoFar,Type,Size,MinSize,WF) :-
261		less_than_direct(0,Idx), %is_index_greater_zero(Idx),
262		not_element_of_wf(int(Idx),IndexesSoFar,WF),
263		check_element_of_wf(X,Type,WF), S1 is Size+1,
264		clpfd_interface:clpfd_leq(Idx,MinSize,_Posted),
265		(var(Tail)
266		-> clpfd_interface:clpfd_domain(MinSize,Low,_Up), % TO DO: ensure that final size at least Low
267		(number(Low),Low>S1 -> Tail = [_|_] % TO DO: proper reification; what if MinSize gets constrained later
268		; expand_seq_if_necessary(Idx,S1,Tail)) % the sequence must be longer; force it
269		; true
270		),
271		is_sequence2(Tail,[int(Idx)|IndexesSoFar],Type,S1,MinSize,WF).
272
273		:- block expand_seq_if_necessary(-,?,-).
274		expand_seq_if_necessary(MinSize,S1,Tail) :- % TO DO: proper reification on MinSize above
275		number(MinSize), MinSize>S1, (var(Tail) ; Tail==[]),
276		!,
277		Tail = [_|_].
278		expand_seq_if_necessary(_,_,_).
279
280		:- block contiguous_set_of_indexes(-,?).
281		contiguous_set_of_indexes([],_).
282		contiguous_set_of_indexes([H|T],Size) :- contiguous_set_of_indexes1(T,H,Size).
283
284		:- block contiguous_set_of_indexes1(-,?,?).
285		contiguous_set_of_indexes1([],int(1),_).
286		contiguous_set_of_indexes1([int(H2)|T],int(H1),Size) :- less_than_equal_direct(H1,Size),
287		less_than_equal_direct(H2,Size), less_than_equal_indexes(T,[H1,H2],Size).
288
289
290		less_than_equal_indexes([],All,_) :- clpfd_interface:clpfd_alldifferent(All).
291		less_than_equal_indexes([int(H)|T],All,Size) :- less_than_equal_direct(H,Size),less_than_equal_indexes(T,[H|All],Size).
292
293		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(4),int(7))],[int(7),int(6)],WF),WF)).
294		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(3),int(8))],[int(7),int(6)],WF),WF)).
295		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(7),int(6)],WF),WF)).
296		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(2),int(6)),(int(3),int(7)),(int(4),int(7))],[int(7),int(6)],WF),WF)).
297		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_is_sequence_wf([(int(1),int(6)),(int(0),int(7)),(int(2),int(7))],[int(7),int(6)],WF),WF)).
298		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_sequence([(int(1),int(22))],[int(22)]))).
299		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
300		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)]))).
301		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_sequence([(int(3),int(33)),(int(1),int(22))],[int(22),int(33),int(44)]))).
302		:- assert_must_fail((not_is_sequence(R,global_set('Name')),R = [])).
303		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
304		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
305		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
306		R = [(int(1),fd(2,'Name'))] )).
307		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
308		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
309		:- assert_must_fail((not_is_sequence(R,global_set('Name')),
310		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
311		:- assert_must_fail((not_is_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
312		global_set('Name')) )).
313		:- assert_must_fail((not_is_sequence(R,[int(1),int(2)]),
314		R = [(int(2),int(2)),(int(1),int(1))] )).
315		:- assert_must_succeed((not_is_sequence(R,[int(1),int(2)]),
316		R = [(int(2),int(2)),(int(3),int(1))] )).
317		:- assert_must_succeed((not_is_sequence(R,[int(1),int(2)]),
318		R = [(int(2),int(2)),(int(1),int(3))] )).
319
320
321		not_is_sequence(X,Type) :- init_wait_flags(WF,[not_is_sequence]),
322		not_is_sequence_wf(X,Type,WF),
323		ground_wait_flags(WF).
324
325		:- block not_is_sequence_wf(-,?,?).
326		not_is_sequence_wf(FF,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
327		% we do not need the Range; this means we can match more closures (e.g., lambda)
328		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
329		not_is_sequence_domain(FFDomain,WF).
330		not_is_sequence_wf(Seq,Range,WF) :-
331		expand_and_convert_to_avl_set_warn(Seq,AER,not_is_sequence_wf,'ARG /: seq(?)',WF),
332		!,
333		(is_avl_sequence(AER) -> is_not_avl_relation_over_range(AER,Range,WF)
334		; true).
335		not_is_sequence_wf(X,Type,WF) :- expand_custom_set_to_list_wf(X,EX,_Done,not_is_sequence_wf,WF),
336		not_is_sequence2(EX,[],Type,WF).
337
338		:- block not_is_sequence2(-,?,?,?).
339	?	not_is_sequence2([],IndexesSoFar,_,_WF) :- not_contiguous_set_of_indexes(IndexesSoFar).
340		not_is_sequence2([(int(Idx),X)|Tail],IndexesSoFar,Type,WF) :-
341		membership_test_wf(IndexesSoFar,int(Idx),MemRes,WF),
342	?	not_is_sequence3(MemRes,Idx,X,Tail,IndexesSoFar,Type,WF).
343
344		:- block not_is_sequence3(-,?,?,?,?,?,?).
345		not_is_sequence3(pred_true,_Idx,_X,_Tail,_IndexesSoFar,_Type,_WF).
346		not_is_sequence3(pred_false,Idx,_X,_Tail,_IndexesSoFar,_Type,_WF) :- nonvar(Idx),Idx<1,!.
347		not_is_sequence3(pred_false,Idx,X,Tail,IndexesSoFar,Type,WF) :-
348		membership_test_wf(Type,X,MemRes,WF),
349	?	not_is_sequence4(MemRes,Idx,Tail,IndexesSoFar,Type,WF).
350
351		:- block not_is_sequence4(-,?,?,?,?,?).
352		not_is_sequence4(pred_false,_Idx,_Tail,_IndexesSoFar,_Type,_WF).
353		not_is_sequence4(pred_true,Idx,Tail,IndexesSoFar,Type,WF) :-
354	?	not_is_sequence2(Tail,[int(Idx)|IndexesSoFar],Type,WF).
355
356		not_contiguous_set_of_indexes(Indexes) :-
357	?	when(ground(Indexes),(sort(Indexes,Sorted),not_contiguous_set_of_indexes2(Sorted,1))).
358		not_contiguous_set_of_indexes2([int(N)|T],N1) :-
359	?	when(?=(N,N1),
360		((N \= N1) ; (N=N1, N2 is N1+1, not_contiguous_set_of_indexes2(T,N2)))).
361
362
363
364
365
366		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:not_is_non_empty_sequence([(int(1),int(22))],[int(22)]))).
367		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:not_is_non_empty_sequence([(int(1),int(2))],[int(22)]))).
368		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),R = [])).
369		:- assert_must_fail((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
370		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
371		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
372		R = [(int(2),fd(1,'Name')),(int(4),fd(2,'Name'))] )).
373		:- assert_must_fail((bsets_clp:not_is_non_empty_sequence(R,global_set('Name')),
374		R = [(int(1),fd(1,'Name')),(int(2),fd(1,'Name'))] )).
375		:- assert_must_succeed((bsets_clp:not_is_non_empty_sequence(R,[int(1),int(2)]),
376		R = [(int(1),int(2)),(int(2),int(3))] )).
377
378		% S /: seq1(T)
379		not_is_non_empty_sequence_wf(S,T,_) :- not_is_non_empty_sequence(S,T).
380		:- block not_is_non_empty_sequence(-,?).
381		not_is_non_empty_sequence([],_) :- !.
382		not_is_non_empty_sequence(X,Type) :-
383		empty_sequence(X) ; not_is_sequence(X,Type).
384
385
386
387		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([(int(1),int(22))],[int(22)],WF),WF)).
388		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
389		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
390		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:injective_sequence_wf([(int(2),int(22)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
391		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_sequence_wf([],global_set('Name'),WF),WF)).
392		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
393		kernel_waitflags:ground_det_wait_flag(WF), R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
394		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
395		ground_det_wait_flag(WF), R = [(int(1),fd(2,'Name'))] )).
396		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
397		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
398		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,global_set('Name'),WF),
399		ground_det_wait_flag(WF), R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
400		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_sequence_wf([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
401		global_set('Name'),WF),WF) ).
402		:- assert_must_succeed((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
403		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(1),int(1))] )).
404		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
405		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(3),int(1))] )).
406		:- assert_must_fail((bsets_clp:injective_sequence_wf(R,[int(1),int(2)],WF),
407		ground_det_wait_flag(WF), R = [(int(2),int(2)),(int(1),int(3))] )).
408
409
410
411		:- block injective_sequence_wf(-,-,?).
412		injective_sequence_wf(Seq,Type,WF) :- /* corresponds to iseq */
413		nonvar(Seq),
414		%expand_and_convert_to_avl_set_warn(Seq,AER,injective_sequence_wf_aux,'ARG : iseq(?)',WF),
415		Seq=avl_set(AER),
416		!,
417		is_avl_sequence(AER),
418		is_injective_avl_relation(AER,_ExactRange), % Should we check _ExactRange <: Type ??
419		is_avl_relation_over_range(AER,Type,WF).
420		injective_sequence_wf(Seq,Type,WF) :-
421		cardinality_as_int_for_wf(Type,MaxCard),
422		custom_explicit_sets:blocking_nr_iseq(MaxCard,ISeqSize),
423		block_get_wait_flag(ISeqSize,injective_sequence_wf,WF,LWF),
424		injective_sequence_wf_aux(Seq,Type,MaxCard,WF,LWF).
425
426		:- block injective_sequence_wf_aux(-,?,?,?,-).
427		injective_sequence_wf_aux(Seq,Type,_,WF,_) :- /* corresponds to iseq */
428		nonvar(Seq),
429		expand_and_convert_to_avl_set_warn(Seq,AER,injective_sequence_wf_aux,'ARG : iseq(?)',WF),!,
430		%Seq=avl_set(AER),
431		!,
432		is_avl_sequence(AER),
433		is_injective_avl_relation(AER,_ExactRange), % Should we check _ExactRange <: Type ??
434		is_avl_relation_over_range(AER,Type,WF).
435		injective_sequence_wf_aux(Seq,Type,MaxCard,WF,LWF) :-
436		expand_custom_set_to_list_wf(Seq,ESeq,_,injective_sequence_wf,WF),
437		is_sequence_wf(ESeq,Type,WF),
438	?	injective_sequence2(ESeq,0,[],Type,WF,MaxCard,LWF).
439
440		:- block injective_sequence2(-,?,?,?,?,?,-),injective_sequence2(-,?,?,?,?,-,?).
441		injective_sequence2([],_,_,_Type,_WF,_MaxCard,_LWF).
442		injective_sequence2([(int(Index),X)|Tail],CardSoFar,SoFar,Type,WF,MaxCard,LWF) :-
443		(number(MaxCard) -> CardSoFar< MaxCard, %less_than_equal_direct(Index,MaxCard) % does not enumerate index
444		in_nat_range_wf(int(Index),int(0),int(MaxCard),WF) % ensures the index gets enumerated, see test 1914, x:iseq(50001..50002) & y:1..100005 & SIGMA(yy).(yy:dom(x)|x(yy)) = y & y>50002
445		; true),
446		check_element_of_wf(X,Type,WF),
447		not_element_of_wf(X,SoFar,WF),
448		add_new_element_wf(X,SoFar,SoFar2,WF),
449		C1 is CardSoFar+1,
450		(C1 == MaxCard -> Tail=[] ; true),
451	?	injective_sequence2(Tail,C1,SoFar2,Type,WF,MaxCard,LWF).
452
453
454		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_injective_sequence([(int(1),int(22))],[int(22)],WF),WF)).
455		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_injective_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
456		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_injective_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
457		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_injective_sequence([(int(2),int(22)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
458		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),_WF),R = [])).
459		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
460		ground_det_wait_flag(WF),
461		R = [(int(2),fd(1,'Name')),(int(1),fd(2,'Name'))] )).
462		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
463		ground_det_wait_flag(WF),
464		R = [(int(1),fd(2,'Name'))] )).
465		:- assert_must_fail((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
466		ground_det_wait_flag(WF),
467		R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
468		:- assert_must_fail((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
469		ground_det_wait_flag(WF),
470		R = [(int(2),int(2)),(int(1),int(1))] )).
471		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,global_set('Name'),WF),
472		ground_det_wait_flag(WF),
473		R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
474		:- assert_must_succeed((bsets_clp:not_injective_sequence([(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))],
475		global_set('Name'),WF),
476		ground_det_wait_flag(WF) )).
477		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
478		ground_det_wait_flag(WF),
479		R = [(int(2),int(2)),(int(3),int(1))] )).
480		:- assert_must_succeed((bsets_clp:not_injective_sequence(R,[int(1),int(2)],WF),
481		ground_det_wait_flag(WF),
482		R = [(int(2),int(2)),(int(1),int(3))] )).
483		:- block not_injective_sequence(-,?,?), not_injective_sequence(?,-,?).
484		not_injective_sequence(Seq,_,_) :- Seq==[],!,fail.
485		not_injective_sequence(Seq,Type,WF) :- nonvar(Seq),
486		expand_and_convert_to_avl_set_warn(Seq,AER,not_injective_sequence,'ARG /: iseq(?)',WF),!,
487		(\+ is_avl_sequence(AER) -> true
488		; is_injective_avl_relation(AER,ExactRange) -> not_subset_of_wf(ExactRange,Type,WF)
489		; true).
490		not_injective_sequence(Seq,Type,WF) :- /* corresponds to Iseq */
491		%get_middle_wait_flag(not_injective_sequence,WF,LWF),
492		ground_value_check(Seq,SV),
493		not_injective_sequence1(Seq,Type,WF,SV).
494		:- block not_injective_sequence1(?,?,?,-).
495		not_injective_sequence1(Seq,Type,WF,_) :-
496		expand_custom_set_to_list_wf(Seq,ESeq,_,not_injective_sequence1,WF),
497		(not_is_sequence_wf(ESeq,Type,WF)
498		; /* CHOICE POINT !! */
499		(is_sequence_wf(ESeq,Type,WF),not_injective_sequence2(ESeq,[],Type,WF))).
500		:- block not_injective_sequence2(-,?,?,?).
501		not_injective_sequence2([(int(_),X)|Tail],SoFar,Type,WF) :-
502		membership_test_wf(SoFar,X,MemRes,WF),
503		not_injective_sequence3(MemRes,X,Tail,SoFar,Type,WF).
504
505		:- block not_injective_sequence3(-,?,?,?,?,?).
506		not_injective_sequence3(pred_true,_X,_Tail,_SoFar,_Type,_WF).
507		not_injective_sequence3(pred_false,X,Tail,SoFar,Type,WF) :-
508		add_new_element_wf(X,SoFar,SoFar2,WF),
509		not_injective_sequence2(Tail,SoFar2,Type,WF).
510
511		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(1),int(22))],[int(22)],WF),WF)).
512		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
513		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
514		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_non_empty_injective_sequence([(int(2),int(33)),(int(1),int(33))],[int(44),int(33),int(22)],WF),WF)).
515		:- assert_must_succeed((bsets_clp:not_non_empty_injective_sequence(R,global_set('Name'),WF),
516		ground_det_wait_flag(WF), R = [])).
517		:- assert_must_fail((bsets_clp:not_non_empty_injective_sequence(R,global_set('Name'),WF),
518		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
519		:- assert_must_succeed((bsets_clp:not_non_empty_injective_sequence(R,[int(1),int(2)],WF),
520		ground_det_wait_flag(WF), R = [(int(2),int(2)),(int(1),int(3))] )).
521
522		:- block not_non_empty_injective_sequence(-,?,?).
523		not_non_empty_injective_sequence([],_Type,_WF) :- !.
524		not_non_empty_injective_sequence(X,Type,WF) :-
525		empty_sequence(X) ; not_injective_sequence(X,Type,WF).
526
527
528		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_non_empty_sequence([(int(1),int(22))],[int(22)],WF),WF)).
529		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective_non_empty_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33),int(44)],WF),WF)).
530		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_non_empty_sequence([(int(2),int(33)),(int(1),int(23))],[int(22),int(33),int(44)],WF),WF)).
531		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:injective_non_empty_sequence([(int(2),int(44)),(int(1),int(44))],[int(22),int(33),int(44)],WF),WF)).
532		:- assert_must_fail((bsets_clp:injective_non_empty_sequence(R,global_set('Name'),WF),
533		ground_det_wait_flag(WF),R = [])).
534		:- assert_must_succeed((bsets_clp:injective_non_empty_sequence(R,global_set('Name'),WF),
535		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
536		:- block injective_non_empty_sequence(-,-,?). /* corresponds to iseq1 */
537		injective_non_empty_sequence(A,Type,WF) :- nonvar(A),A=avl_set(AS), !,
538		injective_sequence_wf(avl_set(AS),Type,WF),is_non_empty_explicit_set_wf(avl_set(AS),WF).
539		injective_non_empty_sequence(Seq,Type,WF) :-
540		((nonvar(Seq),Seq=closure(_,_,_)) -> try_expand_custom_set_wf(Seq,ESeq,injective_non_empty_sequence,WF) ; ESeq=Seq),
541		injective_sequence_wf(ESeq,Type,WF),not_empty_sequence_wf(ESeq,WF).
542
543		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:finite_non_empty_sequence([(int(1),int(22))],[int(22)],WF),WF)).
544		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:finite_non_empty_sequence([(int(1),int(33)),(int(2),int(33))],[int(22),int(33)],WF),WF)).
545		:- assert_must_fail((bsets_clp:finite_non_empty_sequence(R,global_set('Name'),WF),ground_det_wait_flag(WF),ground_det_wait_flag(WF),R = [])).
546		:- assert_must_succeed((bsets_clp:finite_non_empty_sequence(R,global_set('Name'),WF),
547		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
548		:- block finite_non_empty_sequence(-,?,?).
549		finite_non_empty_sequence(Seq,Type,WF) :- /* corresponds to Seq1 */
550		is_sequence_wf_ex(Seq,Type,WF,ESeq),
551		(var(ESeq) -> not_empty_sequence_wf(Seq,WF) ; not_empty_sequence_wf(ESeq,WF)).
552
553
554		:- block test_finite_non_empty_sequence(-,?,-,?).
555		test_finite_non_empty_sequence(Seq,_Type,Res,_WF) :-
556		Seq == [],!, Res=pred_false.
557		test_finite_non_empty_sequence(Seq,Type,Res,WF) :- var(Res),!,
558		ground_value_check(Seq,GrSeq),
559	?	test_finite_non_empty_sequence2(Res,Seq,Type,GrSeq,WF). % will trigger and enumerate Res below
560		% Note: we cannot rely on Res being enumerated; e.g., in case a WD error occurs
561		test_finite_non_empty_sequence(Seq,Type,Res,WF) :-
562		test_finite_non_empty_sequence2(Res,Seq,Type,_,WF).
563
564		% TODO: improve to incrementally check if something is a sequence
565		:- block test_finite_non_empty_sequence2(-,?,?,-,?).
566		test_finite_non_empty_sequence2(pred_true,Seq,Type,_,WF) :-
567		finite_non_empty_sequence(Seq,Type,WF).
568		test_finite_non_empty_sequence2(pred_false,Seq,Type,_,WF) :-
569		not_is_non_empty_sequence_wf(Seq,Type,WF).
570
571
572
573		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:permutation_sequence_wf([(int(1),int(22))],[int(22)],WF),WF)).
574		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:permutation_sequence_wf([(int(2),int(33)),(int(1),int(22))],[int(22),int(33)],WF),WF)).
575		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:permutation_sequence_wf([(int(2),int(33)),(int(1),int(23))],[int(23),int(33),int(44)],WF),WF)).
576		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:permutation_sequence_wf([(int(2),int(44)),(int(1),int(44))],[int(44)],WF),WF)).
577		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1)],WF),
578		ground_det_wait_flag(WF),R = [(int(1),int(1))] )).
579		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
580		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(2))] )).
581		:- assert_must_succeed((bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
582		ground_det_wait_flag(WF),R = [(int(1),int(2)),(int(2),int(1))] )).
583		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[pred_true /* bool_true */,pred_false /* bool_false */],WF), kernel_waitflags:ground_wait_flags(WF), nonvar(R),
584		R = [(int(1),pred_false /* bool_false */),(int(2),pred_true /* bool_true */)] )).
585		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1)],WF),
586		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(1))] )).
587		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),ground_det_wait_flag(WF),R = [])).
588		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
589		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
590		:- assert_must_succeed((bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
591		ground_det_wait_flag(WF),
592		kernel_objects:equal_object(R,[(int(1),fd(1,'Name')),(int(3),fd(2,'Name')),(int(2),fd(3,'Name'))]) )).
593		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(1),int(2)],WF),
594		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(3))] )).
595		:- assert_must_fail((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,global_set('Name'),WF),
596		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
597		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:permutation_sequence_wf(R,[int(4),int(3),int(2),int(1)],WF),
598		ground_det_wait_flag(WF), R=[(int(1),int(1)),(int(2),int(2)),(int(3),int(3)),(int(4),int(4))])).
599
600		:- block permutation_sequence_wf(-,-,?).
601		permutation_sequence_wf(SeqFF,Type,WF) :- nonvar(SeqFF),
602		custom_explicit_sets:dom_range_for_specific_closure(SeqFF,FFDomain,FFRange,function(bijection),WF),!,
603		equal_object_wf(FFRange,Type,permutation_sequence_wf_1,WF),
604		is_sequence_domain(FFDomain,WF).
605		permutation_sequence_wf(Seq,Type,WF) :-
606		expand_and_convert_to_avl_set_warn(Seq,AER,permutation_sequence_wf,'ARG : perm(?)',WF),!,
607		is_avl_sequence(AER),
608		is_injective_avl_relation(AER,Range),
609		kernel_objects:equal_object_wf(Range,Type,permutation_sequence_wf_2,WF).
610		permutation_sequence_wf(Seq,Type,WF) :-
611		try_expand_custom_set_wf(Seq,ESeq,permutation_sequence_wf,WF),
612		cardinality_as_int_wf(Type,int(Card),WF),
613		when(nonvar(Card), (setup_sequence_wf(Card,SkelSeq,perm,WF),
614		CardGround=true,
615		kernel_objects:equal_object_wf(SkelSeq,ESeq,permutation_sequence_wf_3,WF))),
616		%injective_sequence_wf(ESeq,Type,WF,LWF),
617		surjective_iseq_0(SkelSeq,ESeq,Type,WF,Card,CardGround).
618		% quick_all_different_range(ESeq,[],Type,WF).
619
620		:- block surjective_iseq_0(-,-,?,?,?,-).
621		surjective_iseq_0(SkelSeq,_ESeq,Type,WF,_Card,Ground) :-
622		nonvar(Ground),
623		nonvar(SkelSeq),
624		preference(use_clpfd_solver,true), % try and use an optimized version calling global_cardinality in CLPFD module
625	?	get_global_cardinality_list(Type,YType,GCL,_,WF),
626		% this dramatically reduces runtime for NQueens40_perm; maybe we should do this only when necessary, i.e., when surjective_iseq blocks on PreviousRemoveDone
627		% check why it slows down SortByPermutation_v2
628		!,
629		global_cardinality_range(SkelSeq,[],YType,GCL,WF).
630		surjective_iseq_0(_,ESeq,Type,WF,Card,_) :-
631		%quick_propagate_range(ESeq,Type,WF), % ensure that we propagate type information to all elements; p:perm(5..20) & p(10)=21 will fail straightaway (surjective_iseq will block);
632		% but this slows down EulerWay.mch ; maybe because it sets up enumerators ? TO DO: investigate
633		surjective_iseq(ESeq,Type,WF,Card).
634
635		%:- use_module(clpfd_interface,[clpfd_alldifferent/1]).
636		% collect range and then call CLPFD global_cardinality using GCL (Global Cardinality List Ki-Vi)
637		:- use_module(library(clpfd), [global_cardinality/3]).
638		:- block global_cardinality_range(-,?,?,?,?).
639		global_cardinality_range([],Acc,_Type,GCL,WF) :-
640		global_cardinality(Acc,GCL,[consistency(value)]),
641		add_fd_variables_for_labeling(Acc,WF). % this is needed for efficiency for NQueens40_perm !!
642		global_cardinality_range([(_,Y)|T],Acc,Type,GCL,WF) :-
643		get_simple_fd_value(Type,Y,FDYVAL),
644		global_cardinality_range(T,[FDYVAL|Acc],Type,GCL,WF).
645
646
647		:- use_module(library(avl), [avl_domain/2]).
648		:- use_module(b_global_sets,[all_elements_of_type_wf/3,b_integer_set/1]).
649		% try and convert a B set into a list suitable for calling clpfd:global_cardinality
650		% get_global_cardinality_list(avl_set(A) % TO DO: extend to integer_lists
651		get_global_cardinality_list(global_set(G),Type,GCL,list,WF) :- !,
652		all_elements_of_type_wf(G,Values,WF), % can only work for finite sets, not for STRING, NATURAL, REAL, ...
653		(b_integer_set(G) -> Type=integer ; Type=global(G)),
654		findall(X-1,(get_simple_fd_value(Type,VV,X),member(VV,Values)),GCL).
655		get_global_cardinality_list(avl_set(A),Type,GCL,list,_WF) :- !,
656		A = node(TopValue,_True,_,_,_),
657	?	get_simple_fd_value(Type,TopValue,_), % we have CLPFD values
658		avl_domain(A,Values),
659		findall(X-1,(get_simple_fd_value(Type,VV,X),member(VV,Values)),GCL).
660		get_global_cardinality_list(Set,integer,GCL,interval(L1,U1),_WF) :- nonvar(Set),
661		is_interval_closure_or_integerset(Set,L1,U1), number(L1),number(U1),
662		global_cardinality_list_interval(L1,U1,GCL).
663
664		global_cardinality_list_interval(From,To,[]) :- From>To, !.
665		global_cardinality_list_interval(From,To,[From-1|T]) :-
666		F1 is From+1, global_cardinality_list_interval(F1,To,T).
667
668		%try_get_simple_fd_value(Type,V,Val) :- nonvar(V),get_simple_fd_value(Type,V,Val).
669		get_simple_fd_value(integer,int(X),X).
670		get_simple_fd_value(global(T),fd(X,T),X).
671		% try_get_simple_fd_value(pred_false,0). try_get_simple_fd_value(pred_true,1). ??
672		% TO DO: maybe also treat pairs ? but we need complete values; see module clpfd_lists !
673
674		setup_sequence_wf(0,R,_,_) :- !, R=[].
675		setup_sequence_wf(Card,_,PP,WF) :- \+ number(Card), !,
676		add_error_wf(infinite_sequence,'Cannot generate infinite sequence for', PP,unkown,WF). % triggered in test 1979
677		setup_sequence_wf(Card,[(int(1),_)|T] ,_PP,_WF) :- Card>0, C1 is Card-1,
678		setup_sequence(C1,T,2).
679		setup_sequence(0,R,_) :- !, R=[].
680		setup_sequence(Card,[(int(Nr),_)|T], Nr ) :- Card>0, C1 is Card-1,
681		N1 is Nr+1,
682		setup_sequence(C1,T,N1).
683
684		:- block surjective_iseq(?,?,?,-),surjective_iseq(?,-,?,?), surjective_iseq(-,?,?,?).
685		surjective_iseq(avl_set(S),Type,WF,Done) :-
686		expand_custom_set_wf(avl_set(S),ES,surjective_iseq,WF),
687		surjective_iseq(ES,Type,WF,Done).
688		surjective_iseq(closure(P,T,B),Type,WF,Done) :-
689		expand_custom_set_wf(closure(P,T,B),ES,surjective_iseq,WF),
690		surjective_iseq(ES,Type,WF,Done).
691		% no case for global_set: cannot be a relation
692		surjective_iseq([],T,WF,_) :- empty_set_wf(T,WF).
693		surjective_iseq([(int(_Nr),El)|Tail],Type,WF,_PreviousRemoveDone) :-
694		remove_element_wf(El,Type,NType,WF,Done),
695		surjective_iseq(Tail,NType,WF,Done).
696		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_permutation_sequence([(int(1),int(22))],[int(22)],WF),WF)).
697		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(33)),(int(1),int(22))],[int(22),int(33)],WF),WF)).
698		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(33)),(int(1),int(23))],[int(23),int(33),int(44)],WF),WF)).
699		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_permutation_sequence([(int(2),int(44)),(int(1),int(44))],[int(44)],WF),WF)).
700		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1)],WF),
701		ground_det_wait_flag(WF),R = [(int(1),int(1))] )).
702		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
703		ground_det_wait_flag(WF),R = [(int(2),int(2)),(int(1),int(1))] )).
704		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
705		ground_det_wait_flag(WF),R = [(int(1),int(2)),(int(2),int(1))] )).
706		:- assert_must_fail((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
707		ground_det_wait_flag(WF), R = [(int(1),fd(1,'Name')),(int(3),fd(2,'Name')),(int(2),fd(3,'Name'))] )).
708		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,[int(1)],WF),
709		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(1))] )).
710		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),_WF),R = [])).
711		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
712		ground_det_wait_flag(WF),R = [(int(1),fd(1,'Name')),(int(2),fd(2,'Name'))] )).
713		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,[int(1),int(2)],WF),
714		ground_det_wait_flag(WF),R = [(int(1),int(1)),(int(2),int(3))] )).
715		:- assert_must_succeed((bsets_clp:not_permutation_sequence(R,global_set('Name'),WF),
716		ground_det_wait_flag(WF),R = [(int(2),fd(1,'Name')),(int(1),fd(1,'Name'))] )).
717		:- block not_permutation_sequence(-,?,?).
718		not_permutation_sequence(SeqFF,Type,WF) :- nonvar(SeqFF),
719		custom_explicit_sets:dom_range_for_specific_closure(SeqFF,FFDomain,FFRange,function(bijection),WF),!,
720		equality_objects_wf(FFRange,Type,Result,WF),
721		when(nonvar(Result),(Result=pred_false -> true ; not_is_sequence_domain(FFDomain,WF))).
722		not_permutation_sequence(Seq,Type,WF) :-
723		ground_value_check(Seq,SV),
724	?	not_permutation_sequence1(Seq,Type,SV,WF).
725		:- block not_permutation_sequence1(?,-,?,?), not_permutation_sequence1(?,?,-,?).
726		not_permutation_sequence1(avl_set(A),Type,_,WF) :- is_ground_set(Type), !, Seq=avl_set(A),
727		if(not_injective_sequence(Seq,Type,WF),
728		true, % no backtracking required; we could even use regular if with ->
729		not_surj_avl(Seq,Type,WF)
730		).
731		not_permutation_sequence1(avl_set(A),Type,_,WF) :- !, Seq=avl_set(A),
732		(not_injective_sequence(Seq,Type,WF)
733		; injective_sequence_wf(Seq,Type,WF),
734		not_surj_avl(Seq,Type,WF)).
735		not_permutation_sequence1(Seq,Type,_,WF) :-
736		expand_custom_set_to_list_wf(Seq,ESeq,Done,not_permutation_sequence1,WF),
737	?	not_permutation_sequence2(ESeq,Type,WF,Done).
738
739		not_surj_avl(Seq,Type,WF) :- range_wf(Seq,Range,WF),
740		not_equal_object_wf(Range,Type,WF). % TO DO: one could even just check cardinality as Seq is inj
741		%expand_custom_set_to_list_wf(Seq,ESeq,_,not_permutation_sequence1,WF),
742		% not_surjective_seq(ESeq,Type,WF).
743		% check if it is a ground set that cannot be instantiated
744		is_ground_set(V) :- var(V),!,fail.
745		is_ground_set(avl_set(_)).
746		is_ground_set(global_set(_)).
747		is_ground_set([]).
748
749		% here we could have a choice point in WF0
750		:- block not_permutation_sequence2(?,?,?,-).
751		not_permutation_sequence2(Seq,Type,WF,_) :- not_injective_sequence(Seq,Type,WF).
752		not_permutation_sequence2(Seq,Type,WF,_) :-
753		injective_sequence_wf(Seq,Type,WF), not_surjective_seq(Seq,Type,WF).
754
755		:- block not_surjective_seq(-,?,?).
756		not_surjective_seq([],T,WF) :- not_empty_set_wf(T,WF).
757		not_surjective_seq([(int(_),El)|Tail],Type,WF) :-
758		delete_element_wf(El,Type,NType,WF),
759		not_surjective_seq(Tail,NType,WF).
760
761		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(1),int(22))],int(1),_WF))).
762		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22))],int(2),_WF))).
763		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22))],int(3),_WF))).
764		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:size_of_sequence([(int(2),int(22)),(int(1),int(22)),(int(3),int(33))],int(3),_WF))).
765		:- assert_must_succeed((bsets_clp:size_of_sequence(X,R,_WF),
766		X = [(int(1),int(2)),(int(2),int(1))],
767		R = int(2))).
768		:- assert_must_succeed((preferences:preference(use_clpfd_solver,false) -> true
769		; preferences:preference(use_smt_mode,false) -> true
770		; bsets_clp:size_of_sequence(X,R,_WF), R=int(RI),
771		clpfd_interface:clpfd_geq2(RI,2,_), nonvar(X), X = [(I1,_),(I2,_)|T],
772		I1==int(1), I2==int(2), T=[], RI==2 )).
773		:- assert_must_succeed((bsets_clp:size_of_sequence(X,R,_WF),X = [(int(1),_),(int(2),_)],R = int(2))).
774		:- assert_must_succeed((bsets_clp:size_of_sequence(X,_R,_WF),X =[(int(1),_),(int(2),_)] )).
775		:- assert_must_succeed_any((bsets_clp:size_of_sequence(X,int(2),_WF),nonvar(X),X=[_|Y],nonvar(Y),Y=[_|Z],Z==[])).
776		:- assert_must_succeed((bsets_clp:size_of_sequence([],int(0),_WF))).
777		:- assert_must_succeed((bsets_clp:size_of_sequence([],int(0),_WF))).
778		:- assert_must_succeed((bsets_clp:size_of_sequence([(int(1),int(4))],int(1),_WF))).
779		:- assert_must_succeed((bsets_clp:size_of_sequence([],_,_WF))).
780		:- assert_must_fail((bsets_clp:size_of_sequence(X,int(1),_WF),
781		X = [(int(1),_),(int(2),_)|_])).
782		:- block size_of_sequence(-,-,?).
783	?	size_of_sequence(Seq,int(Res),WF) :- size_of_sequence1(Seq,Res,WF),
784		set_up_sequence_skel(Seq,Res,WF).
785
786		% setup sequence skeleton if we have some CLPFD bounds information about the size
787		% currently still quite limited: only sets up if sequence is a variable; + does the setup only once
788		:- use_module(library(clpfd), [(#<=>)/2]).
789		:- use_module(clpfd_interface,[clpfd_domain/3]).
790		set_up_sequence_skel(Seq,Res,WF) :-
791		var(Seq), % to do: also deal with cases when Seq partially instantiated
792		var(Res),
793		preferences:preference(use_clpfd_solver,true),
794		!,
795		clpfd_interface:clpfd_geq2(Res,0,_), % assert that size must not be negative
796		clpfd_interface:try_post_constraint((Res#>0) #<=> Trigger), % generate reified trigger for when we can instantiate Seq
797		set_up_sequence_skel_aux(Seq,Res,Trigger,WF).
798		set_up_sequence_skel(_,_,_). % TO DO: check if Size interval shrinks
799		:- block set_up_sequence_skel_aux(-,?,-,?).
800		set_up_sequence_skel_aux(Seq,_Res,_Trigger,_WF) :-
801		nonvar(Seq),
802		!. % to do: also deal with cases when Seq partially instantiated
803		set_up_sequence_skel_aux(Seq,Res,_Trigger,_WF) :-
804		(number(Res) ; preferences:preference(use_smt_mode,true)),
805		!,
806		gen_seq_for_res(Res,Seq).
807		set_up_sequence_skel_aux(Seq,Res,_Trigger,WF) :-
808		get_large_finite_wait_flag(set_up_sequence_skel,WF,LWF), % delay, avoid costly unification with partially instantaited list skeleton; TO DO: in future we may use the kernel_cardinality attribute instead
809		when((nonvar(LWF) ; nonvar(Seq) ; nonvar(Res)), (nonvar(Seq) -> true ; gen_seq_for_res(Res,Seq))).
810
811		gen_seq_for_res(Res,Seq) :-
812		clpfd_domain(Res,FDLow,FDUp), % FDLow could also be 0
813		gen_sequence_skeleton(1,FDLow,FDUp,S),
814		Seq=S.
815		gen_sequence_skeleton(N,M,FDUp,S) :- N>M,!,(FDUp==M -> S=[] ; true).
816		gen_sequence_skeleton(N,Max,FDUp,[(int(N),_)|T]) :-
817		N1 is N+1,
818		gen_sequence_skeleton(N1,Max,FDUp,T).
819
820		:- block size_of_sequence1(-,-,?).
821		size_of_sequence1(Seq,ResInt,WF) :-
822		nonvar(Seq),is_custom_explicit_set_nonvar(Seq),
823		size_of_custom_explicit_set(Seq,Size,WF),!,
824	?	equal_object_wf(Size,int(ResInt),size_of_sequence1,WF).
825		/* TO DO: CHECK BELOW: would it not be better to use cardinality ?? */
826		/*
827		size_of_sequence1(Seq,Size,WF) :- !,kernel_cardinality_attr:finite_cardinality_as_int_wf(Seq,int(Size),WF), check_indexes(Seq,Size).
828
829		construct_interval_closure(1,Size,Domain),
830		total_function_wf(FF,Domain,Range,_WF)
831		% we could also call total_function 1..Size --> _RangeType; would setup domain ?
832		:- block check_indexes(-,?).
833		check_indexes([],_) :- !.
834		check_indexes([(int(X),_)|T],Size) :- !,
835		less_than_equal_direct(X,Size), check_indexes(T,Size).
836		check_indexes(_,_).
837		*/
838	?	size_of_sequence1(Seq,Size,_WF) :- Size==0,!, empty_sequence(Seq).
839		size_of_sequence1(Seq,Size,WF) :-
840		expand_custom_set_to_list_wf(Seq,ESeq,_,size_of_sequence1,WF),
841	?	(var(ESeq),nonvar(Size) -> size_of_var_seq(ESeqR,0,Size),
842		ESeqR=ESeq % unify after to do propagation in one go, without triggering coroutines inbetween
843	?	; size_of_seq2(ESeq,0,Size),
844		(var(Size),var(ESeq) -> less_than_equal_direct(0,Size) % propagate that Size is positive
845		; true)
846		).
847		/* small danger of expanding closure while still var !*/
848		:- block size_of_seq2(-,?,-).
849		size_of_seq2([],Size,Size).
850		size_of_seq2([I|Tail],SizeSoFar,Res) :-
851		S2 is SizeSoFar + 1,
852	?	check_index(I,Res), % don't instantiate I yet; allow other kernel_predicates to freely instantiate it
853		less_than_equal_direct(S2,Res),
854		%(ground(Res) -> safe_less_than_equal(size_of_seq2,S2,Res) ; true),
855	?	size_of_seq2(Tail,S2,Res).
856		size_of_var_seq([],Size,Size).
857		size_of_var_seq([(int(S2),_)|Tail],SizeSoFar,Res) :-
858		S2 is SizeSoFar + 1,safe_less_than_equal(size_of_var_seq,S2,Res),
859	?	(var(Tail) -> size_of_var_seq(Tail,S2,Res) ; size_of_seq2(Tail,S2,Res)).
860
861
862		:- block check_index(-,?).
863	?	check_index((I,_),Res) :- check_index1(I,Res).
864		:- block check_index1(-,?).
865	?	check_index1(int(Idx),Res) :- less_than_equal_direct(Idx,Res).
866
867		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(1),int(22))],[(int(2),int(22)),(int(1),int(33))],WF),WF)).
868		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[],[(int(1),int(33))],WF),WF)).
869		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(2),int(44)),(int(1),int(22))],[(int(1),int(33)),(int(3),int(44)),(int(2),int(22))],WF),WF)).
870		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:prepend_sequence(int(33),[(int(1),int(22))],[(int(1),int(22)),(int(2),int(33))],WF),WF)).
871		:- assert_must_succeed((bsets_clp:prepend_sequence(int(7),[],[(int(1),int(7))],_WF))).
872		:- assert_must_succeed((bsets_clp:prepend_sequence(int(7),X,R,_WF),
873		X = [(int(2),int(4)),(int(1),int(3))],
874		kernel_objects:equal_object(R,[(int(1),int(7)),(int(2),int(3)),(int(3),int(4))]))).
875		% code for insert_front operator: El -> Seq
876		:- block prepend_sequence(?,-,-,?).
877		prepend_sequence(El,Seq,Res,_WF) :- Seq==[],!,
878		equal_object_optimized([(int(1),El)],Res,prepend_sequence).
879		prepend_sequence(El,Seq,Res,WF) :- nonvar(Seq),is_custom_explicit_set(Seq,prepend_sequence),
880		prepend_custom_explicit_set(Seq,El,ERes),!,
881		equal_sequence(Res,ERes,WF).
882		prepend_sequence(El,Seq,Res,WF) :- nonvar(Res),is_custom_explicit_set(Res,prepend_sequence),
883		tail_sequence_custom_explicit_set(Res,First,Tail,unknown,WF),!,
884		equal_object_wf(El,First,prepend_sequence,WF),
885		equal_sequence(Seq,Tail,WF).
886		prepend_sequence(El,Seq,Res,WF) :-
887		equal_cons_wf(Res,(int(1),El),ShiftSeq,WF),
888		shift_seq_indexes(Seq,1,ShiftSeq,WF).
889
890		:- block shift_seq_indexes(-,-,?,?),shift_seq_indexes(-,?,-,?).
891		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :-
892		Offset == 0,!, equal_sequence(Seq,ShiftedSeq,WF).
893		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :- nonvar(Seq),!,
894		expand_custom_set_to_list_wf(Seq,ESeq,_,shift_seq_indexes,WF),
895		shift_seq_indexes2(ESeq,Offset,ShiftedSeq,WF,Done),
896		(Done == done
897		-> true
898		; % also propagate in the other way: TO DO: make a more efficient fine-grained two-ways propagation; maybe using CHR
899		NegOffset is -Offset,
900		expand_custom_set_to_list_wf(ShiftedSeq,ESeq1,_,shift_seq_indexes,WF),
901		shift_seq_indexes2(ESeq1,NegOffset,ESeq,WF,_)).
902		shift_seq_indexes(Seq,Offset,ShiftedSeq,WF) :- NegOffset is -Offset,
903		% compute in the other direction; TO DO: make a more efficient fine-grained two-ways propagation; maybe using CHR
904		expand_custom_set_to_list_wf(ShiftedSeq,ESeq,_,shift_seq_indexes,WF),
905		shift_seq_indexes2(ESeq,NegOffset,Seq,WF,Done),
906		(Done == done
907		-> true
908		; % also propagate in the original way:
909		expand_custom_set_to_list_wf(Seq,ESeq1,_,shift_seq_indexes,WF),
910		shift_seq_indexes2(ESeq1,Offset,ESeq,WF,_)).
911
912		:- block shift_seq_indexes2(-,?,?,?,?).
913	?	shift_seq_indexes2([],_,R,WF,Done) :- !, Done = done, empty_set_wf(R,WF).
914		shift_seq_indexes2([Pair|Tail],Offset,Res,WF,Done) :- !,
915		Pair = (int(N),El),
916	?	equal_cons_wf(Res,(int(NewN),El),ShiftTail,WF),
917		int_plus(int(N),int(Offset),int(NewN)),
918		shift_seq_indexes2(Tail,Offset,ShiftTail,WF,Done).
919		shift_seq_indexes2(Seq,Offset,Res,WF,Done) :-
920		add_internal_error('Unexpected set argument: ',shift_seq_indexes2(Seq,Offset,Res,WF,Done)), fail.
921
922		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([(int(1),int(22))],int(33),[(int(2),int(33)),(int(1),int(22))],WF),WF)).
923		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([],int(33),[(int(1),int(33))],WF),WF)).
924		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:append_sequence([(int(2),int(44)),(int(1),int(22))],int(33),[(int(1),int(22)),(int(3),int(33)),(int(2),int(44))],WF),WF)).
925		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:append_sequence([(int(1),int(22))],int(33),[(int(1),int(33)),(int(2),int(22))],WF),WF)).
926		:- assert_must_succeed((bsets_clp:append_sequence([],int(7),[(int(1),int(7))],_WF))).
927		:- assert_must_succeed((bsets_clp:append_sequence(X,int(7),R,_WF),
928		X = [(int(2),int(4)),(int(1),int(3))],
929		kernel_objects:equal_object(R,[(int(1),int(3)),(int(2),int(4)),(int(3),int(7))]))).
930
931		% code for the insert_tail operator Seq<-El
932		:- block append_sequence(-,?,-,?).
933		append_sequence(Seq,El,Res,_WF) :- Seq==[],!,
934		equal_object_optimized([(int(1),El)],Res,append_sequence).
935		append_sequence(Seq,El,Res,WF) :-
936		nonvar(Seq),is_custom_explicit_set_nonvar(Seq),
937		append_custom_explicit_set(Seq,El,ERes,WF),!,
938		equal_sequence(Res,ERes,WF).
939		append_sequence(Seq,El,Res,WF) :-
940		nonvar(Res),is_custom_explicit_set_nonvar(Res),
941		% we know result: deconstruct into last El and front Seq
942		front_sequence_custom_explicit_set(Res,Last,Front), !,
943		equal_object_wf(El,Last,append_sequence,WF),
944		equal_sequence(Seq,Front,WF).
945		append_sequence(Seq,El,Res,WF) :-
946		(var(Seq) -> size_of_sequence(Res,INewSize,WF), INewSize=int(NewSize) ; true),
947		equal_cons_wf(Res,(int(NewSize),El),ResT,WF),
948		append_sequence2(Seq,ResT,NewSize,WF).
949
950		:- block append_sequence2(-,?,-,?).
951		append_sequence2(Seq,ResT,_NewSize,WF) :- var(Seq),!,
952		equal_sequence(Seq,ResT,WF).
953		append_sequence2(Seq,ResT,NewSize,WF) :-
954		try_expand_custom_set_wf(Seq,ESeq,append_sequence2,WF),
955		equal_sequence(ESeq,ResT,WF),
956		size_of_sequence(ESeq,Size,WF),
957		int_plus(Size,int(1),int(NewSize)).
958
959		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:prefix_sequence([(int(1),int(22))],int(1),[(int(1),int(22))]))).
960		:- assert_must_succeed(exhaustive_kernel_succeed_check(bsets_clp:prefix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(2),[(int(1),int(11)),(int(2),int(22))]))).
961		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:prefix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(3),[(int(1),int(11)),(int(2),int(22))]))).
962		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(1),X),X = [(int(1),int(1))])).
963		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(0),[]),X = [(int(1),int(1))])).
964		:- assert_must_abort_wf((bsets_clp:prefix_sequence_wf(X,int(-1),_R,WF),X = [(int(1),int(1))]),WF).
965		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(2),Y),
966		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
967		kernel_objects:equal_object(Y,[(int(1),int(1)),(int(2),int(3))]) )).
968		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(1),Y),
969		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
970		kernel_objects:equal_object(Y,[(int(1),int(1))]) )).
971		:- assert_must_succeed((bsets_clp:prefix_sequence(X,int(3),Y),
972		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
973		kernel_objects:equal_object(Y,X) )).
974
975		prefix_sequence(Seq,N,R) :- init_wait_flags(WF,[prefix_sequence]),
976	?	prefix_sequence_wf(Seq,N,R,WF),
977	?	ground_wait_flags(WF).
978
979		% Prefix of a sequence (s /|\ n)
980		prefix_sequence_wf(Seq,int(Num),Res,WF) :-
981	?	prefix_sequence1(Seq,Num,Res,WF),
982	?	propagate_size(Res,Num,WF).
983
984		% the size of the result of (s /|\ n) is the number n
985		:- block propagate_size(-,-,?).
986		propagate_size(Res,Num,WF) :-
987		var(Res),!,
988		(Num<0 -> preferences:preference(disprover_mode,false) % don't do anything; we may want to generate WD error
989	?	; Num < 4 -> size_of_sequence(Res,int(Num),WF)
990		; Prio is 1+Num // 100,
991		get_wait_flag(Prio,propagate_size,WF,LWF), % avoid setting up very large skeletons too early
992		block_size_of_sequence(LWF,Res,int(Num),WF)
993		).
994		propagate_size(_,Num,_) :- number(Num), !. % no need to propagate
995		propagate_size(_,_Num,_) :- \+ preferences:preference(find_abort_values,false),
996		!. % do not propagate as we could prevent detection of WD errors below
997		propagate_size([],Num,_WF) :- !,
998		Num=0. % Note: this could prevent a wd-error being detected
999		propagate_size(avl_set(A),Num,WF) :- var(Num),
1000		% with partially instantated sets we get slowdowns (SimpleCSGGrammar2_SlowCLPFD)
1001		% TO DO: treat list skeletons
1002		!,
1003	?	size_of_sequence(avl_set(A),int(Num),WF). % Note: this could prevent a wd-error being detected
1004		propagate_size(_,_,_). % should we also propagate the other way around ?
1005
1006		:- block block_size_of_sequence(-,?,?,?).
1007		block_size_of_sequence(_,Seq,Size,WF) :- size_of_sequence(Seq,Size,WF).
1008
1009		:- block prefix_sequence1(-,?,?,?), prefix_sequence1(?,-,?,?).
1010		prefix_sequence1(_Seq,Num,Res,WF) :- Num==0,!, empty_set_wf(Res,WF).
1011		prefix_sequence1(_Seq,Num,_Res,WF) :- Num<0,!, % according to version 1.8.8 of Atelier-B manual Num must be in 0..size(_Seq)
1012		add_wd_error('negative index in prefix_sequence (/|\\)! ', Num,WF).
1013		prefix_sequence1(Seq,Num,Res,WF) :-
1014		is_custom_explicit_set(Seq,prefix),
1015		prefix_of_custom_explicit_set(Seq,Num,ERes,WF),!, % TO DO: check Num <= size(Seq)
1016		equal_object_wf(Res,ERes,prefix_sequence1,WF).
1017		prefix_sequence1(Seq,Num,Res,WF) :-
1018		expand_custom_set_to_list_wf(Seq,ESeq,_,prefix_sequence1,WF),
1019		unify_same_index_elements(Res,ESeq,WF),
1020		unify_same_index_elements(Seq,Res,WF),
1021	?	prefix_seq(ESeq,Num,0,Res,WF).
1022		:- block prefix_seq(-,?,?,?,?).
1023		prefix_seq([],Max,Sze,Res,WF) :-
1024		(less_than_direct(Sze,Max)
1025		-> add_wd_error('index larger than size of sequence in prefix_sequence (/|\\)! ', (Max,Sze),WF)
1026		; true),
1027		empty_set_wf(Res,WF).
1028		%(less_than(int(_Sze),int(_Max))
1029		% -> (print_message('Index bigger than sequence size in prefix_sequence (/|\\) !'),
1030		% print_message(Max))
1031		% /* in the AtelierB book this is allowed, in Wordsworth + AMN on web it is not ?? */
1032		% ; true).
1033		prefix_seq([(int(N),El)|Tail],Max,SizeSoFar,Res,WF) :-
1034	?	prefix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF).
1035		:- block prefix_seq2(-,?,?,?,?,?,?).
1036		prefix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF) :- % SizeSoFar is always ground
1037	?	(less_than_equal_direct(N,Max), equal_cons_wf(Res,(int(N),El),PTail,WF)
1038		;
1039		less_than_direct(Max,N), equal_object_wf(Res,PTail,prefix_seq2,WF)
1040		),
1041		( SizeSoFar<N -> NewSizeSoFar=N ; NewSizeSoFar = SizeSoFar ),
1042	?	prefix_seq(Tail,Max,NewSizeSoFar,PTail,WF).
1043
1044		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:suffix_sequence([(int(1),int(22))],int(0),[(int(1),int(22))],WF),ground_det_wait_flag(WF))).
1045		:- assert_must_succeed(exhaustive_kernel_succeed_check(bsets_clp:suffix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(1),[(int(1),int(22)),(int(2),int(33))],_WF))).
1046		:- assert_must_succeed(exhaustive_kernel_fail_check(bsets_clp:suffix_sequence([(int(2),int(22)),(int(3),int(33)),(int(1),int(11))],int(2),[(int(1),int(22)),(int(2),int(33))],_WF))).
1047		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(0),X,_WF),X = [(int(1),int(1))])).
1048		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(1),[],_WF),X = [(int(1),int(1))])).
1049		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(2),Y,_WF),
1050		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1051		kernel_objects:equal_object(Y,[(int(1),int(4))]) )).
1052		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(1),Y,_WF),
1053		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1054		kernel_objects:equal_object(Y,[(int(1),int(3)),(int(2),int(4))]) )).
1055		:- assert_must_succeed((bsets_clp:suffix_sequence(X,int(2),Y,_WF),
1056		X = [(int(2),int(3)),(int(3),int(4)),(int(1),int(1))],
1057		kernel_objects:equal_object(Y,[(int(1),int(4))]) )).
1058		:- assert_must_abort_wf(bsets_clp:suffix_sequence([(int(1),int(11)),(int(2),int(22))],int(-1),_R,WF),WF).
1059		:- assert_must_abort_wf(bsets_clp:suffix_sequence([(int(1),int(11)),(int(2),int(22))],int(3),_R,WF),WF).
1060
1061		% kernel_waitflags:assert_must_abort2_wf(bsets_clp:suffix_sequence([int(11),int(22)],int(-1),_R,WF),WF)
1062
1063		% suffix of a sequence (s \|/ n); also called restrict at tail (Atelier B) or Drop
1064		:- block suffix_sequence(-,?,?,?).
1065		suffix_sequence(Seq,int(Num),Res,WF) :-
1066	?	suffix_sequence1(Seq,Num,Res,WF).
1067		:- block suffix_sequence1(?,-,?,?).
1068		suffix_sequence1(Seq,Num,Res,WF) :- Num==0, !, equal_object_wf(Res,Seq,suffix_sequence1_1,WF).
1069		suffix_sequence1(_Seq,Num,_Res,WF) :- Num<0, !, add_wd_error('negative index in suffix_sequence (\\|/)! ', Num,WF).
1070		suffix_sequence1(Seq,Num,Res,WF) :- is_custom_explicit_set(Seq,suffix),
1071		suffix_of_custom_explicit_set(Seq,Num,ERes,WF),!,
1072		equal_object_wf(Res,ERes,suffix_sequence1_2,WF).
1073		suffix_sequence1(Seq,Num,Res,WF) :-
1074	?	expand_custom_set_to_list_wf(Seq,ESeq,_,suffix_sequence,WF), suffix_seq(ESeq,Num,0,Res,WF).
1075		:- block suffix_seq(-,?,?,?,?).
1076		suffix_seq([],Max,Sze,Res,WF) :-
1077		(less_than_direct(Sze,Max)
1078		-> add_wd_error('index larger than size of sequence in suffix_sequence (\\|/)! ', '>'(Max,Sze),WF)
1079		; true), empty_set_wf(Res,WF).
1080		suffix_seq([(int(N),El)|Tail],Max,SizeSoFar,Res,WF) :-
1081	?	suffix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF).
1082		:- block suffix_seq2(-,?,?,?,?,?,?).
1083		suffix_seq2(N,El,Tail,Max,SizeSoFar,Res,WF) :-
1084		(less_than_equal_direct(N,Max), equal_object_wf(Res,PTail,suffix_seq2,WF)
1085		;
1086		less_than_direct(Max,N),int_minus(int(N),int(Max),int(NN)),
1087		equal_cons_wf(Res,(int(NN),El),PTail,WF)
1088		),
1089		(N>SizeSoFar -> (NewSizeSoFar=N)
1090		; (NewSizeSoFar = SizeSoFar)),
1091	?	suffix_seq(Tail,Max,NewSizeSoFar,PTail,WF).
1092
1093
1094
1095		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:concat_sequence([],[(int(1),int(33))],[(int(1),int(33))],WF),WF)).
1096		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:concat_sequence([(int(1),int(22)),(int(2),int(33))],[(int(1),int(33)),(int(2),int(44))],[(int(2),int(33)),(int(3),int(33)),(int(1),int(22)),(int(4),int(44))],WF),WF)). % not wfdet because of pending label_el_nr from clpfd_lists
1097		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:concat_sequence([(int(1),int(22))],[(int(1),int(33))],[(int(2),int(33)),(int(1),int(22))],WF),WF)).
1098		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:concat_sequence([(int(1),int(22))],[(int(1),int(33))],[(int(2),int(22)),(int(1),int(33))],WF),WF)).
1099		:- assert_must_succeed((bsets_clp:concat_sequence([],X,Y,_WF),
1100		X = [(int(1),int(1))], Y==X)).
1101		:- assert_must_succeed((bsets_clp:concat_sequence(X,[],Y,_WF), X = [(int(1),int(1))], Y==X)).
1102		:- assert_must_succeed((bsets_clp:concat_sequence([(int(1),int(1))],[],Y,_WF), Y==[(int(1),int(1))])).
1103		:- assert_must_succeed((bsets_clp:concat_sequence(X,X,Y,_WF),
1104		X = [(int(1),int(1))], kernel_objects:equal_object(Y,[(int(1),int(1)),(int(2),int(1))]))).
1105		:- assert_must_succeed((bsets_clp:concat_sequence(X,X,Y,_WF),
1106		X = [(int(2),int(88)),(int(1),int(77))],
1107		kernel_objects:equal_object(Y,[(int(1),int(77)),(int(2),int(88)),(int(3),int(77)),(int(4),int(88))]))).
1108
1109		:- block /* concat_sequence(-,-,?,?), */
1110		concat_sequence(?,-,-,?), concat_sequence(-,?,-,?).
1111		concat_sequence(S1,S2,Res,WF) :- Res==[],!, empty_set_wf(S1,WF), empty_set_wf(S2,WF).
1112		concat_sequence(S1,S2,Res,WF) :-
1113		(var(S1),var(S2) -> get_wait_flag(2,concat,WF,LWF) % we have at least two solutions; TODO maybe use cardinality as wait_flag?
1114		; LWF=1),
1115	?	concat_sequence2(LWF,S1,S2,Res,WF).
1116
1117		:- block concat_sequence2(-,?,-,?,?), concat_sequence2(-,-,?,?,?).
1118	?	concat_sequence2(_,S1,S2,Res,WF) :- S1==[],!,equal_sequence(S2,Res,WF).
1119		concat_sequence2(_,S1,S2,Res,WF) :- S2==[],!,equal_sequence(S1,Res,WF).
1120		concat_sequence2(LWF,S1,S2,Res,WF) :-
1121		try_expand_and_convert_to_avl_with_check(S1,AS1,concat1),
1122		try_expand_and_convert_to_avl_with_check(S2,AS2,concat2),
1123	?	concat_sequence3(LWF,AS1,AS2,Res,WF).
1124
1125		concat_sequence3(_,S1,S2,Res,WF) :- nonvar(S1),is_custom_explicit_set(S1,concat_sequence),
1126		concat_custom_explicit_set(S1,S2,ERes,WF),!,
1127		equal_sequence(Res,ERes,WF).
1128		concat_sequence3(_LWF,S1,S2,Res,WF) :-
1129		%try_expand_custom_set_wf(S1,ES1,concat,WF),
1130		size_of_sequence(S1,int(Size1),WF),
1131		(number(Size1) -> true
1132		; size_of_sequence(S2,Size2,WF),
1133		size_of_sequence(Res,SizeRes,WF),
1134	?	int_minus(SizeRes,Size2,int(Size1)),
1135	?	in_nat_range_wf(int(Size1),int(0),SizeRes,WF)
1136		% is this required: ?? ,in_nat_range_wf(Size2,int(0),SizeRes,WF)
1137		),
1138	?	concat_sequence_aux(Size1,S1,S2,Res,WF).
1139
1140
1141		:- assert_must_succeed( (bsets_clp:equal_sequence([(int(1),A)|T1],[(int(1),int(22))|T2],_WF),
1142		A==int(22),T2=[],T1==[] )) .
1143		:- assert_must_succeed( (bsets_clp:equal_sequence([(int(1),A)|T],avl_set(node((int(2),string(a)),true,0,node((int(1),string(c)),true,0,empty,empty),node((int(3),string(b)),true,0,empty,empty))),_WF),
1144		check_eqeq(A,string(c)),
1145		kernel_objects:equal_object(T,[(int(2),B)|T2]), check_eqeq(B,string(a)),
1146		kernel_objects:equal_object(T2,[(int(3),C)]), check_eqeq(C,string(b))) ).
1147		% equal_object optimized for sequences; can infer that pairs with same index have same value
1148		% TO DO: complete and make more efficient
1149		%equal_sequence(X,Y,_WF) :- nonvar(X),nonvar(Y),
1150		% is_custom_explicit_set(X,eval_sequence), is_custom_explicit_set(Y,eval_sequence),!,
1151		% equal_explicit_sets(X,Y).
1152		equal_sequence(X,Y,WF) :- nonvar(X),nonvar(Y),
1153		get_seq_head(X,XI,XEl,XT), get_seq_head(Y,YI,YEl,YT), XI==YI,!,
1154		% THIS CURRENTLY ONLY CHECKS FRONTMOST indexes
1155		equal_object_wf(XEl,YEl,equal_sequence_1,WF),
1156		equal_sequence(XT,YT,WF).
1157		equal_sequence(X,Y,WF) :-
1158		/* (is_custom_explicit_set(Y) -> monitor_equal_sequence_againts_custom_set(X,Y,WF)
1159		; is_custom_explicit_set(X) -> monitor_equal_sequence_againts_custom_set(Y,X,WF)
1160		; true), does not seem to buy anything; equal_object already powerful enough */
1161	?	equal_object_wf(X,Y,equal_sequence_2,WF).
1162
1163		% enforces the constraint that there is only one possible elemenent per index:
1164		%:- block monitor_equal_sequence_againts_custom_set(-,?,?).
1165		%monitor_equal_sequence_againts_custom_set([],_,_) :- !.
1166		%monitor_equal_sequence_againts_custom_set([El|T],CS,WF) :- !,
1167		% element_of_custom_set_wf(El,CS,WF),
1168		% monitor_equal_sequence_againts_custom_set(T,CS,WF).
1169		%monitor_equal_sequence_againts_custom_set(_,_,_).
1170
1171		get_seq_head([(Idx,El)|Tail],Idx,El,Tail).
1172		%get_seq_head(avl_set(AVL),Idx,El,Tail) :- does not seem to buy anything; equal_object already powerful enough
1173		% custom_explicit_sets:avl_min_pair(AVL,Idx,El),
1174		% custom_explicit_sets:direct_remove_element_from_avl(AVL,(Idx,El),Tail). % TO DO: only compute if indexes ==
1175
1176
1177		:- block concat_sequence_aux(-,?,?,?,?).
1178		concat_sequence_aux(Size1,_S1,_S2,Res,WF) :- nonvar(Res),Res=avl_set(_),
1179		size_of_custom_explicit_set(Res,int(RSize),WF), number(RSize),
1180		Size1 > RSize,!, % S1 is longer than Res; no solution (prevent WD error below)
1181		fail.
1182		concat_sequence_aux(Size1,S1,S2,Res,WF) :- nonvar(Res),Res=avl_set(_),
1183		% split Result into prefix and suffix
1184		prefix_of_custom_explicit_set(Res,Size1,Prefix,WF), % we could call versions which do not check WD
1185		suffix_of_custom_explicit_set(Res,Size1,Postfix,WF),
1186		!,
1187		equal_sequence(S1,Prefix,WF), equal_sequence(S2,Postfix,WF).
1188		concat_sequence_aux(Size1,S1,S2,Res,WF) :-
1189		shift_seq_indexes(S2,Size1,NewS2,WF),
1190		% We can do something stronger than disjoint union: we know that the indexes are disjoint!
1191		% Hence: if (int(X),Y) : Res & (int(X),Z) : S1 => Y=Z
1192		% Hence: if (int(X),Y) : Res & (int(X),Z) : S2 => Y=Z
1193		unify_same_index_elements(S1,Res,WF),
1194		unify_same_index_elements(Res,S1,WF),
1195		unify_same_index_elements(NewS2,Res,WF),
1196		unify_same_index_elements(Res,NewS2,WF),
1197	?	disjoint_union_wf(S1,NewS2,Res,WF).
1198
1199		% Check if (int(X),Y) pairs in Seq2 have a matching (int(X),Z) in Seq1 and then unify(Y,Z)
1200		:- block unify_same_index_elements(-,?,?).
1201		unify_same_index_elements(avl_set(A),Seq,WF) :- !,
1202		unify_same_index_elements_aux(Seq,A,WF).
1203		unify_same_index_elements(_,_,_). % TO DO: maybe also support other partially instantiated lists
1204
1205		:- block unify_same_index_elements_aux(-,?,?).
1206		unify_same_index_elements_aux([],_,_) :- !.
1207		unify_same_index_elements_aux([(int(Idx),El)|T],A,WF) :- !,
1208		try_find_index_element(Idx,El,A,WF),
1209		unify_same_index_elements_aux(T,A,WF).
1210		unify_same_index_elements_aux(_,_,_).
1211
1212		:- block try_find_index_element(-,?,?,?).
1213		try_find_index_element(Idx,El,AVL,WF) :-
1214	?	avl_fetch_pair(int(Idx),AVL,AvlEl),
1215		!,
1216		% We have found an entry with the same index: El and AvlEl must be identical:
1217		equal_object_wf(El,AvlEl,try_find_index_element,WF).
1218		try_find_index_element(_Idx,_El,_AVL,_WF). % :- print(not_found(_Idx,_AVL)),nl.
1219
1220		% TO DO: add waitflags + use within partition_wf
1221		% computes union of two sets which are guaranteed to be disjoint: means that if two of three sets known the other one can be determined
1222
1223		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([int(3)],[int(2),int(1)],[int(1),int(3),int(2)],WF),WF)).
1224		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([],[int(2),int(1)],[int(1),int(2)],WF),WF)).
1225		:- assert_must_succeed(exhaustive_kernel_check_wf([commutative],bsets_clp:disjoint_union_wf([int(1),int(2)],[],[int(2),int(1)],WF),WF)).
1226		:- assert_must_succeed((bsets_clp:disjoint_union_wf([int(1)],[int(2)],Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]))).
1227		:- assert_must_succeed((bsets_clp:disjoint_union_wf(A,B,[int(1)],_WF),B=[H],H==int(1),A==[])).
1228
1229		% a union where we know that Set1 and Set2 are disjoint
1230		% this means we can propagate elements of Set1/2 more easily to result
1231		disjoint_union_wf(Set1,Set2,Res,WF) :-
1232		(var(Res)
1233		-> disjoint_union_wf0(Set1,Set2,DRes,DRes,WF),
1234		equal_object_optimized(Res,DRes) % try and convert result to AVL
1235	?	; disjoint_union_wf0(Set1,Set2,Res,Res,WF)).
1236
1237		% disjoint_union_wf0(Set1,Set2,UnionOfSet1Set2, SuperSet, WF)
1238		:- block disjoint_union_wf0(-,-,-,?,?).
1239		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Set1==[],!,equal_object_wf(Set2,Res,disjoint_union_wf0_1,WF).
1240		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Set2==[],!,equal_object_wf(Set1,Res,disjoint_union_wf0_2,WF).
1241		disjoint_union_wf0(Set1,Set2,Res,_,WF) :- Res==[],!,empty_set_wf(Set1,WF), empty_set_wf(Set2,WF).
1242		disjoint_union_wf0(Set1,Set2,Res,FullRes,WF) :-
1243		((nonvar(Set1);nonvar(Set2)) -> true ; get_cardinality_powset_wait_flag(Res,disjoint_union_wf0,WF,_Card,CWF)),
1244	?	disjoint_union0(Set1,Set2,Res,FullRes,WF,CWF).
1245
1246		:- block disjoint_union0(-,-,?,?,?,-), disjoint_union0(-,?,-,-,?,?), disjoint_union0(?,-,-,-,?,?).
1247		disjoint_union0(Set1,Set2,Res,_,WF,_) :- Set1==[],!,equal_object_wf(Set2,Res,disjoint_union0_1,WF).
1248		disjoint_union0(Set1,Set2,Res,_,WF,_) :- Set2==[],!,equal_object_wf(Set1,Res,disjoint_union0_2,WF).
1249		disjoint_union0(S1,S2,Res,_F,WF,_CWF) :-
1250		ground_value(Res),
1251		( ground_value(S1) -> !,
1252		check_subset_of_wf(S1,Res,WF), % TO DO: check if we can merge the check_subset and difference set in one predicate
1253		difference_set_wf(Res,S1,S2,WF)
1254		; ground_value(S2) -> !,
1255		check_subset_of_wf(S2,Res,WF),
1256	?	difference_set_wf(Res,S2,S1,WF)
1257		; var(S1),var(S2) -> !, % CWF nonvar
1258		% see test 1408; allows to generate subsets of Res and avoid enumeration warnings
1259		check_subset_of_wf(S1,Res,WF),
1260		%check_subset_of(S1,Res), % without waitflag: will generate ground version
1261		difference_set_wf(Res,S1,S2,WF)
1262		).
1263		disjoint_union0(Set1,Set2,Res,_,WF,_) :- nonvar(Set1),
1264		is_custom_explicit_set_nonvar(Set1),
1265		union_of_explicit_set(Set1,Set2,Union), !, % TODO: if it fails: copy/propagate values to result?
1266	?	equal_object_wf(Union,Res,disjoint_union0_3,WF).
1267		disjoint_union0(Set1,Set2,Res,Full,WF,_) :-
1268		expand_custom_set_to_list_no_dups_wf(Set1,ESet1,_,disjoint_union0_1,WF),
1269		expand_custom_set_to_list_no_dups_wf(Set2,ESet2,_,disjoint_union0_2,WF),
1270	?	disj_union1(ESet1,ESet2,Res,Full,WF).
1271
1272		:- block disj_union1(-,-,?,?,?).
1273		disj_union1(X,Y,Res,FullRes,WF) :-
1274	?	var(X) -> disj_union2(Y,X,Res,FullRes,WF) ; disj_union2(X,Y,Res,FullRes,WF).
1275
1276		disj_union2([],Y,Res,_,_WF) :- equal_object_optimized(Y,Res,disj_union2).
1277		disj_union2([X|TX],Y,Res,FullRes,WF) :-
1278	?	remove_element_wf(X,Res,TR,WF), % was: equal_cons_wf(Res,X,TR,WF) but error was that it could force X to be a certain value
1279		ground_value_check(X,XV),
1280		(nonvar(XV) -> equal_cons_wf(Res,X,TR,WF)
1281		; check_element_of_wf(X,FullRes,WF), % ensure that we set up proper constraints for X; e.g., for x \/ y = 1..10 & x /\ y = {}
1282		when(nonvar(XV), equal_cons_wf(Res,X,TR,WF))
1283		), % ensure that we instantiate Res if TR known; otherwise we may get pending co-routines, e.g. test 506, SyracuseGrammar
1284		disj_union3(TX,Y,TR,FullRes,WF).
1285
1286		:- block disj_union3(-,-,-,?,?).
1287		disj_union3(X,Y,Res,_,WF) :- Res==[],!,empty_set_wf(X,WF),empty_set_wf(Y,WF).
1288		disj_union3(X,Y,Res,FullRes,WF) :- disj_union1(X,Y,Res,FullRes,WF).
1289
1290
1291		:- block disjoint_union_generalized_wf(-,?,?).
1292		%disjoint_union_generalized_wf([Set1|T],Res,_WF) :- T==[],!, % just one set; probably not covered at the moment (ast_cleanup simplifies partition with single set
1293		% equal_object(Set1,Res).
1294		disjoint_union_generalized_wf(ListOfSets,Res,WF) :-
1295		%expand_custom_set_to_list_wf(SetsOfSets,ESetsOfSets,_,disjoint_union_generalized_wf,WF), % this is a list of sets
1296		disjoint_union_generalized2(ListOfSets,[],Res,WF).
1297		:- block disjoint_union_generalized2(-,?,?,?).
1298		disjoint_union_generalized2([],S,Res,WF) :- !, equal_object_optimized_wf(S,Res,disjoint_union_generalized2,WF).
1299		disjoint_union_generalized2([H|T],UnionSoFar,Res,WF) :- !,
1300		disjoint_union_wf0(H,UnionSoFar,UnionSoFar2,Res,WF),
1301		%% print_message(called_disjoint_union(H,UnionSoFar,UnionSoFar2)), %%
1302		disjoint_union_generalized2(T,UnionSoFar2,Res,WF).
1303		disjoint_union_generalized2(L,S,Res,WF) :-
1304		add_internal_error('Not a list: ',disjoint_union_generalized2(L,S,Res,WF)),fail.
1305		% TO DO: if there are more than two sets: it may be interesting to set up constraint that
1306		% each set is a subset of the full set;
1307		% (would avoid enumeration warning in, e.g., x \/ y \/ z = 1..10 & x /\ y = {} & x /\ z = {} & y /\ z = {} & card(x)=card(y)+2 )
1308
1309		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:concatentation_of_sequences([(int(1),[]),(int(3),[(int(1),int(22)),(int(2),int(33))]),(int(2),[(int(1),int(11))])],
1310		[(int(1),int(11)),(int(2),int(22)),(int(3),int(33))],_WF))).
1311		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:concatentation_of_sequences([(int(1),[]),(int(2),[(int(1),int(33))])],[(int(1),int(33))],_WF))).
1312		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:concatentation_of_sequences([(int(1),[]),(int(2),[(int(1),int(55))])],Res,WF),
1313		kernel_waitflags:ground_wait_flags(WF),
1314		kernel_objects:equal_object(Res,[(int(1),int(55))]) )).
1315		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:concatentation_of_sequences([(int(1),[(int(1),int(22))]),(int(2),[(int(1),int(55))])],Res,WF),
1316		kernel_waitflags:ground_wait_flags(WF),
1317		kernel_objects:equal_object(Res,[(int(1),int(22)),(int(2),int(55))]) )).
1318		:- block concatentation_of_sequences(-,?,?).
1319		concatentation_of_sequences(SeqOfSeq,Res,WF) :-
1320		try_expand_and_convert_to_avl_with_check(SeqOfSeq,ES,conc),
1321	?	concs2(ES,Res,WF).
1322
1323		concs2(SeqOfSeq,Res,WF) :- is_custom_explicit_set(SeqOfSeq,conc),
1324		conc_custom_explicit_set(SeqOfSeq,CRes),!,
1325		equal_object_wf(CRes,Res,concs2,WF).
1326		concs2(SeqOfSeq,Res,WF) :- empty_set_wf(SeqOfSeq,WF),empty_set_wf(Res,WF).
1327		concs2(SeqOfSeq,Res,WF) :- not_empty_set_wf(SeqOfSeq,WF),
1328		front_sequence(SeqOfSeq,Front,WF),
1329	?	concatentation_of_sequences(Front,FrontRes,WF),
1330	?	last_sequence(SeqOfSeq,Last,WF),
1331	?	concat_sequence(FrontRes,Last,Res,WF).
1332
1333		:- assert_must_abort_wf(bsets_clp:tail_sequence([],_R,unknown,WF),WF).
1334		:- assert_must_abort_wf(bsets_clp:tail_sequence([],[],unknown,WF),WF).
1335		:- assert_must_succeed(exhaustive_kernel_succeed_check(
1336		bsets_clp:tail_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(5))],unknown,_WF)) ).
1337		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:tail_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],
1338		[(int(1),int(5)),(int(2),int(6))],unknown,_WF)) ).
1339		:- assert_must_succeed((bsets_clp:tail_sequence(X,R,unknown,_),
1340		X = [(int(1),int(6)),(int(2),int(5))],
1341		kernel_objects:equal_object(R,[(int(1),int(5))]) )).
1342		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(1),int(5))],unknown,_),
1343		X = [(int(1),int(6)),(int(2),int(5))] )).
1344		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(1),int(5)),(int(2),int(7))],unknown,_),
1345		X = [(int(1),int(6)),(int(2),int(5)),(int(3),int(7))] )).
1346		:- assert_must_succeed((bsets_clp:tail_sequence(X,[(int(2),int(7)),(int(1),int(5))],unknown,_),
1347		X = [(int(1),int(6)),(int(2),int(5)),(int(3),int(7))] )).
1348		:- block tail_sequence(-,?,?,?).
1349		tail_sequence(S1,Res,Span,WF) :- is_custom_explicit_set(S1,tail_sequence),
1350		tail_sequence_custom_explicit_set(S1,_,TRes,Span,WF),!,
1351		equal_object_wf(TRes,Res,tail_sequence,WF).
1352		tail_sequence(S1,Res,Span,WF) :- expand_custom_set_to_list_wf(S1,ES1,_,tail_sequence,WF),
1353		tail2(ES1,Res,Span,WF).
1354
1355		tail2([],_,Span,WF) :-
1356		add_wd_error_span('tail applied to empty sequence!',[],Span,WF).
1357		tail2([H|T],Res,_Span,WF) :- domain_subtraction_wf([int(1)],[H|T],IntRes,WF),
1358		shift_seq_indexes(IntRes,-1,Res,WF).
1359
1360
1361		:- assert_must_abort_wf(bsets_clp:first_sequence([],_R,unknown,WF),WF).
1362		:- assert_must_abort_wf(bsets_clp:first_sequence([],int(1),unknown,WF),WF).
1363		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:first_sequence([(int(1),int(4)),(int(2),int(5))],int(4),unknown,_WF)) ).
1364		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:first_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],int(4),unknown,_WF)) ).
1365		:- assert_must_succeed((bsets_clp:first_sequence(X,R,unknown,_WF),
1366		X = [(int(1),int(2)),(int(2),int(1))],
1367		R = int(2))).
1368
1369		:- block first_sequence(-,?,?,?).
1370		first_sequence([],_,Span,WF) :- !,add_wd_error_span('first applied to empty sequence!',[],Span,WF).
1371		first_sequence(Seq,Res,Span,WF) :- apply_to(Seq,int(1),Res,Span,WF).
1372
1373
1374
1375		:- assert_must_abort_wf(bsets_clp:front_sequence([],_R,WF),WF).
1376		:- assert_must_abort_wf(bsets_clp:front_sequence([],[],WF),WF).
1377		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:front_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(4))],_WF)) ).
1378		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:front_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],[(int(1),int(4)),(int(2),int(5))],_WF)) ).
1379		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:front_sequence(X,R,WF),
1380		X = [(int(1),int(2)),(int(2),int(55))],kernel_waitflags:ground_wait_flags(WF),
1381		kernel_objects:equal_object(R,[(int(1),int(2))]))).
1382		:- assert_must_succeed((kernel_waitflags:init_wait_flags(WF),bsets_clp:front_sequence(X,R,WF),
1383		X = [(int(3),int(33))|R], kernel_waitflags:ground_wait_flags(WF),
1384		kernel_objects:equal_object(R,[(int(1),int(2)),(int(2),int(55))]) )).
1385
1386	?	front_sequence(Seq,Res,WF) :- front_sequence(Seq,Res,unknown,WF).
1387		:- block front_sequence(-,?,?,?).
1388		front_sequence(S1,Res,_Span,WF) :-
1389		is_custom_explicit_set(S1,front_sequence),
1390		front_sequence_custom_explicit_set(S1,_,FRes),!,
1391		equal_object_wf(FRes,Res,front_sequence,WF).
1392		front_sequence(Seq,Res,Span,WF) :- expand_custom_set_to_list_wf(Seq,ESeq,_,front_sequence,WF),
1393	?	front2(ESeq,Res,Span,WF).
1394		front2([],_,Span,WF) :- add_wd_error_span('front applied to empty sequence!',[],Span,WF).
1395		front2([H|T],Res,_Span,WF) :- size_of_sequence([H|T],int(Size),WF),
1396	?	(number(Size) -> true ; size_of_sequence(Res,SizeRes,WF), int_plus(SizeRes,int(1),int(Size))),
1397	?	when(ground(Size), domain_subtraction_wf([int(Size)],[H|T],Res,WF)).
1398
1399
1400		:- assert_must_abort_wf(bsets_clp:last_sequence([],_R,WF),WF).
1401		:- assert_must_abort_wf(bsets_clp:last_sequence([],int(1),WF),WF).
1402		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:last_sequence([(int(1),int(4)),(int(2),int(5))],int(5),_WF)) ).
1403		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:last_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],int(6),_WF)) ).
1404		:- assert_must_succeed((bsets_clp:last_sequence(X,R,_WF),
1405		X = [(int(1),int(2)),(int(2),int(55))],R = int(55))).
1406		:- assert_must_succeed((bsets_clp:last_sequence(X,R,_WF), X = [(int(1),int(55))], R = int(55))).
1407		:- assert_must_succeed((bsets_clp:last_sequence([(int(1),[(int(1),int(22))]),(int(2),[(int(1),int(55))])],R,_WF), R == [(int(1),int(55))])).
1408
1409	?	last_sequence(Seq,Res,WF) :- last_sequence(Seq,Res,unknown,WF).
1410		:- block last_sequence(-,?,?,?).
1411		last_sequence(Seq,Res,_Span,WF) :-
1412		is_custom_explicit_set(Seq,last_sequence),
1413		last_sequence_explicit_set(Seq,Last), !,
1414		equal_object_wf(Last,Res,last_sequence,WF).
1415		last_sequence([],_,Span,WF) :- !,add_wd_error_span('last applied to empty sequence!',[],Span,WF).
1416		last_sequence(Seq,Res,Span,WF) :-
1417		size_of_sequence(Seq,int(Size),WF),
1418	?	last_sequence_aux(Size,Seq,Res,Span,WF).
1419		:- block last_sequence_aux(-,?,?,?,?).
1420		last_sequence_aux(Size,Seq,Res,Span,WF) :-
1421	?	apply_to(Seq,int(Size),Res,Span,WF).
1422
1423
1424		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4)),(int(2),int(5))],[(int(1),int(5)),(int(2),int(4))],WF),WF )).
1425		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4))],[(int(1),int(4))],WF),WF )).
1426		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([],[],WF),WF )).
1427		:- assert_must_succeed(exhaustive_kernel_check_wfdet( bsets_clp:rev_sequence([(int(1),int(4)),(int(3),int(6)),(int(2),int(5))],[(int(1),int(6)),(int(3),int(4)),(int(2),int(5))],WF),WF )).
1428		:- assert_must_succeed((bsets_clp:rev_sequence([],[],_WF))).
1429		:- assert_must_succeed((bsets_clp:rev_sequence(X,R,_WF),
1430		X = [(int(1),int(2)),(int(2),int(1))],
1431		kernel_objects:equal_object(R,[(int(2),int(2)),(int(1),int(1))]) )).
1432		:- assert_must_succeed((bsets_clp:rev_sequence(X,R,_WF),
1433		X = [(int(1),int(23)),(int(2),int(1)),(int(3),int(55))],
1434		kernel_objects:equal_object(R,[(int(3),int(23)),(int(2),int(1)),(int(1),int(55))]) )).
1435		:- assert_must_succeed((bsets_clp:rev_sequence(R,X,_WF),
1436		X = [(int(1),int(23)),(int(2),int(1)),(int(3),int(55))],
1437		kernel_objects:equal_object(R,[(int(3),int(23)),(int(2),int(1)),(int(1),int(55))]) )).
1438		:- assert_must_succeed((bsets_clp:rev_sequence(X,_R,_WF),
1439		X = [(int(2),int(1)),(int(1),int(23)),(int(3),int(55))] )).
1440		:- assert_must_succeed((bsets_clp:rev_sequence(_R,X,_WF),
1441		X = [(int(3),int(55)),(int(1),int(23)),(int(2),int(1))] )).
1442
1443		/* reverse of sequence */
1444		:- block rev_sequence(-,-,?).
1445		rev_sequence(S1,Res,WF) :-
1446	?	(nonvar(S1) -> rev_sequence2(S1,Res,WF)
1447		; rev_sequence2(Res,S1,WF)).
1448
1449		rev_sequence2(S1,Res,WF) :- reverse_custom_explicit_set(S1,RS1),!,
1450		equal_object_wf(Res,RS1,WF).
1451		rev_sequence2(S1,Res,WF) :-
1452		expand_custom_set_to_list_wf(S1,ES1,_,rev_sequence2,WF),
1453		size_of_sequence(ES1,int(Size1),WF),
1454		% TO DO: we could also try and get the size from the result Res
1455	?	rev_sequence3(ES1,Size1,Res,WF).
1456
1457		:- block rev_sequence3(?,-,-,?).
1458		rev_sequence3(E,_Size,Res,WF) :- nonvar(Res), reverse_custom_explicit_set(Res,RevRes),!,
1459		equal_object_wf(E,RevRes,WF).
1460		rev_sequence3(E,Size,Res,WF) :- var(Size), !,
1461		% try to obtain size from result as well
1462	?	size_of_sequence(Res,int(Size),WF), rev_sequence3b(E,Size,Res,WF).
1463	?	rev_sequence3(E,S,R,WF) :- rev_sequence4(E,S,R,WF).
1464
1465		:- block rev_sequence3b(?,-,?,?).
1466		rev_sequence3b(E,S,R,WF) :- rev_sequence4(E,S,R,WF).
1467
1468		:- block rev_sequence4(-,?,?,?).
1469		rev_sequence4([],_,Res,WF) :- empty_set_wf(Res,WF).
1470		rev_sequence4([(int(N),El)|Tail],Size1,Res,WF) :-
1471	?	equal_cons_wf(Res,(NewN,El),RTail,WF),
1472		% compute NewN = Size - (N-1)
1473		int_minus(int(N),int(1),N1),
1474		int_minus(int(Size1),N1,NewN),
1475		(ground(NewN) -> true ; in_nat_range(NewN,int(0),int(Size1))),
1476	?	rev_sequence4(Tail,Size1,RTail,WF).
1477
1478
1479		/* --------- */
1480		/* RELATIONS */
1481		/* --------- */
1482
1483		%maplet(X,Y,(X,Y)).
1484
1485		% relation([]).
1486		% relation([(_X,_Y)|T]) :- relation(T).
1487
1488		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:relation_over_wf([(int(1),int(6)),(int(2),int(7)),(int(1),int(7))],[int(1),int(2)],[int(7),int(6)],WF),WF)).
1489		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:relation_over([],[int(1),int(2)],[int(2)]) )).
1490		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(int(1),int(2))],[int(1),int(2)],[int(2)]) )).
1491		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([([int(1)],[int(2)])],[[int(1)],[],[int(2)]],[[int(2)]]) )).
1492		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(pred_true /* bool_true */,pred_false /* bool_false */)],[pred_false /* bool_false */,pred_true /* bool_true */],[pred_false /* bool_false */]) )).
1493		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((pred_true /* bool_true */,int(2)),fd(1,'Name'))],[(pred_false /* bool_false */,int(1)),(pred_true /* bool_true */,int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1494		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((rec([field(a,fd(1,'Name'))]),int(2)),fd(1,'Name'))],[(rec([field(a,fd(1,'Name'))]),int(1)),(rec([field(a,fd(1,'Name'))]),int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1495		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((rec([field(a,fd(2,'Name')),field(b,fd(1,'Name'))]),int(2)),fd(1,'Name'))],[(rec([field(a,fd(1,'Name')),field(b,fd(1,'Name'))]),int(1)),(rec([field(a,fd(1,'Name')),field(b,fd(2,'Name'))]),int(2)),(rec([field(a,fd(2,'Name')),field(b,fd(1,'Name'))]),int(2))],[fd(2,'Name'),fd(1,'Name')]) )).
1496		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([((pred_true /* bool_true */,int(2)),string('STRING1'))],[(pred_false /* bool_false */,int(1)),(pred_true /* bool_true */,int(2))],[string('STRING2'),string('STRING1')]) )).
1497		:- assert_must_succeed(exhaustive_kernel_succeed_check( /* multiple solutions !!*/ bsets_clp:relation_over([(int(1),int(2)),(int(2),int(2))],[int(1),int(2)],[int(2)]) )).
1498		:- assert_must_succeed(exhaustive_kernel_succeed_check( bsets_clp:relation_over([(int(1),int(2)),(int(1),int(3))],[int(1),int(2)],[int(3),int(2)]) )).
1499		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:relation_over([(int(1),int(2)),(int(2),int(1))],[int(1),int(2)],[int(2)]) )).
1500		:- assert_must_fail(( bsets_clp:relation_over([(int(1),int(1))],[int(1),int(2)],[int(2)]) )).
1501		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(3)]),
1502		X==[(int(1),int(3))] )).
1503		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(3)]),
1504		X==[(int(1),int(3)),(int(2),int(3))] )).
1505		:- assert_must_succeed(( bsets_clp:relation_over(X,[int(1),int(2)],[int(4),int(5)]),
1506		X==[(int(2),int(4)),(int(2),int(5))] )).
1507
1508		relation_over(R,Dom,Ran) :- init_wait_flags(WF,[relation_over]),
1509	?	relation_over_wf(R,Dom,Ran,WF),
1510	?	ground_wait_flags(WF).
1511
1512		:- block relation_over_wf(-,-,-,?).
1513		relation_over_wf(R,Dom,Ran,WF) :-
1514		kernel_equality:get_cardinality_relation_over_wait_flag(Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels),
1515	?	relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels).
1516
1517		:- block relation_over1(-,?,?,?,-,?,?).
1518		relation_over1(FF,Domain,Range,WF,_WFR,_MaxCard,_MaxNrOfRels) :-
1519		nonvar(FF),
1520		custom_explicit_sets:is_definitely_maximal_set(Range),
1521		% we do not need the Range; this means we can match more closures (e.g., lambda)
1522		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,_,WF),!,
1523		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF).
1524		relation_over1(FF,Domain,Range,WF,_WFR,_MaxCard,_MaxNrOfRels) :- nonvar(FF),
1525		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,_,WF),!,
1526		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
1527		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
1528		relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels) :- var(R),!,
1529		expand_custom_set_to_list_wf(R,ER,_,relation_over1,WF),
1530	?	relation_over2(ER,[],Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels,none).
1531		relation_over1(R,Domain,Range,WF,_WFR,_MaxCard,_) :-
1532		expand_and_convert_to_avl_set_catch(R,AER,relation_over1,'ARG : ? <-> ?',ResultStatus,WF),!,
1533		(ResultStatus=avl_set
1534	?	-> is_avl_relation_over_domain(AER,Domain,WF),
1535		is_avl_relation_over_range(AER,Range,WF)
1536		; (debug_mode(on) -> add_message_wf(relation_over,'SYMBOLIC <-> check: ',R,unknown,WF) ; true),
1537		symbolic_domain_subset_check(R,Domain,WF),
1538		symbolic_range_subset_check(R,Range,WF)
1539		).
1540		relation_over1(R,Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels) :-
1541		expand_custom_set_to_list_wf(R,ER,_,relation_over1,WF),
1542	?	relation_over2(ER,[],Dom,Ran,WF,WFR,MaxCard,MaxNrOfRels,none).
1543
1544		% check the domain of a symbolic closure value FF whose domain is FFDomain and expected domain is Domain:
1545		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF) :-
1546		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
1547		[b_operator(domain,[FF]),Domain],unknown),WF2),
1548		check_subset_of_wf(FFDomain,Domain,WF2).
1549		% ditto for range
1550		check_range_subset_for_closure_wf(FF,FFRange,Range,WF) :-
1551		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
1552		[b_operator(range,[FF]),Range],unknown),WF2),
1553		check_subset_of_wf(FFRange,Range,WF2).
1554
1555
1556		% try and expand set to AVL and catch enumeration warning exceptions and set OK result value
1557		% if it succeeds with OK = avl_set -> we have an avl_set
1558		% if it fails: it cannot be expanded at the moment
1559		% if it retuns keep_symbolic: expansion cannot be performed and can never be performed; keep set symbolic
1560		expand_and_convert_to_avl_set_catch(R,_AS,_Origin,_Operator,_ResultStatus,_WF) :- var(R),!,fail.
1561		expand_and_convert_to_avl_set_catch(R,_AS,_Origin,_Operator,ResultStatus,_WF) :-
1562		is_infinite_explicit_set(R),!, % we could also use is_infinite_or_symbolic_closure
1563		ResultStatus=keep_symbolic.
1564		expand_and_convert_to_avl_set_catch(R,AS,Origin,Operator,ResultStatus,WF) :-
1565		catch(
1566		(expand_and_convert_to_avl_set(R,AS,Origin,Operator),ResultStatus=avl_set),
1567		enumeration_warning(_,_,_,_,_),
1568		(add_message_wf(Origin,'Attempting symbolic treatment, enumeration warning occured while expanding ARG for ',
1569		Operator,b(value(R),any,[]),WF),
1570		ResultStatus=keep_symbolic)).
1571
1572		expand_and_convert_to_avl_set_warn(R,_AS,_Origin,_Operator,_WF) :- var(R),!,fail.
1573		expand_and_convert_to_avl_set_warn(R,AS,Origin,Operator,WF) :-
1574		% TO DO: check for not fully instantiated closures, like memoization closures where ID not yet known
1575		% it is used before a cut: we need to expand straightaway without choice points
1576		(is_symbolic_closure(R)
1577		-> add_message_wf(Origin,'Expanding symbolic set argument ARG for predicate ',Operator,b(value(R),any,[]),WF)
1578		; true),
1579		% TODO: instead of observe_enumeration_warnings we could push onto the call-stack and pass WF
1580		observe_enumeration_warnings(expand_and_convert_to_avl_set(R,AS,Origin,Operator),
1581		add_message_wf(Origin,'Enumeration warning occured while expanding argument ARG for predicate ',
1582		Operator,b(value(R),any,[]),WF)).
1583		%expand_and_convert_to_avl_set(R,AS,_,Operator,Values) :-
1584		% observe_enumeration_warnings(expand_and_convert_to_avl_set(R,AS,,),
1585		% display_warning_message(Operator,Values)).
1586		%display_warning_message(Operator,Values) :-
1587		% format(user_error,'Enumeration Warning for Operator ~w~n',[Operator]),
1588		% maplist(translate:print_bvalue,Values),nl.
1589
1590		:- block relation_over2(-,?,?,?,?,-,?,?,?).
1591		relation_over2([],_,_,_,_WF,_WFR,_MaxCard,_MaxNrOfRels,_LastPair).
1592		relation_over2(REL,SoFar,Domain,Range,WF,WFR,MaxCard,MaxNrOfRels,LastPair) :-
1593		(var(REL) -> NewLastPair=(X,Y) ; NewLastPair=none), %remember whether we freely chose X,Y
1594		REL = [(X,Y)|T],
1595	?	(number(MaxCard)
1596		-> MaxCard>0,C1 is MaxCard-1 ,(C1=0 -> T=[] ; true)
1597		; C1=MaxCard),
1598		% TO DO: try to enumerate elements in order
1599	?	ordered_pair(LastPair,X,Y,not_equal),
1600	?	check_element_of_wf(X,Domain,WF),
1601	?	check_element_of_wf(Y,Range,WF),
1602	?	not_element_of_wf((X,Y),SoFar,WF),
1603		update_waitflag(MaxNrOfRels,WFR,NewWFR,WF),
1604	?	relation_over2(T,[(X,Y)|SoFar],Domain,Range,WF,NewWFR,C1,MaxNrOfRels,NewLastPair).
1605
1606		% check that new pair is greater than previous pair, if that pair was freely chosen
1607		ordered_pair(none,_,_,_).
1608		ordered_pair((LastX,LastY),NewX,NewY,Eq) :- ordered_value(LastX,NewX,EqualX),
1609		check_second_component(EqualX,LastY,NewY,Eq).
1610
1611		:- block check_second_component(-,?,?,?).
1612		check_second_component(equal,X,Y,EqRes) :- ordered_value(X,Y,EqRes).
1613		check_second_component(not_equal,_X,_Y,not_equal). % no need to check 2nd component
1614
1615		:- block ordered_value(-,?,?), ordered_value(?,-,?).
1616		ordered_value(pred_true /* bool_true */,B,Eq) :- !, (B=pred_true /* bool_true */ -> Eq=equal ; Eq=not_equal).
1617		ordered_value(pred_false /* bool_false */,B,Eq) :- !, B=pred_false /* bool_false */, Eq=equal.
1618		ordered_value(int(X),int(Y),Eq) :- !,
1619		kernel_objects:less_than_equal_direct(X,Y), equal_atomic_term(X,Y,Eq).
1620		ordered_value(fd(NrX,T),fd(NrY,T),Eq) :- !,
1621		kernel_objects:less_than_equal_direct(NrX,NrY),
1622		equal_atomic_term(NrX,NrY,Eq).
1623		ordered_value((X1,X2),(Y1,Y2),Eq) :- !, ordered_pair((X1,X2),Y1,Y2,Eq).
1624		ordered_value(string(X),string(Y),Eq) :- !, less_equal_atomic_term(X,Y,Eq).
1625		ordered_value(rec(FX),rec(FY),Eq) :- !,
1626		ordered_fields(FX,FY,Eq).
1627		ordered_value([],Y,Eq) :- !, (Y==[] -> Eq=equal ; Eq=not_equal). % empty set is the smallest set
1628		ordered_value(avl_set(A),Y,Eq) :- !,
1629		(Y==[] -> fail
1630		; Y=avl_set(B) -> (A @< B -> Eq=not_equal ; A@>B -> fail ; Eq=equal)
1631		; print(assuming_strictly_ordered(avl_set(A),Y)),nl,
1632		Eq=not_equal). % TO DO: treat sets better
1633		ordered_value([H|T],Y,Eq) :- !, ordered_value_cons(Y,H,T,Eq).
1634		ordered_value(term(floating(F1)),term(floating(F2)),Eq) :- !,
1635		kernel_reals:real_less_than_equal_wf(term(floating(F1)),term(floating(F2)),no_wf_available),
1636		equal_atomic_term(F1,F2,Eq).
1637		ordered_value(A,B,not_equal) :- print(assuming_strictly_ordered(A,B)),nl.
1638
1639		ordered_value_cons([],_,_,_) :- !,fail.
1640		ordered_value_cons([H2|T2],H,T,Eq) :- !,ordered_pair((H,T),H2,T2,Eq). % Note: order different than for avl_sets!
1641		ordered_value_cons(Y,H,T,not_equal) :- write(assuming_strictly_ordered([H|T],Y)),nl.
1642
1643		:- block ordered_fields(-,?,?).
1644		ordered_fields([],RHS,Eq) :- !,RHS=[], Eq=equal.
1645		ordered_fields([field(Name,ValX)|TX],RHS,Eq) :- !,RHS=[field(Name,ValY)|TY],
1646	?	ordered_value(ValX,ValY,Equal1), check_next_field(Equal1,TX,TY,Eq).
1647		ordered_fields(FX,FY,Eq) :- add_internal_error('Unknown fields: ',ordered_fields(FX,FY,Eq)), Eq=not_equal.
1648
1649		:- block check_next_field(-,?,?,?).
1650	?	check_next_field(equal,TX,TY,EqRes) :- ordered_fields(TX,TY,EqRes).
1651		check_next_field(not_equal,_X,_Y,not_equal). % no need to check next field
1652
1653		:- block less_equal_atomic_term(-,?,?), less_equal_atomic_term(?,-,?).
1654		less_equal_atomic_term(A,B,Res) :- (A==B -> Res=equal ; A @<B, Res=not_equal).
1655
1656		:- block equal_atomic_term(-,?,?), equal_atomic_term(?,-,?).
1657		equal_atomic_term(A,B,Res) :- (A==B -> Res=equal ; Res=not_equal).
1658
1659
1660		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:not_relation_over([(int(1),int(2)),(int(2),int(1))],[int(1),int(2)],[int(2)],_WF) )).
1661		:- assert_must_succeed(exhaustive_kernel_check( bsets_clp:not_relation_over([(int(1),int(2))],[],[int(2)],_WF) )).
1662		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:not_relation_over([(int(1),pred_true)],[int(1)],[pred_true],_WF) )).
1663		:- assert_must_succeed(exhaustive_kernel_fail_check( bsets_clp:not_relation_over([],[int(1)],[pred_true],_WF) )).
1664		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(2))],[int(3)],[int(1),int(2)],_) ).
1665		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(2))],[int(1)],[int(3)],_) ).
1666		:- assert_must_succeed( bsets_clp:not_relation_over([(int(1),int(3)),(int(1),int(2))],[int(1)],[int(3)],_) ).
1667		:- assert_must_fail( bsets_clp:not_relation_over([(int(1),int(3))],[int(1)],[int(3)],_) ).
1668		:- assert_must_fail( bsets_clp:not_relation_over([],[int(1)],[int(3)],_) ).
1669		:- assert_must_fail( bsets_clp:not_relation_over([],[],[],_) ).
1670		:- block not_relation_over(-,?,?,?).
1671
1672		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
1673		% we do not need the Range; this means we can match more closures (e.g., lambda)
1674		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,_,WF),!,
1675		not_subset_of_wf(FFDomain,Domain,WF).
1676		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),
1677		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,_,WF),!,
1678		not_both_subset_of(FFDomain,FFRange,Domain,Range,WF).
1679		/* could be slightly more efficient: but not clear if warrants additional complexity in code:
1680		not_relation_over(FF,Domain,Range,WF) :- nonvar(FF),
1681		check_element_can_be_decided(Domain), % ensures that check_element_of_wf will not block below
1682		check_element_can_be_decided(Range), % ensures that check_element_of_wf will not block below
1683		expand_and_convert_to_avl_set(FF,AER,no_relation_over,''),!,
1684		(is_avl_relation_over_domain(AER,Domain,WF)
1685		-> \+ is_avl_relation_over_range(AER,Range,WF)
1686		; true).
1687		check_element_can_be_decided(X) :- var(X),!,fail.
1688		check_element_can_be_decided(avl_set(_)).
1689		check_element_can_be_decided([]).
1690		check_element_can_be_decided(closure(P,T,B)) :-
1691		custom_explicit_sets:is_interval_closure_or_integerset(closure(P,T,B),Low,Up),
1692		ground(Low), ground(Up).
1693		*/
1694		not_relation_over(R,Dom,Ran,WF) :-
1695		expand_custom_set_to_list_wf(R,ER,_,not_relation_over,WF),
1696		%% print(not_rel(ER,Dom,Ran)),nl,
1697		not_relation_over2(ER,Dom,Ran,WF).
1698
1699
1700		%not_relation_over2(R,_,_) :- when(nonvar(R), (R\=[], R\=[_|_])) . % TYPE ERROR !
1701		:- block not_relation_over2(-,?,?,?).
1702		not_relation_over2([(X,Y)|T],Domain,Range,WF) :-
1703		membership_test_wf(Domain,X,MemRes,WF),
1704		not_relation_over3(MemRes,Y,T,Domain,Range,WF).
1705
1706		:- block not_relation_over3(-,?,?,?,?,?).
1707		not_relation_over3(pred_false,_Y,_T,_Domain,_Range,_WF).
1708		not_relation_over3(pred_true,Y,T,Domain,Range,WF) :-
1709		membership_test_wf(Range,Y,MemRes,WF),
1710		not_relation_over4(MemRes,T,Domain,Range,WF).
1711
1712		:- block not_relation_over4(-,?,?,?,?).
1713		not_relation_over4(pred_false,_T,_Domain,_Range,_WF).
1714		not_relation_over4(pred_true,T,Domain,Range,WF) :-
1715		not_relation_over2(T,Domain,Range,WF).
1716
1717
1718
1719		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf([],[],WF),WF)).
1720		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf([(int(1),int(3))],[int(1)],WF),WF)).
1721		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf(
1722		[(int(0),int(55)),(int(2),int(3)),(int(1),int(3))],[int(1),int(2),int(0)],WF),WF)).
1723		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:domain_wf(
1724		[(int(99),int(55)),(int(2),int(3)),(int(99),int(4))],[int(2),int(99)],WF),WF)).
1725		:- assert_must_succeed((bsets_clp:domain_wf([],Res,_WF),Res=[])).
1726		:- assert_must_succeed((bsets_clp:domain_wf([(int(1),int(2))],Res,_WF),
1727		kernel_objects:equal_object(Res,[int(1)]))).
1728		:- assert_must_succeed((bsets_clp:domain_wf([(int(1),int(2)),(int(1),int(1))],Res,_WF),
1729		kernel_objects:equal_object(Res,[int(1)]))).
1730		:- assert_must_succeed((bsets_clp:domain_wf([(int(2),int(2)),(int(1),int(2))],Res,_WF),
1731		kernel_objects:equal_object(Res,[int(1),int(2)]))).
1732		:- assert_must_succeed((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(3),int(2)]),
1733		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(3),int(2))]))).
1734		:- assert_must_succeed((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]),
1735		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(1),int(2))]))).
1736		:- assert_must_fail((bsets_clp:domain_wf(X,Res,_WF),kernel_objects:equal_object(Res,[int(1),int(2)]),
1737		kernel_objects:equal_object(X,[(int(2),int(2)),(int(1),int(1)),(int(3),int(2))]))).
1738
1739		:- block domain_wf(-,-,?).
1740		domain_wf(Rel,Res,WF) :- Res == [],!,
1741		empty_set_wf(Rel,WF).
1742		domain_wf(Rel,Res,WF) :- var(Rel),!, % hence Res must me nonvar
1743		(is_custom_explicit_set(Res,domain_wf)
1744		-> expand_custom_set_to_list_wf(Res,Res2,_,propagate_result_to_input2,WF) % avoid expanding twice
1745		; Res2 = Res),
1746		propagate_result_to_input(Res2,Rel,domain,WF),
1747		domain_wf1(Rel,Res2,WF).
1748	?	domain_wf(Rel,Res,WF) :- domain_wf1(Rel,Res,WF).
1749
1750
1751		% propagate result of domain/range back to original relation
1752		propagate_result_to_input(Result,OriginalRel,DomOrRange,WF) :-
1753		propagate_empty_set_wf(Result,result,OriginalRel,WF), % this will trigger before LWF ground
1754		(preferences:preference(use_smt_mode,true)
1755		-> propagate_result_to_input1(Result,OriginalRel,1,DomOrRange)
1756		% hopefully full CHR implementation will avoid the need for this hack
1757		% ; kernel_objects:is_marked_to_be_computed(OriginalRel) -> true % get_last_wait_flag(propagate_result_to_input,WF,LWF)
1758		;
1759		get_wait_flag(2000,propagate_result_to_input,WF,LWF), % TO DO: determine right value for Priority ?
1760		% higher number for data_validation mode seems slightly counterproductive (on private_source_not_available tests)
1761		propagate_result_to_input1(Result,OriginalRel,LWF,DomOrRange) % this slows down test 289 if not guarded, 1088 if guarded
1762		).
1763
1764		:- block propagate_result_to_input1(-,?,?,?), propagate_result_to_input1(?,-,-,?).
1765		% Note: if arg 2 (Rel) is known we will not propagate
1766		propagate_result_to_input1([],Rel,_,_) :- !, empty_set(Rel).
1767		propagate_result_to_input1(Result,Input,LWF,DomOrRange) :-
1768		(kernel_objects:is_marked_to_be_computed(Input) -> true
1769		; propagate_result_to_input2(Result,Input,LWF,DomOrRange)).
1770
1771		%:- block propagate_result_to_input2(-,?).
1772		:- block propagate_result_to_input2(-,?,?,?), propagate_result_to_input2(?,-,-,?).
1773		% maybe do in CHR in future: x:dom(R) => #z.(x,z) : R
1774		% TO DO: make stronger; also support avl_set ...
1775		propagate_result_to_input2([],_Rel,_,_) :- !. % nothing can be said; we could have repeated entries for previous domain elements
1776		propagate_result_to_input2([D|T],Rel,LWF,DomOrRange) :- %print(propagate_result_to_input2([D|T],Rel,LWF,DomOrRange)),nl,
1777		!,
1778		(Rel == [] -> fail % we would need more relation elements to generate the domain/range
1779		; nonvar(Rel) -> true % no propagation
1780		; (DomOrRange=domain -> Rel = [(D,_)|RT] ; Rel = [(_,D)|RT]),
1781		propagate_result_to_input2(T,RT,LWF,DomOrRange)
1782		).
1783		propagate_result_to_input2(CS,Rel,LWF,DomOrRange) :- var(Rel), is_custom_explicit_set(CS),!,
1784		expand_custom_set_to_list(CS,Res,_,propagate_result_to_input2),
1785		propagate_result_to_input2(Res,Rel,LWF,DomOrRange).
1786		propagate_result_to_input2(_1,_2,_LWF,_DomOrRange).
1787
1788		:- block domain_wf1(-,?,?).
1789		domain_wf1(Rel,Res,WF) :- is_custom_explicit_set(Rel,domain_wf),
1790		domain_of_explicit_set_wf(Rel,Dom,WF), !,
1791	?	equal_object_wf(Dom,Res,domain_wf1,WF).
1792		domain_wf1(Rel,Res,WF) :-
1793		expand_custom_set_to_list_wf(Rel,Relation,_,domain_wf,WF),
1794	?	newdomain1(Relation,[],Res,WF),
1795		quick_propagate_domain(Relation,Res,WF).
1796
1797		:- block quick_propagate_domain(-,?,?).
1798		quick_propagate_domain([],_,_WF).
1799		quick_propagate_domain([(X,_)|T],FullRes,WF) :-
1800		quick_propagation_element_information(FullRes,X,WF,FullRes1), % should we use a stronger check ?
1801		quick_propagate_domain(T,FullRes1,WF).
1802
1803		%:- block newdomain1(-,?,-,?). % why was this commented out ?
1804		:- block newdomain1(-,?,?,?).
1805		/* newdomain1(Rel,SoFar,Res,WF) :- var(Rel), !,
1806		domain_propagate_result(Res,Rel,SoFar,WF). */
1807	?	newdomain1(Dom,SoFar,Res,WF) :- newdomain2(Dom,SoFar,Res,WF).
1808
1809		%:- block newdomain2(-,?,?,?).
1810	?	newdomain2([],_SoFar,Res,WF) :- empty_set_wf(Res,WF).
1811		newdomain2([(X,Y)|T],SoFar,Res,WF) :-
1812		(Res==[]
1813		-> MemRes=pred_true, % no new elements can appear, all Xs must already be in SoFar
1814		check_element_of_wf(X,SoFar,WF)
1815		; membership_test_wf(SoFar,X,MemRes,WF),
1816		% now check that card of Relation is greater or equal to Result; if equal set MemRes to pred_false
1817		% if card(Result)=card(dom(Result)) => all elements in Result must be fresh domain elements
1818		card_greater_equal_check([(X,Y)|T],Res,MemRes)
1819		),
1820	?	newdomain3(MemRes,X,T,SoFar,Res,WF).
1821
1822		:- block newdomain3(-,?,?,?,?,?).
1823		newdomain3(pred_true,_,T,SoFar,Res,WF) :- newdomain1(T,SoFar,Res,WF).
1824		newdomain3(pred_false,X,T,SoFar,Res,WF) :-
1825		kernel_objects:mark_as_non_free(X,domain), % X is linked to a particular Y -> it is not free
1826		add_element_wf(X,SoFar,SoFar2,WF),
1827	?	equal_cons_wf(Res,X,Res2,WF),
1828	?	newdomain1(T,SoFar2,Res2,WF).
1829
1830
1831		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_domain_wf(int(2),[(int(2),int(7))],WF),WF)).
1832		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_wf(int(2),[(int(1),int(6)),(int(2),int(7))],WF),WF)). % used to be wfdet; but dom_symbolic can create existential quantifier, not all co-routines/... evaluated in wfdet
1833		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_domain_wf(int(22),[(int(1),int(6)),(int(22),int(7)),(int(33),int(7))],WF),WF)). % used to be wfdet (see above)
1834		:- assert_must_succeed((bsets_clp:in_domain_wf(int(1),[(int(1),int(2))],_))).
1835		:- assert_must_succeed((bsets_clp:in_domain_wf(int(3),[(int(1),int(2)),(int(3),int(4))],_))).
1836		:- assert_must_fail((bsets_clp:in_domain_wf(int(3),[],_))).
1837		:- assert_must_fail((bsets_clp:in_domain_wf(int(3),[(int(1),int(2))],_))).
1838		/* a more efficient version than using element_of and computing domain */
1839
1840		% just like not_empty_set_wf but instantiates with (El,_) as first element
1841		in_domain_wf(El,S,WF) :- var(S),!, force_in_domain_wf(El,S,WF).
1842	?	in_domain_wf(El,Rel,WF) :- in_domain_wf_lazy(El,Rel,WF).
1843
1844		:- use_module(kernel_non_empty_attr,[mark_var_set_as_non_empty/1]).
1845		% next is also used in apply_to/6
1846		force_in_domain_wf(El,S,WF) :-
1847		(preferences:preference(use_smt_mode,true) -> get_wait_flag0(WF,WF0),
1848		when(ground(WF0),delayed_force_in_domain_wf(El,S,WF))
1849		; % TO DO: non-empty flag
1850		mark_var_set_as_non_empty(S),
1851		get_enumeration_starting_wait_flag(not_empty_domain_wf,WF,LWF), in_domain_lwf(El,S,LWF,WF)).
1852		% delay instantiating S somewhat: it can mess up many other optimisations
1853		% fixes trying to deconstruct infinite set enum warning for test 2022
1854		delayed_force_in_domain_wf(El,S,_WF) :- var(S),!, S=[(El,_)|_]. % TODO: mark _ as irrelevant
1855		delayed_force_in_domain_wf(El,Rel,WF) :- in_domain_wf_lazy(El,Rel,WF).
1856
1857		:- block in_domain_lwf(-,-,-,?).
1858		% was :- block in_domain_lwf(-,?,-,?). but this prevents instantiating El in case Rel becomes known ! see e.g. private_examples/ClearSy/ComparePv10Pv11/DebugPv10/ test 1952, 2270
1859		%:- block in_domain_lwf(-,-,?,?),in_domain_lwf(?,-,-,?). % this annotation fails test 1703
1860		in_domain_lwf(El,Rel,LWF,WF) :- % tools_printing:print_term_summary(in_domain_lwf(El,Rel,LWF)),
1861		(var(Rel) -> ground_value_check(El,GrVal),
1862		in_domain_lwf2(El,Rel,LWF,GrVal,WF) % we could also wait at least until WF0 is fully grounded?
1863		; not_empty_set_unless_closure_wf(Rel,WF),
1864		in_domain_wf_lazy(El,Rel,WF)).
1865
1866		:- block in_domain_lwf2(?,-,-,-,?).
1867		in_domain_lwf2(El,Rel,_LWF,_Grval,WF) :- % tools_printing:print_term_summary(in_domain_lwf2(El,Rel,_LWF,_Grval)),
1868		(var(Rel) -> Rel = [(El,_)|_]
1869		% can create a choice point when unifying with large avl_set:, see rule_Rule_DB_PSR_0003_C
1870		% maybe we should delay even further
1871		; not_empty_set_unless_closure_wf(Rel,WF),
1872		in_domain_wf_lazy(El,Rel,WF)).
1873
1874		not_empty_set_unless_closure_wf(closure(_,_,_),_) :- !. % do not check this; in_domain_wf or other call will find a solution anyway; no need to set up closure constraints twice
1875		not_empty_set_unless_closure_wf(Rel,WF) :- not_empty_set_wf(Rel,WF).
1876
1877		% does not instantiate to [(El,_)|_]
1878		:- block in_domain_wf_lazy(?,-,?).
1879		in_domain_wf_lazy(_DomainElement,[],_WF) :- !,fail.
1880		in_domain_wf_lazy(DomainElement,avl_set(A),_WF) :-
1881		ground_value(DomainElement), !,
1882		check_in_domain_of_avlset(DomainElement,A).
1883		% TO DO: check for infinite closures
1884		in_domain_wf_lazy(DomainElement,ES,WF) :-
1885		is_custom_explicit_set(ES,in_domain_wf_lazy),
1886		domain_of_explicit_set_wf(ES,Dom,WF),!,
1887	?	check_element_of_wf(DomainElement,Dom,WF).
1888		in_domain_wf_lazy(El,Rel,WF) :-
1889		expand_custom_set_to_list_wf(Rel,Relation,Done,in_domain_wf_lazy,WF),
1890		get_binary_choice_wait_flag(in_domain_wf_lazy(El),WF,LWF), % TO DO: get_pow2_binary_choice_priority(Len,Prio), get_binary_choice_wait_flag_exp_backoff
1891		% if Done == true -> we can use maybe clpfd_inlist or clpfd:element or quick_propagate
1892		quick_propagation_domain_element_list(Done,Relation,El,WF),
1893		in_domain2(El,Relation,WF,LWF).
1894
1895		% a custom implementation of quick_propagation_element_information for checking domain elements and lists only
1896		:- use_module(clpfd_lists,[try_in_fd_value_list_check/4]).
1897		:- block quick_propagation_domain_element_list(-,?,?,?).
1898		quick_propagation_domain_element_list(_,_,_,_) :- preferences:preference(use_clpfd_solver,false),!.
1899		quick_propagation_domain_element_list(_,_,El,_) :- ground(El),!.
1900		quick_propagation_domain_element_list(_,RelList,El,WF) :-
1901		try_in_fd_value_list_check(RelList,(El,_),couple_left(_),WF). % use couple_left to ignore range values
1902
1903
1904		:- block in_domain2(?,-,?,?).
1905		in_domain2(El,[(X,_Y)|T],WF,LWF) :-
1906		(T==[]
1907		-> equal_object_wf(El,X,in_domain2,WF)
1908		; kernel_objects:equality_objects_lwf(El,X,EqRes,LWF,WF),
1909		in_domain3(EqRes,El,T,WF,LWF)
1910		).
1911
1912		:- block in_domain3(-,?,?,?,?).
1913		in_domain3(pred_true,_El,_T,_WF,_LWF).
1914		in_domain3(pred_false,El,T,WF,LWF) :-
1915		get_new_subsidiary_wait_flag(LWF,in_domain2(El,T),WF,NewLWF), % not necessary if T only has single element
1916		in_domain2(El,T,WF,NewLWF).
1917
1918
1919		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[],WF),WF)).
1920		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(2),int(7))],WF),WF)).
1921		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(2),int(7)),(int(4),int(3))],WF),WF)).
1922		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:not_in_domain_wf(int(4),[(int(2),int(7)),(int(4),int(3))],WF),WF)).
1923		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(1),[(int(1),int(2))],_))).
1924		:- assert_must_fail((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2)),(int(3),int(4))],_))).
1925		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[],_))).
1926		:- assert_must_succeed((bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2))],_))).
1927		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_domain_wf(int(3),[(int(1),int(2)),(int(2),int(3))],WF),WF)).
1928		/* a more efficient version than using not_element_of and computing domain */
1929
1930
1931		:- block not_in_domain_wf(?,-,?).
1932		not_in_domain_wf(DomainElement,ES,WF) :- is_custom_explicit_set(ES,not_in_domain),
1933		domain_of_explicit_set_wf(ES,Dom,WF),!,
1934		not_element_of_wf(DomainElement,Dom,WF).
1935		not_in_domain_wf(El,Rel,WF) :-
1936		expand_custom_set_to_list_wf(Rel,Relation,_,not_in_domain,WF),
1937		not_in_domain2(Relation,El,WF).
1938		:- block not_in_domain2(-,?,?).
1939		not_in_domain2([],_,_WF).
1940		not_in_domain2([(X,_Y)|T],E,WF) :- not_equal_object_wf(E,X,WF), not_in_domain2(T,E,WF).
1941
1942
1943
1944
1945		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([],[],WF),WF)).
1946		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf([(int(1),int(3))],[int(3)],WF),WF)).
1947		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:range_wf(
1948		[(int(0),int(55)),(int(2),int(3)),(int(1),int(3))],[int(3),int(55)],WF),WF)).
1949		:- assert_must_succeed((bsets_clp:range_wf([],Res,_WF),Res=[])).
1950		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2))],Res,_WF),
1951		kernel_objects:equal_object(Res,[int(2)]))).
1952		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(1)),(int(2),int(1))],Res,_WF),
1953		kernel_objects:equal_object(Res,[int(1)]))).
1954		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1))],Res,_WF),
1955		kernel_objects:equal_object(Res,[int(1),int(2)]))).
1956		:- assert_must_succeed((bsets_clp:range_wf([(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],Res,_WF),
1957		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1958		:- assert_must_succeed((bsets_clp:range_wf(X,Res,_WF),
1959		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(3))],
1960		kernel_objects:equal_object(Res,[int(1),int(3),int(2)]))).
1961		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1962		X = [(int(1),int(2)),(int(1),int(1)),(int(2),int(2))])).
1963		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1964		X = [(int(2),int(1)),(int(1),int(2)),(int(2),int(2))])).
1965		:- assert_must_succeed((bsets_clp:range_wf(X,Res,WF), bsets_clp:domain_wf(X,Res2,WF), kernel_objects:equal_object(Res,Res2),
1966		X = [])).
1967		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1968		([int(0),int(1)],[int(0),int(1)]),([int(0),int(2)],[int(0),int(2)]),
1969		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1970		([int(1),int(2)],[int(1),int(2)]),([int(1),int(3)],[int(1),int(3)]),
1971		([int(1),int(4)],[int(1),int(4)]),([int(2)],[int(2)]),([int(2),int(3)],[int(2),int(3)]),
1972		([int(2),int(4)],[int(2),int(4)]),([int(3)],[int(3)]),([int(3),int(4)],
1973		[int(3),int(4)]),([int(4)],[int(4)])],_Res,_WF))).
1974		:- assert_must_succeed((bsets_clp:range_wf([([],[]),([int(0)],[int(0)]),
1975		([int(0),int(1)],[int(0),int(1)]),
1976		([int(0),int(3)],[int(0),int(3)]),([int(0),int(4)],[int(0),int(4)]),([int(1)],[int(1)]),
1977		([int(1),int(2)],[int(1),int(2)])],_Res,_WF))).
1978
1979
1980		:- block range_wf(-,-,?).
1981		range_wf(Rel,Res,WF) :- Res ==[],!, empty_set_wf(Rel,WF).
1982		range_wf(Rel,Res,WF) :- Rel ==[],!, empty_set_wf(Res,WF).
1983	?	range_wf(Rel,Res,WF) :- range_wf1(Rel,Res,WF),
1984		propagate_result_to_input(Res,Rel,range,WF).
1985
1986		:- block range_wf1(-,?,?).
1987		range_wf1(Rel,Res,WF) :-
1988		is_custom_explicit_set(Rel,range_wf1),
1989		range_of_explicit_set_wf(Rel,Range,WF), !,
1990	?	equal_object_wf(Range,Res,range_wf1,WF).
1991		range_wf1(Rel,Res,WF) :-
1992		% TO DO : propagate information that card of Res <= card of Rel; similar thing for domain
1993		expand_custom_set_to_list_wf(Rel,Relation,_,range_wf1,WF),
1994	?	newrange2(Relation,[],Res,WF),
1995		quick_propagate_range(Relation,Res,WF).
1996
1997
1998		:- block quick_propagate_range(-,?,?).
1999		quick_propagate_range([],_,_WF).
2000		quick_propagate_range([(_,Y)|T],FullRes,WF) :-
2001		quick_propagation_element_information(FullRes,Y,WF,FullRes1), % should we use a stronger check ?
2002		quick_propagate_range(T,FullRes1,WF).
2003
2004		:- block newrange2(-,?,?,?).
2005		newrange2([],_SoFar,Res,WF) :-
2006	?	empty_set_wf(Res,WF).
2007		newrange2([(X,Y)|T],SoFar,Res,WF) :-
2008		(Res==[]
2009		-> MemRes=pred_true, check_element_of_wf(Y,SoFar,WF)
2010		; membership_test_wf(SoFar,Y,MemRes,WF),
2011		card_greater_equal_check([(X,Y)|T],Res,MemRes), % check that card of Relation is greater or equal to Result; if equal set MemRes to pred_false
2012		(var(MemRes) -> prop_empty_pred_true(Res,MemRes) %,print(delay_range(Y,T)),nl
2013		% TO DO: we could look further in T if we can decide membership for other elements in T ?
2014		; true)
2015		),
2016	?	newrange3(MemRes,Y,T,SoFar,Res,WF).
2017
2018		:- block prop_empty_pred_true(-,?).
2019		prop_empty_pred_true([],R) :- !, R=pred_true.
2020		prop_empty_pred_true(_,_).
2021
2022		% card_greater_equal_check(Set1,Set2,EqFlag) : check that cardinality of Set1 is greater or equal to that of Set2; set EqFlag to pred_false if they are equal
2023		% checking is stopped if EqFlag becomes nonvar
2024		% tested by testcase 1061
2025		:- block card_greater_equal_check(-,?,-), card_greater_equal_check(?,-,-).
2026		card_greater_equal_check(_,_,Flag) :- nonvar(Flag),!. % no longer required; even though we could prune failure !? done later in newrange2/newdomain2 ??!!
2027		card_greater_equal_check([],Set2,Flag) :- !,empty_set(Set2),
2028		Flag=pred_false. % Flag set indicates that both sets have same size
2029		card_greater_equal_check(_,[],_) :- !.
2030		card_greater_equal_check([_|T],[_|R],Flag) :- !, card_greater_equal_check(T,R,Flag).
2031		% To do: deal with AVL args as Result + also use efficient_card_for_set for closures
2032		%card_greater_equal_check([_|T],Set,Flag) :- efficient_card_for_set(B,CardB,CodeB),!,
2033		% f: 1..7 -->> 1..n & n>=7 & n<10 still does not work well
2034		% TO DO: can we merge code with check_card_greater_equal
2035		card_greater_equal_check(_,_,_).
2036
2037
2038		:- block newrange3(-,?,?,?,?,?).
2039	?	newrange3(pred_true,_Y,T,SoFar,Res,WF) :- newrange2(T,SoFar,Res,WF).
2040		newrange3(pred_false,Y,T,SoFar,Res,WF) :-
2041		kernel_objects:mark_as_non_free(Y,range), % Y is linked to a particular X -> it is not free
2042		add_element_wf(Y,SoFar,SoFar2,WF),
2043	?	equal_cons_wf(Res,Y,Res2,WF),
2044	?	newrange2(T,SoFar2,Res2,WF).
2045
2046
2047		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([],Res,_WF),Res=[])).
2048		:- assert_must_succeed((bsets_clp:identity_relation_over_wf([int(1),int(2)],Res,_WF),
2049		Res=[(int(1),int(1)),(int(2),int(2))])).
2050		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:identity_relation_over_wf([int(2),int(4)],[(int(4),int(4)),(int(2),int(2))],WF),WF)).
2051		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:identity_relation_over_wf([int(1),int(2),int(4)],[(int(4),int(4)),(int(2),int(2)),(int(1),int(1))],WF),WF)).
2052		:- assert_must_fail((bsets_clp:identity_relation_over_wf([int(1)|_],_,_WF),fail)). /* check: no loop */
2053
2054		:- block identity_relation_over_wf(-,?,?).
2055		identity_relation_over_wf(Set1,IDRel,WF) :-
2056		expand_custom_set_to_list_wf(Set1,ESet1,_,identity_relation_over_wf,WF),
2057		identity_relation_over2(ESet1,IDRel,WF).
2058
2059		:- block identity_relation_over2(-,?,?).
2060		identity_relation_over2([],Res,WF) :- empty_set_wf(Res,WF).
2061		identity_relation_over2([X|T1],Res,WF) :- equal_cons_wf(Res,(X,X),T2,WF), % equal_object([(X,X)|T2],Res),
2062		identity_relation_over2(T1,T2,WF).
2063
2064
2065
2066		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:in_identity((int(1),int(1)),[int(1),int(2)],WF),WF)).
2067		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[int(1),int(2)],_WF))).
2068		:- assert_must_fail((bsets_clp:in_identity((int(3),int(3)),[int(1),int(2)],_WF))).
2069		:- assert_must_fail((bsets_clp:in_identity((int(1),int(2)),[],_WF))).
2070		in_identity((X,Y),Domain,WF) :-
2071	?	equal_object_wf(X,Y,in_identity,WF), check_element_of_wf(X,Domain,WF).
2072
2073		:- assert_must_fail((bsets_clp:not_in_identity((int(1),int(1)),[int(1),int(2)],_WF))).
2074		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(1),int(2)),[int(1),int(2)],WF),WF)).
2075		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:not_in_identity((int(3),int(3)),[int(1),int(2)],WF),WF)).
2076		:- assert_must_succeed((bsets_clp:not_in_identity((int(1),int(2)),[],_WF))).
2077		not_in_identity((X,Y),Domain,WF) :-
2078		equality_objects_wf(X,Y,Eq,WF),
2079		not_in_id2(Eq,X,Domain,WF).
2080
2081		:- block not_in_id2(-,?,?,?).
2082		not_in_id2(pred_true,X,Domain,WF) :- not_element_of_wf(X,Domain,WF).
2083		not_in_id2(pred_false,_,_,_).
2084
2085
2086		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([(int(1),int(2)),(int(3),int(4)),(int(5),int(6))], [(int(6),int(5)),(int(2),int(1)),(int(4),int(3))],WF),WF)).
2087		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([(int(1),int(2))], [(int(2),int(1))],WF),WF)).
2088		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:invert_relation_wf([], [],WF),WF)).
2089		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [])).
2090		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,X,_),X = [(int(2),int(2))])).
2091		:- assert_must_succeed((bsets_clp:invert_relation_wf(X,[(int(1),int(2)),(int(7),int(6))],_WF),
2092		X = [(int(2),int(1)),(int(6),int(7))])).
2093		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],X,_WF),
2094		X = [(int(2),int(1)),(int(6),int(7))])).
2095		:- assert_must_succeed((bsets_clp:invert_relation_wf([(int(1),int(2)),(int(7),int(6))],
2096		[(int(6),int(7)),(int(2),int(1))],_WF))).
2097		:- assert_must_succeed((bsets_clp:invert_relation_wf(closure([a,b],[string,boolean],b(truth,pred,[])),
2098		closure([b,a],[boolean,string],b(truth,pred,[])),_WF))).
2099
2100		:- block invert_relation_wf(-,-,?).
2101		invert_relation_wf(R,IR,WF) :-
2102		% (nonvar(R) -> invert_relation2(R,IR) ; invert_relation2(IR,R)).
2103		invert_relation2(R,IR,WF). % , print_term_summary(invert_relation(R,IR)).
2104		/* Optimization for some types of closures: Instead of expanding the closures, we just
2105		swap the parameters. This does not work with closures wich have only one parameter
2106		wich is a pair */
2107		invert_relation2(CS,R,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2108		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_1,WF).
2109		invert_relation2(R,CS,WF) :- nonvar(CS),is_custom_explicit_set_nonvar(CS),!,
2110		invert_explicit_set(CS,ICS), equal_object_wf(R,ICS,invert_relation2_2,WF).
2111		%invert_relation2(closure([P1,P2],[T1,T2],Clo),closure([P2,P1],[T2,T1],Clo)) :- !.
2112		invert_relation2(R,IR,WF) :- %try_expand_custom_set_wf(R,ER,invert,WF),
2113		% (nonvar(R) -> invert_relation3(R,IR)
2114		% ; invert_relation3(IR,R),(ground(IR)-> true ; invert_relation3(R,IR))).
2115		invert_relation3(R,IR,WF,1), invert_relation3(IR,R,WF,1).
2116
2117		:- block invert_relation3(-,?,?,?).
2118		invert_relation3(closure(P,T,B),Res,WF,_) :- invert_explicit_set(closure(P,T,B),ICS),
2119		equal_object_wf(Res,ICS,invert_relation3_1,WF).
2120		invert_relation3(avl_set(S),Res,WF,_) :- invert_explicit_set(avl_set(S),ICS),
2121		equal_object_wf(Res,ICS,invert_relation3_2,WF).
2122	?	invert_relation3([],Res,WF,_) :- empty_set_wf(Res,WF).
2123		invert_relation3([(X,Y)|T],Res,WF,Depth) :-
2124		D1 is Depth+1, get_wait_flag(D1,invert_relation3,WF,LWF),
2125		equal_cons_lwf(Res,(Y,X),IT,LWF,WF),
2126		invert_relation3(T,IT,WF,D1).
2127
2128
2129
2130
2131		tuple_of(X,Y,R) :- check_element_of((X,Y),R).
2132		%tuple_of_wf(X,Y,R,WF) :- check_element_of_wf((X,Y),R,WF).
2133
2134
2135		% RELATIONAL COMPOSITION (;)
2136
2137		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2138		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2139		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2140		[(int(11),int(12)),(int(11),int(33))],
2141		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2142		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(12)),
2143		[(int(11),int(12)),(int(11),int(33))],
2144		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2145		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:in_composition_wf((int(11),int(22)),
2146		[(int(11),[int(33),int(32)])],
2147		[([int(32),int(33)],int(22))],WF),WF)).
2148		:- assert_must_succeed(exhaustive_kernel_fail_check_wfdet(bsets_clp:in_composition_wf((int(11),int(33)),
2149		[(int(11),int(12)),(int(11),int(33))],
2150		[(int(33),int(12)),(int(33),int(22))],WF),WF)).
2151		% check if (X,Y) element of (F ; G)
2152		in_composition_wf((X,Y),F,G,WF) :-
2153		check_element_of_wf((X,Z1),F,WF), % no need to enumerate Z (TODO: check)
2154		equal_object_wf(Z1,Z2,check_element_of_wf,WF),
2155		check_element_of_wf((Z2,Y),G,WF).
2156
2157		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(11),int(33)),
2158		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2159		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:not_in_composition_wf((int(33),int(22)),
2160		[(int(11),int(33))],[(int(33),int(22))],WF),WF)).
2161
2162		% just evaluates arguments; TODO: improve or at least pass Type (for symbolic composition)
2163		not_in_composition_wf(Couple,F,G,WF) :-
2164		rel_composition_wf(F,G,Comp,_UnknownType,WF),
2165		not_element_of_wf(Couple,Comp,WF).
2166
2167		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([(int(1),int(2)),(int(3),int(4)),(int(5),int(6))], [(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],
2168		[(int(1),int(1)),(int(5),int(7)),(int(3),int(33))]))).
2169		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([], [(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],[]))).
2170		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([(int(6),int(7)),(int(2),int(1)),(int(22),int(22)),(int(4),int(33))],[],[]))).
2171		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_composition([],[],[]))).
2172		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2173		[(int(1),int(11))],X),X = [])).
2174		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],[],X),X = [])).
2175		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(6))],
2176		[(int(2),int(11))],X),
2177		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2178		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(2))],[(int(2),int(11))],X),
2179		ground(X), bsets_clp:equal_object(X,[(int(1),int(11)),(int(7),int(11))]))).
2180		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(7),int(5))],
2181		[(int(2),int(11)),(int(2),int(4))],X),
2182		kernel_objects:equal_object(X,[(int(1),int(11)),(int(1),int(4))]))).
2183		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),int(2)),(int(1),int(5))],
2184		[(int(2),int(11)),(int(5),int(11))],X),
2185		kernel_objects:equal_object(X,[(int(1),int(11))]))).
2186		:- assert_must_succeed((bsets_clp:rel_composition([(int(1),[int(1)]),(int(1),[int(2),int(5)])],
2187		[([int(1),int(2)],int(13)),([int(5),int(2)],int(12))],X),
2188		kernel_objects:equal_object(X,[(int(1),int(12))]))).
2189
2190		rel_composition(Rel1,Rel2,Comp) :- % only used in unit_tests above
2191		init_wait_flags(WF,[rel_composition]),
2192		rel_composition_wf(Rel1,Rel2,Comp,_UnknownType,WF),
2193	?	ground_wait_flags(WF).
2194
2195		:- block rel_composition_wf(-,-,?,?,?).
2196		rel_composition_wf(Rel1,Rel2,Comp,_,WF) :-
2197		(Rel1==[] ; Rel2==[]),
2198		!,
2199		empty_set_wf(Comp,WF).
2200		rel_composition_wf(Rel1,Rel2,Comp,Type,WF) :- rel_composition1(Rel1,Rel2,Comp,Type,WF).
2201
2202		:- block rel_composition1(-,?,?,?,?),rel_composition1(?,-,?,?,?).
2203		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2204		(Rel1==[] ; Rel2==[]),!, empty_set_wf(Comp,WF).
2205	?	rel_composition1(Rel1,Rel2,Comp,Type,WF) :- keep_symbolic(Rel1),
2206		(Rel2 = avl_set(_) -> SYMBOLIC=false ; SYMBOLIC=symbolic),
2207		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3),
2208		!,
2209		equal_object_wf(Comp,Rel3,rel_composition1_0,WF).
2210		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2211		rel_composition_for_explicit_set(Rel1,Rel2,Res),!, % treats finite Rel1 and avl_set for Rel2
2212		equal_object_wf(Res,Comp,rel_composition1_1,WF).
2213		rel_composition1(Rel1,Rel2,Comp,Type,WF) :- Rel2=closure(_,_,_),
2214	?	keep_symbolic(Rel2),
2215		(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
2216		-> !,
2217		expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1,WF),
2218		rel_compose_with_inf_fun(Relation1,Domain,Rel2,Comp,WF) % this is like map Rel2 over Rel1 in functional programmming
2219		; symbolic_composition(Rel1,Rel2,false,Type,Rel3),
2220		!,
2221		expand_custom_set_wf(Rel3,CRes,rel_composition,WF),% do we need to expand ?
2222	?	equal_object_optimized(CRes,Comp,rel_composition1_3)
2223		).
2224		rel_composition1(Rel1,Rel2,Comp,_,WF) :-
2225		expand_custom_set_to_list_wf(Rel1,Relation1,_,rel_composition1_2,WF),
2226		expand_custom_set_to_list_wf(Rel2,Relation2,_,rel_composition1_3,WF),
2227	?	rel_compose2(Relation1,Relation2,Comp,WF).
2228
2229		:- use_module(btypechecker, [l_unify_types_strict/2]).
2230		symbolic_composition(Rel1,Rel2,SYMBOLIC,Type,Rel3) :-
2231		get_set_type(Type,couple(TX,TZ)),
2232		get_relation_types(Rel1,TX1,TY1),
2233		get_relation_types(Rel2,TY2,TZ2),
2234		(l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2]) -> true
2235		; add_internal_error('Could not unify range/domain types: ',l_unify_types_strict([TX1,TY1,TZ],[TX,TY2,TZ2])),
2236		fail
2237		),
2238		ground((TX1,TY1,TZ)), % avoid creating a closure with non-ground type list
2239		rel_comp_closure(Rel1,Rel2,TX1,TY1,TZ,SYMBOLIC,Rel3).
2240		% generate a closure for {xx,zz | #(yy).(xx|->yy : Rel1 & yy|->zz : Rel2)}
2241		% TO DO: maybe detect special cases: Rel1 is a function/cartesian product, e.g., (((0 .. 76) * (0 .. 76)) * {FALSE}) ; {(FALSE|->0),(TRUE|->1)}
2242		:- use_module(bsyntaxtree, [safe_create_texpr/4]).
2243		rel_comp_closure(Rel1,Rel2,TX,TY,TZ,SYMBOLIC,closure(Args,Types,CBody)) :-
2244		Args = ['_rel_comp1','_rel_comp2'], Types = [TX,TZ],
2245		couple_member_pred('_rel_comp1',TX,'_zzzz_unary',TY,Rel1, Pred1),
2246		couple_member_pred('_zzzz_unary',TY,'_rel_comp2',TZ,Rel2, Pred2),
2247		UsedIds = ['_rel_comp1','_rel_comp2','_zzzz_unary'], % avoid having to call find_identifier_uses
2248		%conjunct_predicates([Pred1,Pred2],P12a), bsyntaxtree:check_computed_used_ids(P12a,UsedIds),
2249		safe_create_texpr(conjunct(Pred1,Pred2),pred,[used_ids(UsedIds)],P12),
2250		%b_interpreter_components:create_unsimplified_exists([b(identifier('_zzzz_unary'),TY,[])],P12,Body),
2251		bsyntaxtree:create_exists_opt_liftable([b(identifier('_zzzz_unary'),TY,[])],P12,Body), % cf Thales_All/rule_zcpa2 test 2287
2252		(SYMBOLIC==symbolic
2253		-> mark_bexpr_as_symbolic(Body,CBody)
2254		; CBody=Body).
2255
2256		% generate predicate for X|->Y : Rel
2257		couple_member_pred(X,TX,Y,TY,Rel, Pred) :-
2258		Pred = b(member(b(couple(b(identifier(X),TX,[]),
2259		b(identifier(Y),TY,[])),couple(TX,TY),[]),
2260		b(value(Rel),set(couple(TX,TY)),[])),pred,[]).
2261
2262
2263
2264		:- block rel_compose2(-,?,?,?).
2265		rel_compose2([],_,Out,WF) :- empty_set_wf(Out,WF).
2266		rel_compose2([(X,Y)|T],Rel2,Out,WF) :-
2267	?	rel_extract(Rel2,X,Y,OutXY,[],WF),
2268		% rel_extract(Rel2,X,Y,Out,OutRem),
2269	?	rel_compose2(T,Rel2,OutRem,WF),
2270	?	union_wf(OutRem,OutXY,Out,WF). % used to call union wihout wf; makes test 1394 fail
2271
2272		:- block rel_extract(-,?,?,?,?,?).
2273		rel_extract([],_,_,Rem,Rem,_WF). % should we use equal_object here ?????
2274		rel_extract([(Y1,Z)|T],X,Y,Res,Rem,WF) :-
2275	?	rel_extract(T,X,Y,CT,Rem,WF),
2276	?	equality_objects_wf(Y1,Y,EqRes,WF),
2277		rel_extract2(EqRes,Z,X,CT,Res).
2278
2279		:- block rel_extract2(-,?,?,?,?).
2280		rel_extract2(pred_true, Z, X,CT,Res) :- add_element((X,Z),CT,Res).
2281		rel_extract2(pred_false,_Z,_X,CT,Res) :- Res = CT.
2282
2283
2284		% relational composition of a finite relation with an infinite or symbolic function
2285		rel_compose_with_inf_fun(R,Dom,Fun,CompRes,WF) :- !,
2286		rel_compose_with_inf_fun_acc(R,Dom,Fun,[],CompRes,WF).
2287		:- block rel_compose_with_inf_fun_acc(-,?,?,?,?,?).
2288		rel_compose_with_inf_fun_acc([],_Dom,_Rel2,Acc,Comp,WF) :-
2289		equal_object_wf(Comp,Acc,rel_compose_with_inf_fun_acc,WF).
2290		rel_compose_with_inf_fun_acc([(X,Y)|T],Dom,Fun,Acc,CompRes,WF) :-
2291		membership_test_wf(Dom,Y,MemRes,WF), % check if Y is in the domain of the symbolic relation
2292		rel_compose_with_inf_fun_acc_aux(MemRes,X,Y,T,Dom,Fun,Acc,CompRes,WF).
2293
2294		:- block rel_compose_with_inf_fun_acc_aux(-,?,?,?, ?,?,?,?, ?).
2295		rel_compose_with_inf_fun_acc_aux(pred_true,X,Y,T,Dom,Fun,Acc,CompRes,WF) :-
2296		apply_to(Fun,Y,FY,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
2297		add_element_wf((X,FY),Acc,NewAcc,WF),
2298		rel_compose_with_inf_fun_acc(T,Dom,Fun,NewAcc,CompRes,WF).
2299		rel_compose_with_inf_fun_acc_aux(pred_false,_X,_Y,T,Dom,Fun,Acc,Comp,WF) :-
2300		rel_compose_with_inf_fun_acc(T,Dom,Fun,Acc,Comp,WF).
2301
2302		% TO DO: if we obtain a list such as [(int(1),X),...] in Acc rather than an avl_set,
2303		% we may still be able to sort and avoid quadratic comparisons if e.g.
2304		% first component is a data-type where equality can be decided by unification (integer, bool, global(GS), ...)
2305		% we could put the optimisation into add_element_wf ?
2306		% TO DO: special version for avl_set as relation?
2307
2308		/*
2309		Note: old version; has performance problem, 2021/02_Feb/CDS
2310		the add_element_wf calls below can only construct/instantiate result when empty_set_wf reached
2311		and a lot of pending co-routines pile up for long relation lists
2312
2313		:- block rel_compose_with_inf_fun(-,?,?,?,?).
2314		rel_compose_with_inf_fun([],_Dom,_Rel2,Comp,WF) :- empty_set_wf(Comp,WF).
2315		rel_compose_with_inf_fun([(X,Y)|T],Dom,Fun,CompRes,WF) :-
2316		membership_test_wf(Dom,Y,MemRes,WF), rel_compose_with_inf_fun_aux(MemRes,X,Y,T,Dom,Fun,CompRes,WF).
2317
2318		:- block rel_compose_with_inf_fun_aux(-,?,?,?, ?,?,?,?).
2319		rel_compose_with_inf_fun_aux(pred_true,X,Y,T,Dom,Fun,CompRes,WF) :-
2320		apply_to(Fun,Y,FY,WF),
2321		add_element_wf((X,FY),CT,CompRes,WF),
2322		rel_compose_with_inf_fun(T,Dom,Fun,CT,WF).
2323		rel_compose_with_inf_fun_aux(pred_false,_X,_Y,T,Dom,Fun,Comp,WF) :-
2324		rel_compose_with_inf_fun(T,Dom,Fun,Comp,WF).
2325		*/
2326
2327		:- assert_must_abort_wf(bsets_clp:rel_iterate_wf([],int(-1),_R,set(couple(integer,integer)),WF),WF).
2328		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([], int(2),[],set(couple(integer,integer)),_WF))).
2329		:- 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))).
2330		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:rel_iterate_wf([(pred_true,pred_true)], int(0),
2331		[(pred_true,pred_true),(pred_false,pred_false)],set(couple(boolean,boolean)),_WF))).
2332		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:rel_iterate_wf([(int(1),int(2)),
2333		(int(2),int(4)),(int(4),int(6))], int(2),[(int(1),int(4)),(int(2),int(6))],
2334		set(couple(integer,integer)),WF),WF)).
2335		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF), R=[],
2336		bsets_clp:equal_object(X,R))).
2337		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(1),X,set(couple(integer,integer)),_WF),
2338		R=[(int(1),int(2)),(int(2),int(3))],
2339		bsets_clp:equal_object(X,R))).
2340		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(2),X,set(couple(integer,integer)),_WF),
2341		R=[(int(1),int(2)),(int(2),int(3))],
2342		bsets_clp:equal_object(X,[(int(1),int(3))]))).
2343		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2344		R=[(int(1),int(2)),(int(2),int(3))],
2345		bsets_clp:equal_object(X,[]))).
2346		:- assert_must_succeed((bsets_clp:rel_iterate_wf(R,int(3),X,set(couple(integer,integer)),_WF),
2347		R=[(int(1),int(2)),(int(2),int(3)),(int(1),int(1))],
2348		bsets_clp:equal_object(X,[(int(1),int(1)),(int(1),int(2)),(int(1),int(3))]))).
2349
2350		rel_iterate_wf(Rel,int(Nr),Res,Type,WF) :-
2351		opt_push_wait_flag_call_stack_info(WF,b_operator_call(iterate,
2352		[Nr,Rel],unknown),WF2),
2353		rel_iterate1(Nr,Rel,Res,Type,WF2).
2354
2355		:- block rel_iterate1(-,?,?,?,?).
2356		rel_iterate1(X,Rel,Res,Type,WF) :-
2357		%value_variables(Rel,GrV),
2358		rel_iterate2(X,Rel,Res,Type,WF).
2359
2360		rel_iterate2(X,Rel,Res,Type,WF) :-
2361		( X=1 -> equal_object_wf(Res,Rel,rel_iterate2,WF)
2362		; X>1 -> X1 is X-1,
2363		rel_iterate2(X1,Rel,R1,Type,WF),
2364		rel_composition_wf(Rel,R1,Res,Type,WF)
2365		; X=0 -> rel_iterate0(Rel,Type,Res,WF)
2366		; add_wd_error('negative index in iterate',X,WF)
2367		).
2368
2369		:- use_module(bsyntaxtree,[get_set_type/2]).
2370		:- block rel_iterate0(?,-,?,?).
2371		rel_iterate0(_Rel,EType,Res,WF) :-
2372		get_set_type(EType,couple(Type,Type)),
2373		event_b_identity_for_type(Type,Res,WF).
2374
2375		:- use_module(typing_tools,[is_infinite_type/1]).
2376		event_b_identity_for_type(Type,Res,WF) :-
2377		create_texpr(identifier('_zzzz_unary'),Type,[],TIdentifier1), % was [generated]
2378		create_texpr(identifier('_zzzz_binary'),Type,[],TIdentifier2), % was [generated]
2379		(is_infinite_type(Type) -> Info = [prob_annotation('SYMBOLIC')] ; Info =[]),
2380		create_texpr(equal(TIdentifier1,TIdentifier2),pred,Info,TPred),
2381		construct_closure(['_zzzz_unary','_zzzz_binary'],[Type,Type],TPred,CRes),
2382		% for small types we could do: all_objects_of_type(Type,All), identity_relation_over_wf(All,CRes,WF)
2383		%, print(constructed_eventb_identity(Res)),nl
2384		equal_object_wf(Res,CRes,WF).
2385
2386
2387		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([],[(int(1),int(11))],[],_WF))).
2388		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2389		[(int(1),int(11))],[(int(1),(int(2),int(11)))],_WF))).
2390		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2391		[(int(2),int(11))],[],_WF))).
2392		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2393		[(int(2),int(11))],X,_WF),
2394		X = [])).
2395		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(7),int(6))],
2396		[(int(1),int(11))],X,_WF),
2397		kernel_objects:equal_object(X,[(int(1),(int(2),int(11)))]))).
2398		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(1),int(6))],
2399		[(int(1),int(11))],X,_WF),
2400		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(6),int(11)))]))).
2401		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2402		[(int(1),int(11)),(int(1),int(12))],X,_WF),
2403		kernel_objects:equal_object(X,[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))]))).
2404		:- assert_must_succeed((bsets_clp:direct_product_wf([(int(1),int(2)),(int(2),int(6))],
2405		[(int(1),int(11)),(int(1),int(12))],
2406		[(int(1),(int(2),int(11))),(int(1),(int(2),int(12)))],_WF))).
2407		:- 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))),
2408		avl_set(node((fd(1,'Name'),fd(2,'Name')),true,1,empty,node((fd(2,'Name'),fd(3,'Name')),true,0,empty,empty))),
2409		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)))
2410		,_WF))).
2411
2412		:- block direct_product_wf(-,?,?,?),direct_product_wf(?,-,?,?).
2413		direct_product_wf(Rel1,Rel2,Prod,WF) :-
2414		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),
2415		try_expand_and_convert_to_avl_with_check(Rel2,E2,direct_product),
2416	?	direct_product_wf1(E1,E2,Prod,WF).
2417
2418		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2419		direct_product_explicit_set(Rel1,Rel2,Res),!,
2420		equal_object_wf(Prod,Res,direct_product_wf1,WF).
2421		direct_product_wf1(Rel1,Rel2,Prod,WF) :-
2422		expand_custom_set_to_list_wf(Rel1,Relation1,_,direct_product_wf1_1,WF),
2423		expand_custom_set_to_list_wf(Rel2,Relation2,_,direct_product_wf1_2,WF),
2424	?	direct_product2(Relation1,Relation2,Prod,WF),
2425	?	direct_product_backwards(Relation1,Relation2,Prod,WF).
2426
2427		:- block direct_product2(-,?,?,?).
2428		direct_product2([],_,Out,WF) :- equal_object_wf(Out,[],direct_product2,WF).
2429		direct_product2([(X,Y)|T],Rel2,Out,WF) :-
2430	?	direct_product_tuple(Rel2,X,Y,Out,OutRem,WF),
2431		direct_product2(T,Rel2,OutRem,WF).
2432
2433		:- block direct_product_tuple(-,?,?,?,?,?).
2434		direct_product_tuple([],_,_,Res,Rem,WF) :- equal_object_optimized_wf(Res,Rem,direct_product_tuple,WF).
2435		direct_product_tuple([(X2,Z)|T],X,Y,Res,Rem,WF) :-
2436		direct_product_tuple(T,X,Y,CT,Rem,WF),
2437		equality_objects_wf(X2,X,EqRes,WF),
2438	?	direct_product_tuple3(EqRes,X,Y,Z,CT,Res,WF).
2439
2440		:- block direct_product_tuple3(-,?,?,?,?,?,?).
2441		direct_product_tuple3(pred_true,X,Y,Z,CT,Res,WF) :-
2442	?	equal_cons_wf(Res,(X,(Y,Z)),CT,WF). /* no need for add_element as output uniquely determines X,Y,Z !?*/
2443		direct_product_tuple3(pred_false,_X,_Y,_Z,CT,Res,WF) :- equal_object_optimized_wf(Res,CT,direct_product_tuple3,WF).
2444
2445		:- block direct_product_backwards(?,?,-,?).
2446		% Propagate information backwards from result to arguments
2447		direct_product_backwards(R1,R2,Prod,WF) :-
2448		((ground_value(R1) ; ground_value(R2)) -> true
2449		; expand_custom_set_to_list_wf(Prod,ProdList,_,direct_product_backwards,WF),
2450	?	direct_product_propagate_back(ProdList,R1,R2,WF)
2451		).
2452
2453		:- block direct_product_propagate_back(-,?,?,?).
2454		direct_product_propagate_back([],_,_,_WF).
2455		direct_product_propagate_back([(X,(Y,Z))|T],R1,R2,WF) :-
2456	?	check_element_of_wf((X,Y),R1,WF), check_element_of_wf((X,Z),R2,WF),
2457		direct_product_propagate_back(T,R1,R2,WF).
2458
2459		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([],[(int(3),int(4))],[]))).
2460		:- assert_must_succeed(exhaustive_kernel_check(bsets_clp:parallel_product([(int(1),int(2))],
2461		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2462		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2463		[(int(3),int(4))],X), ground(X),
2464		equal_object(X,[((int(1),int(3)),(int(2),int(4)))]))).
2465		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))],
2466		[(int(3),int(4))],[((int(1),int(3)),(int(2),int(4)))]))).
2467		:- assert_must_succeed((bsets_clp:parallel_product([(int(1),int(2))], [],X),X == [])).
2468		:- assert_must_succeed((bsets_clp:parallel_product([], [(int(3),int(4))],X),X == [])).
2469
2470	?	parallel_product(Rel1,Rel2,Prod) :- parallel_product_wf(Rel1,Rel2,Prod,no_wf_available).
2471
2472		:- block parallel_product_wf(-,?,?,?),parallel_product_wf(?,-,?,?).
2473		% NOTE: we now have in_parallel_product; as such parallel products are kept symbolic
2474		%parallel_product_wf(Rel1,Rel2,Prod,WF) :- (keep_symbolic(Rel1) -> true ; keep_symbolic(Rel2)),
2475		% print_term_summary(parallel_product(Rel1,Rel2,Prod)),nl,
2476		%% % TO DO: generate closure
2477		% %{xy,mn|#(x,y,m,n).(xy=(x,y) & mn=(m,n) & (x,m):S & (y,n):R)}
2478		% fail.
2479		parallel_product_wf(Rel1,Rel2,Prod,WF) :-
2480		expand_custom_set_to_list_wf(Rel1,Relation1,_,parallel_product_1,WF),
2481		expand_custom_set_to_list_wf(Rel2,Relation2,_,parallel_product_2,WF),
2482		parallel_product2(Relation1,Relation2,ProdRes,WF),
2483	?	equal_object_optimized_wf(ProdRes,Prod,parallel_product,WF).
2484
2485		:- use_module(kernel_equality,[conjoin_test/4]).
2486		%(Rel1||Rel2) = {(x,y),(m,n)| (x,m):Rel1 & (y,n):Rel2}
2487
2488		% TO DO: use this in b_interpreter_check:
2489		in_parallel_product_test(((X,Y),(M,N)),Rel1,Rel2,Result,WF) :-
2490	?	conjoin_test(MemRes1,MemRes2,Result,WF),
2491	?	membership_test_wf(Rel1,(X,M),MemRes1,WF),
2492		membership_test_wf(Rel2,(Y,N),MemRes2,WF).
2493
2494		:- 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)).
2495
2496		in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2497		in_parallel_product_test(El,Rel1,Rel2,pred_true,WF).
2498
2499
2500		:- 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))).
2501
2502		not_in_parallel_product_wf(El,Rel1,Rel2,WF) :-
2503	?	in_parallel_product_test(El,Rel1,Rel2,pred_false,WF).
2504
2505
2506		:- block parallel_product2(-,?,?,?).
2507		parallel_product2([],_,Out,WF) :- empty_set_wf(Out,WF).
2508		parallel_product2([(X,Y)|T],Rel2,Out,WF) :-
2509		parallel_product_tuple(Rel2,X,Y,Out,Tail,WF),
2510		parallel_product2(T,Rel2,Tail,WF).
2511
2512		:- block parallel_product_tuple(-,?,?,?,?,?).
2513		parallel_product_tuple([],_,_,Tail1,Tail2,WF) :- equal_object_wf(Tail1,Tail2,parallel_product_tuple,WF).
2514		parallel_product_tuple([(X2,Y2)|T],X,Y,Rel2,Tail,WF) :-
2515		equal_object_wf(Rel2,[((X,X2),(Y,Y2))|RT],parallel_product_tuple,WF),
2516		parallel_product_tuple(T,X,Y,RT,Tail,WF).
2517
2518
2519		% -------------------------------------------------
2520
2521		:- 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
2522		:- 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)).
2523		:- 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)).
2524		:- 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)).
2525		:- 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)).
2526		:- 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)).
2527		:- assert_must_fail((bsets_clp:not_partial_function([],[int(1)],[int(7)],_WF))).
2528		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[int(7)],_WF),
2529		X = [(int(1),int(7))])).
2530		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7)],_WF),
2531		X = [(int(2),int(7)),(int(1),int(7))])).
2532		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2533		[int(7),int(6)],_WF),
2534		X = [([(int(1),int(2))],int(7)),
2535		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2536		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2537		[int(7),int(6)],_WF),
2538		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2539		:- assert_must_fail((bsets_clp:not_partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2540		[int(7),int(6)],_WF),
2541		X = [([(int(1),int(2))],int(7)),
2542		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2543		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1)],[[int(7),int(6)]],_WF),
2544		X = [(int(1),[int(6),int(7)])])).
2545		:- assert_must_fail((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2546		X = [(int(2),int(7)),(int(1),int(7))])).
2547		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2548		X = [(int(2),int(7)),(int(2),int(6))])).
2549		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2550		X = [(int(2),int(7)),(int(1),int(2))])).
2551		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2552		X = [(int(2),int(7)),(int(3),int(6))])).
2553		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2554		X = [(int(2),int(7)),(int(2),int(5))])).
2555		:- assert_must_succeed((bsets_clp:not_partial_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
2556		X = [(int(1),int(7)),(int(2),int(6)),(int(2),int(7))])).
2557		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2558		X = [(int(1),int(7)),(int(5),int(75))])).
2559		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2560		X = [(int(1),int(7)),(int(0),int(7))])).
2561		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2562		X = [(int(1),int(7)),(int(-1),int(7))])).
2563		:- assert_must_succeed((bsets_clp:not_partial_function(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2564		X = [(int(1),int(7)),(int(0),int(7))])).
2565		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('Name'),global_set('Code'),_WF),
2566		X = [(fd(1,'Name'),fd(1,'Code'))])).
2567		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2568		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2569		:- assert_must_fail((bsets_clp:not_partial_function(X,global_set('NATURAL'),global_set('Code'),_WF),
2570		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2571		:- assert_must_succeed((bsets_clp:not_partial_function([(fd(1,'Code'),int(1)),(fd(1,'Code'),int(2))],
2572		global_set('Code'),global_set('NAT1'),_WF) )).
2573
2574		:- block not_partial_function(-,?,?,?).
2575		not_partial_function([],_Domain,_Range,_WF) :- !,fail.
2576		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
2577		% we do not need the Range; this means we can match more closures (e.g., lambda)
2578		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2579		not_subset_of_wf(FFDomain,Domain,WF).
2580		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF),
2581		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2582		not_both_subset_of(FFDomain,FFRange,Domain,Range,WF).
2583		not_partial_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2584		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
2585		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2586		get_range_id_expression(P,T,TRangeID),!,
2587		subset_test(FFDomain,Domain,SubRes,WF),
2588		when(nonvar(SubRes),
2589		(SubRes=pred_false -> true % not a subset -> it is not a partial function over the domain
2590		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
2591		not_partial_function(R,Domain,Range,WF) :-
2592		expand_and_convert_to_avl_set_warn(R,AER,not_partial_function,'ARG /: ? +-> ?',WF),!,
2593		% TO DO: expand_and_convert_to_avl_set_catch and provide symbolic treatment similar to partial_function
2594		% e.g., to support f = NATURAL1 * {22,33} & not(f: NATURAL1 +-> NATURAL)
2595		is_not_avl_partial_function(AER,Domain,Range,WF).
2596		not_partial_function(R,Domain,Range,WF) :-
2597		expand_custom_set_to_list_wf(R,ER,_,not_partial_function,WF),
2598		not_pf(ER,[],Domain,Range,WF).
2599
2600		is_not_avl_partial_function(AER,Domain,Range,WF) :-
2601		(is_avl_partial_function(AER)
2602		-> is_not_avl_relation_over_domain_range(AER,Domain,Range,WF)
2603		; true
2604		).
2605
2606		:- block not_pf(-,?,?,?,?).
2607		not_pf([],_,_,_,_) :- fail.
2608		not_pf([(X,Y)|T],SoFar,Dom,Ran,WF) :-
2609		membership_test_wf_with_force(SoFar,X,MemRes,WF),
2610		not_pf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2611
2612		:- block not_pf2(-,?,?,?,?,?,?,?).
2613		not_pf2(pred_true,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* then not a function */
2614		not_pf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
2615		membership_test_wf_with_force(Dom,X,MemRes,WF), % creates a choice point in SMT mode
2616		not_pf2a(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
2617
2618		:- block not_pf2a(-,?,?,?,?,?,?,?).
2619		not_pf2a(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF). /* function, but domain wrong */
2620		not_pf2a(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
2621		remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,_LWF,Done), %% provide _LWF ??
2622		not_pf2b(Done,X,Y,T,SoFar,Dom2,Ran,WF).
2623
2624		:- block not_pf2b(-, ?,?,?, ?,?,?, ?).
2625		not_pf2b(_Done, X,Y,T, SoFar,Dom2,Ran, WF) :-
2626		add_element_wf(X,SoFar,SoFar2,WF),
2627		(T==[] -> not_element_of_wf(Y,Ran,WF)
2628		; membership_test_wf_with_force(Ran,Y,MemRes,WF),
2629		prop_empty_pred_false(T,MemRes), % if T=[] -> Y must not be in Ran
2630		not_pf3(MemRes,T,SoFar2,Dom2,Ran,WF)).
2631
2632		:- block prop_empty_pred_false(-,?).
2633		prop_empty_pred_false([],R) :- !, R=pred_false.
2634		prop_empty_pred_false(_,_).
2635
2636		:- block not_pf3(-,?,?,?,?,?).
2637		not_pf3(pred_false,_T,_SoFar,_Dom2,_Ran,_WF). /* illegal range */
2638		not_pf3(pred_true,T,SoFar,Dom2,Ran,WF) :-
2639		not_pf(T,SoFar,Dom2,Ran,WF).
2640
2641		:- 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)).
2642		:- 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)).
2643		:- 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)).
2644		:- 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)).
2645		:- assert_must_succeed((bsets_clp:partial_function([],[int(1)],[int(7)]))).
2646		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1)],[int(7)]),
2647		X = [(int(1),int(7))])).
2648		:- assert_must_succeed((bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),
2649		equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2650		:- assert_must_succeed((findall(X,bsets_clp:partial_function(X,[int(1),int(2)],[int(7)]),L),
2651		length(L,Len), Len >= 4,
2652		(preferences:get_preference(convert_comprehension_sets_into_closures,true) -> true ; Len=4) )).
2653		:- assert_must_succeed((bsets_clp:partial_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2654		[int(7),int(6)]),
2655		equal_object(X,[([(int(1),int(2))],int(7)),
2656		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
2657		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2658		[int(7),int(6)],_WF),
2659		X = [([(int(2),int(3)),(int(1),int(3))],int(6))])).
2660		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2661		[int(7),int(6)],_WF),
2662		X = [([(int(1),int(2))],int(7)),
2663		([(int(2),int(3)),(int(1),int(3))],int(6))])).
2664		:- assert_must_succeed((bsets_clp:partial_function_wf(X,[int(1)],[[int(7),int(6)]],_WF),
2665		X = [(int(1),[int(6),int(7)])])).
2666		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2667		X = [(int(1),int(7)),(int(5),int(75))])).
2668		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2669		X = [(int(1),int(7)),(int(0),int(7))])).
2670		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('NATURAL1'),_WF),
2671		X = [(int(1),int(7)),(int(-1),int(7))])).
2672		:- assert_must_fail((bsets_clp:partial_function_wf(X,global_set('NATURAL1'),global_set('NATURAL1'),_WF),
2673		X = [(int(1),int(7)),(int(0),int(7))])).
2674		:- assert_must_fail((bsets_clp:partial_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
2675		X = [(int(2),int(7)),(int(2),int(6))])).
2676		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('Name'),global_set('Code'),_WF),
2677		X = [(fd(1,'Name'),fd(1,'Code'))])).
2678		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2679		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(88),fd(2,'Code'))])).
2680		:- assert_must_succeed((bsets_clp:partial_function_wf(X,global_set('NATURAL'),global_set('Code'),_WF),
2681		X = [(int(1),fd(1,'Code')),(int(0),fd(1,'Code')),(int(2),fd(2,'Code'))])).
2682
2683		partial_function(R,Domain,Range) :- init_wait_flags(WF,[partial_function]),
2684		partial_function_wf(R,Domain,Range,WF),
2685	?	ground_wait_flags(WF).
2686
2687		:- use_module(kernel_equality,[get_cardinality_powset_wait_flag/5]).
2688		:- use_module(closures,[is_lambda_value_domain_closure/5]).
2689		:- block partial_function_wf(-,-,?,?).
2690		partial_function_wf(R,_Domain,_Range,_WF) :- R==[], !.
2691		partial_function_wf(R,Domain,Range,WF) :- (Domain==[] ; Range==[]), !, empty_set_wf(R,WF).
2692		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2693		custom_explicit_sets:is_definitely_maximal_set(Range),
2694		% we do not need the Range; this means we can match more closures (e.g., lambda)
2695		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2696		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF).
2697		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF),
2698		% TODO: this will fail if is_definitely_maximal_set was true above !
2699		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2700		% same as for total_function_wf check
2701		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2702		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
2703		partial_function_wf(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2704		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2705		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2706		get_range_id_expression(P,T,TRangeID),
2707		!,
2708		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
2709		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2710		[b_operator(range,[FF]),Range],unknown),WF3),
2711		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF3). % we could use symbolic_range_subset_check
2712		partial_function_wf(R,Domain,Range,WF) :-
2713		expand_and_convert_to_avl_set_catch(R,AER,partial_function_wf,'ARG : ? +-> ?',ResultStatus,WF),!,
2714		(ResultStatus=avl_set
2715	?	-> is_avl_partial_function_over(AER,Domain,Range,WF)
2716		; % keep symbolic
2717		(debug_mode(off) -> true ; print('SYMBOLIC +-> check : '),translate:print_bvalue(R),nl),
2718		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
2719		symbolic_domain_subset_check(R,Domain,WF),
2720		symbolic_range_subset_check(R,Range,WF),
2721		symbolic_functionality_check(R,WF)
2722		).
2723		partial_function_wf(R,Domain,Range,WF) :-
2724		get_cardinality_powset_wait_flag(Domain,partial_function_wf,WF,Card,CWF),
2725		% probably we should compute real cardinality of set of partial functions over Domain +-> Range ?
2726		% the powset waitflag uses 2^Card as priority; is the number of partial functions when Range contains just a single element
2727		% slows down test 1088: TO DO investigate
2728		% get_cardinality_partial_function_wait_flag(Domain,Range,partial_function_wf,WF,Card,_,CWF),
2729		%% Maybe we should only enumerate partial functions for domain variables ; e.g., not f <+ {x |-> y} : T +-> S
2730		%% print_bt_message(pf_dom_card(Card)),nl, %%%
2731		% probably we should use a special version when R is var
2732		propagate_empty_set_wf(Domain,dom_pf,R,WF),
2733		propagate_empty_set_wf(Range,ran_pf,R,WF),
2734	?	(var(R) -> pf_var_r(R,var,Domain,Range,Card,WF,CWF) ; pf_var_r(R,nonvar,Domain,Range,Card,WF,CWF)).
2735
2736		% symbolic dom(R) <: Domain check for closures
2737		symbolic_domain_subset_check(R,Domain,WF) :-
2738		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2739		[b_operator(domain,[R]),Domain],unknown),WF2),
2740		domain_subtraction_wf(Domain,R,Res,WF2), % works symbolically
2741		(debug_mode(off) -> true ; print('Domain Violations: '),translate:print_bvalue(Res),nl),
2742		empty_set_wf(Res,WF2). % empty_set does a symbolic treatment calling gen_typed_ids and b_not_test_exists:
2743		% symbolic ran(R) <: Range check for closures
2744		symbolic_range_subset_check(R,Range,WF) :-
2745		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2746		[b_operator(range,[R]),Range],unknown),WF2),
2747		range_subtraction_wf(R,Range,Res,WF2), % works symbolically
2748		(debug_mode(off) -> true ; print('Range Violations: '),translate:print_bvalue(Res),nl),
2749		empty_set_wf(Res,WF2). % works symbolically
2750		symbolic_functionality_check(Closure,WF) :-
2751		custom_explicit_sets:symbolic_functionality_check_closure(Closure,ViolationsClosure),!,
2752		(debug_mode(off) -> true ; print('FUNCTIONALITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2753		empty_set_wf(ViolationsClosure,WF). % works symbolically
2754		symbolic_functionality_check(R,WF) :-
2755		add_error_wf(symbolic_functionality_check,'Could not check functionality of:',R,R,WF).
2756
2757		symbolic_injectivity_check(Closure,WF) :-
2758		custom_explicit_sets:symbolic_injectivity_check_closure(Closure,ViolationsClosure),!,
2759		(debug_mode(off) -> true ; print('INJECTIVITY Violations: '),translate:print_bvalue(ViolationsClosure),nl),
2760		empty_set_wf(ViolationsClosure,WF). % works symbolically
2761		symbolic_injectivity_check(R,WF) :-
2762		add_error_wf(symbolic_functionality_check,'Could not check injectivity of:',R,R,WF).
2763
2764
2765		is_avl_partial_function_over(AER,Domain,Range,WF) :-
2766		is_avl_partial_function(AER),
2767	?	is_avl_relation_over_domain(AER,Domain,WF),
2768	?	is_avl_relation_over_range(AER,Range,WF).
2769
2770		% symbolically check that the range of lambda closure is a subset of a given Range
2771		% TRangeID is obtained by calling get_range_id_expression(P,T,TRangeID)
2772		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2773		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
2774		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2775		% CHECK not(#P.(Pred & TRangeID /: Range))
2776		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2777		is_empty_closure_wf(P,T,Pred2,WF2). % do we need to rename _lambda_result_ using rename_lambda_result_id ?
2778		% now the negation thereof:
2779		check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF) :-
2780		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
2781		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2782		% CHECK (#P.(Pred & TRangeID /: Range))
2783		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2784		is_non_empty_closure_wf(P,T,Pred2,WF2).
2785		test_lambda_closure_range(P,T,Pred,TRangeID,Range,Res,WF) :-
2786		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset, % it is actually a reify check
2787		[b_operator(range,[closure(P,T,Pred)]),Range],unknown),WF2),
2788		% reify not(#P.(Pred & TRangeID /: Range))
2789		get_not_in_range_pred_aux(Pred,TRangeID,Range,Pred2),
2790		test_empty_closure_wf(P,T,Pred2,Res,WF2).
2791
2792		get_not_in_range_pred_aux(Pred,TRangeID,Range,NewPred) :- % construct (Pred & TRangeID /: Range)
2793		ExpectedRange = b(value(Range),set(RanT),[]),
2794		get_texpr_type(TRangeID,RanT),
2795		safe_create_texpr(not_member(TRangeID,ExpectedRange),pred,NotMemCheck),
2796		conjunct_predicates([Pred,NotMemCheck],NewPred).
2797
2798
2799		% if first argument is empty, second argument must also be empty
2800		:- block propagate_empty_set_wf(-,?,?,?).
2801		propagate_empty_set_wf([],_PP,A,WF) :- !, %print(prop_empty(_PP,A)),nl,
2802		kernel_objects:empty_set_wf(A,WF). % TO DO: add WF
2803		propagate_empty_set_wf(_,_,_,_).
2804
2805		:- block pf_var_r(-,?,?,?,?,?,-).
2806		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
2807		expand_and_convert_to_avl_set_warn(R,AER,pf_var_r,'ARG : ? +-> ?',WF),!,
2808	?	is_avl_partial_function_over(AER,Domain,Range,WF).
2809		pf_var_r(R,_,Domain,Range,Card,WF,CWF) :-
2810		expand_custom_set_to_list_wf(R,ER,_,partial_function_wf,WF),
2811		%get_last_wait_flag(partial_fun(Domain),WF,LWF),
2812	?	pf_w(ER,[],Domain,Range,Card,_Large,WF,CWF).
2813
2814		pf_w(T,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2815		(Card==0 -> T=[]
2816	?	; pf(T,SoFar,Dom,Ran,Card,Large,WF,LWF)).
2817
2818		:- block pf(-,?,?,?,?,?,?,-).
2819		pf(LIST,_,_,_,_,_WF,_,_LWF) :- LIST==[],!. % avoid leaving choicepoint
2820		pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF) :- nonvar(AVL),AVL=avl_set(_A),
2821		add_internal_error('AVL arg: ',pf(AVL,SoFar,Dom,Ran,Card,Large,WF,LWF)),fail.
2822		pf([],_,_,_,_,_WF,_,_LWF).
2823		pf(LIST,SoFar,Dom,Ran,Card,Large,WF,LWF) :-
2824		(var(LIST) -> ListWasVar = true ; ListWasVar = false), % is ListWasVar = true we are doing the enumeration driven by LWF being ground
2825		LIST = [(X,Y)|T],
2826	?	dec_card(Card,NC),/* Card ensures we do not build too big lists */
2827		Dom \== [],
2828	?	remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done),
2829	?	check_element_of_wf(Y,Ran,WF),
2830	?	pf1(Done, X,Y,T,SoFar,Dom2,Ran,NC,Large,WF,LWF).
2831
2832		:- block dec_card(-,?).
2833		dec_card(inf,NewC) :- !, NewC=inf.
2834		dec_card(inf_overflow,NewC) :- !, NewC=inf_overflow.
2835		dec_card(C,NewC) :- C>0, NewC is C-1.
2836
2837		:- block pf1(-, ?,?,?,?,?,?,?,?,?,?).
2838		pf1(_Done, X,_Y,T,SoFar,Dom2,Ran,Card,Large,WF,LWF) :-
2839		not_element_of_wf(X,SoFar,WF), /* check that it is a function */
2840		%% check_element_of_wf(Y,Ran,WF), % this check is now done above in pf
2841		add_new_element_wf(X,SoFar,SoFar2,WF),
2842	?	pf_w(T,SoFar2,Dom2,Ran,Card,Large,WF,LWF).
2843
2844		remove_domain_element(ListWasVar,X,Y,Dom,Dom2,Large,WF,LWF,Done) :- compute_large(Dom,Large),
2845		((ListWasVar==true,var(X),var(Y),Large==false,
2846		preference(convert_comprehension_sets_into_closures,false), % not in symbolic mode
2847		ground_value(Dom))
2848		-> %% (X, Y are free and we drive the enumeration: we can influence which element is taken from Dom
2849		remove_a_minimal_element(X,Dom,Dom2,WF,Done) %%%%%%%%%% added Jul 15 2008
2850	?	; remove_element_wf_if_not_infinite_or_closure(X,Dom,Dom2,WF,LWF,Done)
2851		).
2852		compute_large(Dom,Large) :- % check if the domain is large; ensure that we compute this only once
2853		(nonvar(Large) -> true
2854		; var(Dom) -> true
2855		; dont_expand_this_explicit_set(Dom) -> Large=large
2856		; Large=false).
2857
2858		:- assert_must_succeed(( bsets_clp:remove_a_minimal_element(X,[int(1)],R,_WF,Done),
2859		X==int(1), Done==true, R=[] )).
2860		:- 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),
2861		X==int(2), Done==true, R=[int(3)] )).
2862		:- 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),
2863		X==int(1), R=[int(2),int(3)], Done==true )).
2864		:- 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),
2865		X==int(3), R=[], Done==true )).
2866		:- assert_must_succeed(( init_wait_flags(WF), CL=closure(['_zzzz_binary'],[integer],b(member( b(identifier('_zzzz_binary'),integer,[]),
2867		b(interval(b(value(int(1)),integer,[]),b(value(int(10)),integer,[])),set(integer),[])),pred,[])),
2868		bsets_clp:remove_a_minimal_element(X,CL,R,WF,Done), ground_wait_flags(WF),
2869		X=int(9), Done==true, kernel_objects:equal_object(R,[int(10)]) )).
2870
2871		/* usage: restrict number of possible choices if element to remove is free */
2872		/* select one element; and disallow all elements appearing before it in the list */
2873		remove_a_minimal_element(X,Set,Res,WF,Done) :-
2874		expand_custom_set_to_list_wf(Set,ESet,EDone,remove_a_minimal_element,WF),
2875		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done).
2876
2877		:- use_module(kernel_equality,[get_cardinality_wait_flag/4]).
2878		:- block remove_a_minimal_element2(?,?,-,?,?,?).
2879		remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done) :- var(ESet),
2880		% should not happen as we wait for EDone
2881		add_internal_error('Illegal call: ',remove_a_minimal_element2(X,ESet,EDone,Res,WF,Done)),
2882		fail.
2883		remove_a_minimal_element2(X,ESet,_EDone,Res,WF,Done) :-
2884		ESet \= [],
2885		(ESet = [El]
2886		-> X=El, empty_set_wf(Res,WF), Done=true % only one choice
2887		; get_cardinality_wait_flag(ESet,remove_a_minimal_element2,WF,CWF),
2888		remove_a_minimal_element3(X,ESet,Res,WF,Done,CWF)
2889		).
2890
2891		:- block remove_a_minimal_element3(?,?,?,?,?,-).
2892		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- var(Res), !,
2893		append(_,[X|TRes],ESet), % WHAT IF Res has been instantiated in the meantime ???
2894		equal_object_wf(Res,TRes,remove_a_minimal_element2_2,WF),Done=true.
2895		remove_a_minimal_element3(X,ESet,Res,WF,Done,_) :- %print(remove_min_nonvar_res(Res)),nl,
2896		equal_cons_wf(ESet,X,Res,WF), Done=true.
2897
2898
2899		% reified version of partial function test partial_function_wf:
2900		:- 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)).
2901		:- 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)).
2902		:- 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)).
2903		:- 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)).
2904		:- 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)).
2905		:- 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)).
2906
2907		:- use_module(kernel_equality,[subset_test/4]).
2908		:- block partial_function_test_wf(-,?,?,-,?), partial_function_test_wf(?,-,-,-,?).
2909		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_true,!,
2910	?	partial_function_wf(FF,Domain,Range,WF).
2911		partial_function_test_wf(FF,Domain,Range,Res,WF) :- Res==pred_false,!,
2912		not_partial_function(FF,Domain,Range,WF). % TO DO: remove not_partial_function to use check_is_partial_function?
2913		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2914		custom_explicit_sets:is_definitely_maximal_set(Range),
2915		% we do not need the Range; this means we can match more closures (e.g., lambda)
2916		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
2917		subset_test(FFDomain,Domain,Res,WF).
2918		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF),
2919		% TODO: this will fail if is_definitely_maximal_set was true above !
2920		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
2921		% same as for total_function_wf check
2922		subset_test(FFDomain,Domain,DomainOk,WF),
2923		(DomainOk==pred_false -> Res = pred_false
2924		; conjoin_test(DomainOk,RangeOk,Res,WF),
2925		subset_test(FFRange,Range,RangeOk,WF)).
2926		partial_function_test_wf(FF,Domain,Range,Res,WF) :- nonvar(FF), FF=closure(P,T,Pred),
2927		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 +-> NATURAL
2928		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
2929		get_range_id_expression(P,T,TRangeID),
2930		!,
2931		subset_test(FFDomain,Domain,DomainOk,WF),
2932		(DomainOk == pred_false -> Res=pred_false
2933		; conjoin_test(DomainOk,RangeOk,Res,WF),
2934		test_lambda_closure_range(P,T,Pred,TRangeID,Range,RangeOk,WF)
2935		).
2936		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2937		expand_and_convert_to_avl_set_warn(R,AER,partial_function_test_wf,'ARG : ? +-> ?',WF),!,
2938		% TO DO: use expand_and_convert_to_avl_set_catch
2939		(is_avl_partial_function(AER)
2940		-> % TO DO: we could do something similar to this instead: is_not_avl_relation_over_domain_range
2941		domain_of_explicit_set_wf(avl_set(AER),FFDomain,WF),
2942		subset_test(FFDomain,Domain,DomainOk,WF),
2943		(DomainOk == pred_false -> Res=pred_false
2944		; range_of_explicit_set_wf(avl_set(AER),FFRange,WF),
2945		conjoin_test(DomainOk,RangeOk,Res,WF),
2946		subset_test(FFRange,Range,RangeOk,WF)
2947		)
2948		; Res=pred_false).
2949		partial_function_test_wf(R,Domain,Range,Res,WF) :-
2950		expand_custom_set_to_list_wf(R,ER,_,partial_function_test_wf,WF),
2951		check_is_partial_function_acc_wf(ER,[],Domain,Range,Res,WF).
2952
2953		:- block check_is_partial_function_acc_wf(-,?,?,?,?,?).
2954		check_is_partial_function_acc_wf([],_,_,_,Res,_WF) :- !, Res=pred_true.
2955		check_is_partial_function_acc_wf([(A,FA)|T],Acc,Dom,Ran,Res,WF) :- !,
2956		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF),
2957		(MemResDomRan==pred_false
2958		-> Res = pred_false
2959		; membership_test_wf(Acc,A,MemResNotFunc,WF),
2960		negate(MemResNotFunc,MemResFunctionality),
2961		conjoin_test(MemResDomRan,MemResFunctionality,PF_Head,WF),
2962		(PF_Head == pred_false -> Res = pred_false
2963		; T==[] -> Res=PF_Head
2964		; add_element_wf(A,Acc,NewAcc,WF),
2965	?	conjoin_test(PF_Head,PF_Tail,Res,WF),
2966		check_is_partial_function_acc_wf(T,NewAcc,Dom,Ran,PF_Tail,WF))
2967		).
2968
2969		check_pair_in_domain_range(A,FA,Dom,Ran,MemResDomRan,WF) :-
2970		membership_test_wf(Dom, A,MemResDom,WF), % use membership_test_wf_with_force for SMT mode ??
2971		(MemResDom == pred_false -> MemResDomRan = pred_false
2972		; membership_test_wf(Ran,FA,MemResRan,WF),
2973		conjoin_test(MemResDom,MemResRan,MemResDomRan,WF)).
2974
2975		:- 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)).
2976		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[int(7)]),
2977		X = [(int(1),int(7))])).
2978		:- assert_must_succeed((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
2979		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
2980		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3))]],[int(7),int(6)]),
2981		kernel_objects:equal_object(X,[([(int(1),int(3))],int(7)),([(int(1),int(2))],int(7))]))).
2982		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2983		[int(7),int(6)]),
2984		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
2985		([(int(2),int(3)),(int(1),int(3))],int(6))]))).
2986		:- assert_must_succeed((bsets_clp:total_function(X,[int(1)],[[int(7),int(6)]]),
2987		kernel_objects:equal_object(X,[(int(1),[int(6),int(7)])]))).
2988		:- assert_must_succeed((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
2989		[[int(7),int(6)]]),
2990		kernel_objects:equal_object(X,[([(int(1),int(2))],[int(6),int(7)]),
2991		([(int(2),int(3)),(int(1),int(3))],[int(6),int(7)])]))).
2992		:- assert_must_succeed((bsets_clp:total_function(X,[ [(int(1),int(3)),(int(2),int(3))]],
2993		[int(6)]),
2994		kernel_objects:equal_object(X,[ ([(int(2),int(3)),(int(1),int(3))], int(6)) ]))).
2995		:- assert_must_succeed((bsets_clp:total_function(X,global_set('Name'),
2996		[[],[fd(1,'Code'),fd(2,'Code')],[fd(1,'Code')],[fd(2,'Code')]]),
2997		kernel_objects:enumerate_basic_type(X,set(couple(global('Name'),set(global('Code'))))),
2998		kernel_objects:equal_object(X,[(fd(3,'Name'),[fd(2,'Code')]),(fd(1,'Name'),[fd(2,'Code')]),(fd(2,'Name'),[])]))).
2999
3000		%:- assert_must_succeed(( kernel_waitflags:init_wait_flags(WF),bsets_clp:total_function_wf(TF,global_set('Code'),
3001		% closure([zzzz],[set(set(couple(integer,boolean)))],
3002		% member(identifier(zzzz),
3003		% pow_subset(value(closure([zzzz],[set(couple(integer,boolean))],
3004		% member('ListExpression'(['Identifier'(zzzz)]),
3005		% 'Seq'(value([pred_true /* bool_true */,pred_false /* bool_false */])))))))),WF),
3006		% 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 */)]]),
3007		% (fd(2,'Code'), [[],[(int(1),pred_true /* bool_true */)],[(int(1),pred_true /* bool_true */),(int(2),pred_true /* bool_true */)]]) ]),
3008		% kernel_waitflags:ground_wait_flags(WF) )).
3009
3010		:- assert_must_succeed((bsets_clp:total_function([],[],[int(7)]))).
3011
3012		:- assert_must_fail((bsets_clp:total_function([],[int(1)],[int(7)]))).
3013		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3014		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
3015		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3016		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(5))]))).
3017		:- assert_must_fail((bsets_clp:total_function(X,[int(1),int(2)],[int(7),int(6)]),
3018		kernel_objects:equal_object(X,[(int(2),int(7))]))).
3019		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3020		[int(7),int(6)]),
3021		kernel_objects:equal_object(X,[([(int(1),int(2))],int(7)),
3022		([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3023		:- assert_must_fail((bsets_clp:total_function(X,[[(int(1),int(2))],[(int(1),int(3)),(int(2),int(3))]],
3024		[int(7),int(6)]),
3025		kernel_objects:equal_object(X,[([(int(1),int(3)),(int(1),int(3))],int(6))]))).
3026
3027		total_function(R,Domain,Range) :- init_wait_flags(WF,[total_function]),
3028		total_function_wf(R,Domain,Range,WF),
3029	?	ground_wait_flags(WF).
3030
3031
3032		:- assert_must_succeed((bsets_clp:total_function_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
3033		nonvar(X),X=[(A,B),(C,D)],A==int(1),C==int(2),\+ ground(B),\+ ground(D), B=int(7),D=int(7) )).
3034
3035		:- block total_function_wf(-,-,-,?).
3036		total_function_wf(FF,Domain,_Range,WF) :- FF == [],!,
3037		empty_set_wf(Domain,WF).
3038		total_function_wf(FF,Domain,Range,WF) :-
3039		Range == [],!,
3040		empty_set_wf(FF,WF), empty_set_wf(Domain,WF).
3041		total_function_wf(FF,Domain,Range,WF) :-
3042		% TO DO: if FF or Domain nonvar but \= [] -> check if other variable becomes []
3043	?	total_function_wf1(FF,Domain,Range,WF).
3044
3045		:- block total_function_wf1(?,-,?,?).
3046		total_function_wf1(FF,Domain,_Range,WF) :-
3047		FF==[],!,
3048		empty_set_wf(Domain,WF).
3049		total_function_wf1(FF,Domain,Range,WF) :-
3050		custom_explicit_sets:is_definitely_maximal_set(Range),
3051		% we do not need the Range; this means we can match more closures (e.g., lambda)
3052		(nonvar(FF),
3053		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF)
3054		-> !,
3055		equal_object_wf(FFDomain,Domain,total_function_wf1_1,WF)
3056		; var(FF),
3057		get_wait_flag1(WF,WF1), var(WF1),
3058		\+ (custom_explicit_sets:get_card_for_specific_custom_set(Domain,Card), number(Card)),
3059		% we have a total_function over a possibly infinite domain,
3060		% better wait: maybe a recursive of other closure will be produced for FF
3061		!,
3062		when( (nonvar(FF) ; nonvar(WF1)), total_function_wf1(FF,Domain,Range,WF))
3063		).
3064		total_function_wf1(FF,Domain,Range,WF) :- nonvar(FF),
3065		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3066		equal_object_wf(FFDomain,Domain,total_function_wf1_2,WF),
3067		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
3068		total_function_wf1(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3069		total_function_avl_set(AEF,Domain,Range,WF).
3070		total_function_wf1(FF,Domain,Range,WF) :-
3071		% want to replace FF by closure: needs to be a variable!
3072		var(FF),
3073		% if the total function can not be build up explicitly (i.e. infinite domain)
3074		% TODO: can / should this be relaxed?
3075		custom_explicit_sets:is_infinite_explicit_set(Domain), % get_card_for_specific_custom_set or is_infinite_or_symbolic_closure
3076		% TO DO: delay if Domain infinite or closure and not yet known and range is type
3077		kernel_objects:infer_value_type(Domain,set(DomT)),
3078		kernel_objects:infer_value_type(Range,set(RanT)),
3079		!,
3080		% IDEA : TF = %x.(x:Domain|DEFAULT) <+ SFF, where SFF is partial function and DEFAULT is some default value
3081		% build up a partial function instead (fulfilling all constraints)
3082		% better? : %x.(x:Domain|IF x:dom(SFF) THEN SFF(x) ELSE DEFAULT)?
3083		partial_function_wf(SFF,Domain,Range,WF),
3084		% next, build up a total function mapping everything to a default value
3085		% this function will be overriden by the partial function to fulfilling
3086		% given constraints
3087		% 1. identifiers for closure
3088		create_texpr(identifier('__domid__'),DomT,[],TDomId),
3089		create_texpr(identifier('__ranid__'),RanT,[],TRanId),
3090		% 2. domain identifier might take all values of the domain
3091		create_texpr(member(TDomId,b(value(Domain),set(DomT),[])),pred,[],DomMember),
3092		% 3. pick a single value for the range identifier
3093		check_element_of_wf(RangeElement,Range,WF),
3094		%% external_functions:observe_value(RangeElement,"range"),external_functions:observe_value(SFF,"pf"),
3095		create_texpr(equal(TRanId,b(value(RangeElement),RanT,[])),pred,[],RanMember),
3096		% 4. conjunct and form closure (should be treated symbolically)
3097		conjunct_predicates([RanMember,DomMember],Pred),
3098		Default = closure(['__domid__','__ranid__'],[DomT,RanT],Pred),
3099		% 5. override default values where needed
3100		override_relation(Default,SFF,FF,WF),
3101		get_last_wait_flag(enum_symb_tf,WF,LastWF),
3102		when(nonvar(LastWF), % if we enum too early test 1619 fails; see also test 2022
3103		% as partial_function_wf does not fully enumerate the new variable SFF we may have to enumerate SFF; see test 2328
3104		(enumerate_basic_type_wf(RangeElement,RanT,WF),
3105		enumerate_basic_type_wf(SFF,set(couple(DomT,RanT)),WF)
3106		)).
3107		total_function_wf1(R,Domain,Range,WF) :-
3108		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
3109		% TO DO: maybe avoid converting intervals which are not fully instantiated ?
3110		% 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
3111		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
3112		propagate_empty_set_wf(Range,tf_range,ER,WF), % if the range of a total function is empty then the function must be empty
3113	?	total_function_wf2(ER,EDomain,Range,WF).
3114
3115		:- block total_function_wf2(?,-,?,?).
3116		total_function_wf2(R,Domain,Range,WF) :- nonvar(R), R=avl_set(AEF), !,
3117		total_function_avl_set(AEF,Domain,Range,WF).
3118		total_function_wf2(R,Domain,Range,WF) :-
3119		cardinality_as_int_wf(Domain,int(Card),WF),
3120	?	total_function_wf3(R,Card,Domain,Range,WF).
3121
3122		:- use_module(kernel_card_arithmetic,[is_inf_or_overflow_card/1]).
3123		total_function_wf3(FF,Card,Domain,Range,WF) :-
3124		nonvar(FF),
3125		(number(Card) -> (Card >= 1000 -> true ; is_symbolic_closure(FF)) ; true),
3126		% note: we can have symbolic closures with a finite domain: /*@symbolic */ %p.(p:BOOL|(%t.(t:NATURAL|t+100)))
3127		custom_explicit_sets:dom_for_lambda_closure(FF,FFDomain),
3128		% we have a lambda closure where we cannot determine the range,
3129		% otherwise dom_range_for_specific_closure would have succeeded
3130		% example: f = %x.(x:NATURAL1|x+1) & f: NATURAL1 --> NATURAL
3131		FF = closure(P,T,Pred),
3132		get_range_id_expression(P,T,TRangeID),
3133		!,
3134		equal_object_wf(FFDomain,Domain,total_function1_closure,WF),
3135		% CHECK not(#P.(Pred & P /: Range))
3136		check_lambda_closure_range(P,T,Pred,TRangeID,Range,WF).
3137		total_function_wf3(R,Card,Domain,Range,WF) :- nonvar(Card),is_inf_or_overflow_card(Card),!,
3138		when(nonvar(R), total_function_symbolic(R,Domain,Range,WF)).
3139		total_function_wf3(R,Card,Domain,Range,WF) :-
3140		card_convert_int_to_peano(Card,PeanoCard),
3141		((nonvar(R);ground(PeanoCard))
3142		-> true
3143		; get_last_wait_flag(total_fun(Domain),WF,WF1)),
3144	?	when((nonvar(R);ground(PeanoCard);
3145		(nonvar(PeanoCard),nonvar(WF1))), /* mal 12/5/04: changed , into ; 17/3/2008: added WF1 */
3146		/* reason for delaying nonvar(Card): Card grounded bit by bit by cardinality; avoid
3147		triggering too early and missing tf_var */
3148		total_function1(R,Card,PeanoCard,Domain,Range,WF
3149		)).
3150
3151		:- use_module(bsyntaxtree,[safe_create_texpr/3]).
3152		:- use_module(library(lists),[last/2]).
3153		% for a closure get the identifier or proj expression that represents range values
3154		get_range_id_expression([PairID],[Type],Res) :- !,
3155		Type = couple(_,TX),
3156		TP = b(identifier(PairID),Type,[]),
3157		safe_create_texpr(second_of_pair(TP),TX,Res). % prj2(PairID) ,
3158		%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
3159		% but currently lambda closure detection in dom_for_lambda_closure cannot handle such closures anyway
3160		get_range_id_expression(P,T,b(identifier(ID),Type,[])) :- last(P,ID), last(T,Type).
3161
3162		total_function_avl_set(AEF,Domain,Range,WF) :-
3163		(Domain = avl_set(Dom) -> is_avl_total_function_over_domain(AEF,Dom)
3164		; is_avl_partial_function(AEF),
3165		domain_of_explicit_set_wf(avl_set(AEF),AEF_Domain,WF),
3166		equal_object_wf(AEF_Domain,Domain,total_function_avl_set,WF)
3167		),
3168		is_avl_relation_over_range(AEF,Range,WF).
3169
3170		total_function_symbolic(FF,Domain,Range,WF) :-
3171		(debug_mode(off) -> true ; print('SYMBOLIC --> check : '),translate:print_bvalue(FF),nl),
3172		% can deal with, e.g., f = %x.(x:NATURAL|x+1) & g = f <+ {0|->0} & g : INTEGER +-> INTEGER
3173		domain_wf(FF,Domain,WF),
3174		symbolic_range_subset_check(FF,Range,WF),
3175		symbolic_functionality_check(FF,WF).
3176
3177		total_function1(FF,Card,PeanoCard,Domain,Range,WF) :- nonvar(Card),is_inf_or_overflow_card(Card),
3178		nonvar(PeanoCard),is_inf_or_overflow_card(PeanoCard),!,
3179		total_function_symbolic(FF,Domain,Range,WF).
3180		total_function1(FF,_,_,Domain,Range,WF) :-
3181		expand_and_convert_to_avl_set_catch(FF,AEF,total_function1,'ARG : ? --> ?',ResultStatus,WF),!,
3182		(ResultStatus=avl_set -> total_function_avl_set(AEF,Domain,Range,WF)
3183		; % keep symbolic
3184		% TO DO: ensure no pending co-routine infinite_peano in card_convert_int_to_peano
3185		total_function_symbolic(FF,Domain,Range,WF)
3186		).
3187		total_function1(R,_,Card,Domain,Range,WF) :-
3188		try_expand_custom_set_wf(R,ER,total_function1,WF),
3189	?	total_function2(ER,Card,Domain,Range,WF).
3190
3191		total_function2(ER,Card,Domain,Range,WF) :-
3192		var(ER),ground(Card),!,
3193		tf_var(TotalFunction,[],Card,Domain,Range,WF),
3194		ER=TotalFunction.
3195		total_function2(ER,Card,Domain,Range,WF) :-
3196		(ground(Card)
3197		-> get_wait_flag(0,tot_fun,WF,LWF) % we seem to know the domain exactly now; see e.g. test 1316
3198		; 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 ?
3199	?	tf(ER,[],Card,Domain,Range,WF,LWF).
3200
3201		:- block tf(-,?,-,?,?,?,?),tf(-,?,?,?,?,?,-).
3202		tf([],_,0,Dom,_,WF,_) :- empty_set_wf(Dom,WF).
3203		tf(FUN,SoFar,s(Card),Dom,Ran,WF,LWF) :- var(FUN),nonvar(Dom), % try setting up skeleton for total fun
3204		remove_exact_first_element(X,Dom,Dom2),not_element_of_wf(X,SoFar,WF),var(FUN),!,
3205	?	FUN = [(X,Y)|T], tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3206		tf([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF,LWF) :-
3207		not_element_of_wf(X,SoFar,WF),
3208		remove_element_wf(X,Dom,Dom2,WF), %mal: 17/3/08 changed to _wf version
3209	?	tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF).
3210		tf(CS,SoFar,Card,Dom,Ran,WF,LWF) :- nonvar(CS), is_custom_explicit_set(CS),
3211		expand_custom_set_to_list_wf(CS,ER,_,tf,WF),
3212		tf(ER,SoFar,Card,Dom,Ran,WF,LWF).
3213		tf1(X,Y,T,SoFar,Card,Dom2,Ran,WF,LWF) :-
3214		check_element_of_wf(Y,Ran,WF),
3215		%when((nonvar(T);nonvar(Card)), /* mal 12/5/04: changed , into ; */
3216		add_new_element_wf(X,SoFar,SoFar2,WF), %%% try_expand_and_convert_to_avl
3217	?	tf(T,SoFar2,Card,Dom2,Ran,WF,LWF).
3218
3219		:- block tf_var(-,?,-,?,?,?).
3220		tf_var(F,_,Card,Dom,_,WF) :- Card==0,!,F=[],empty_set_wf(Dom,WF). % avoid choice point
3221		tf_var([],_,0,Dom,_,WF) :- empty_set_wf(Dom,WF).
3222		tf_var([(X,Y)|T],SoFar,s(Card),Dom,Ran,WF) :-
3223		/* supposes that X + Y are unbound */
3224		/* TO DO: rewrite like enumerate <-------------------------- */
3225		((var(X),var(Y)) -> true ; (print_message(warning,'Nonvar in tf_var: '),
3226		print_message(warning,((X,Y))))),
3227		remove_exact_first_element(X,Dom,Dom2),
3228		not_element_of_wf(X,SoFar,WF),
3229		check_element_of_wf(Y,Ran,WF),
3230		add_new_element_wf(X,SoFar,SoFar2,WF),
3231		tf_var(T,SoFar2,Card,Dom2,Ran,WF).
3232
3233
3234
3235		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1)],[int(7)]),
3236		X = [(int(1),int(7))])).
3237		:- assert_must_succeed((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3238		kernel_objects:equal_object(X,[(int(2),int(8)),(int(1),int(7))]))).
3239		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1)],[int(7),int(3)]),
3240		X = [(int(1),int(7))])).
3241		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(3)]),
3242		X = [(int(1),int(3)),(int(2),int(3))])).
3243		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3244		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
3245		:- assert_must_fail((bsets_clp:total_bijection(X,[int(1),int(2)],[int(7),int(8)]),
3246		X = [(int(1),int(7)),(int(1),int(8))])).
3247
3248
3249
3250		total_bijection(R,Domain,Range) :- init_wait_flags(WF,[total_bijection]),
3251		total_bijection_wf(R,Domain,Range,WF),
3252	?	ground_wait_flags(WF).
3253
3254		:- block total_bijection_wf(?,-,?,?).
3255		total_bijection_wf(FF,Domain,Range,WF) :- nonvar(FF),
3256		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3257		equal_object_wf(FFDomain,Domain,total_bijection_wf_1,WF),
3258		equal_object_wf(FFRange,Range,total_bijection_wf_2,WF).
3259		%(R,Domain,Range,WF) :- Domain==Range,!, print(eq_domain_range),nl, total_injection_wf(R,Domain,Range,WF).
3260		total_bijection_wf(R,Domain,Range,WF) :-
3261		same_cardinality_wf(Domain,Range,WF),
3262		total_injection_wf2(R,Domain,Range,WF). % TO DO: use cardinality_as_int_wf ? makes test 1194 fail
3263
3264		%Note: we used to call custom code: total_bijection_wf2(R,Domain,Card,Range,WF).
3265		% total_injection_wf2 gives a considerable performance boost, e.g., for test 1222 ClearSy/alloc_large.mch or NQueens with >->>
3266
3267		:- 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)).
3268		:- 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)).
3269		:- 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)).
3270		:- 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)).
3271		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1)],[int(7)],_WF),
3272		X = [(int(1),int(7))])).
3273		:- assert_must_fail((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3274		X = [(int(2),int(7)),(int(1),int(7))])).
3275		:- assert_must_succeed((bsets_clp:not_total_function([],[int(1)],[int(7)],_WF))).
3276		:- assert_must_succeed((bsets_clp:not_total_function([],[global_set('NAT1')],[global_set('Name')],_WF))).
3277		:- assert_must_succeed((bsets_clp:not_total_function([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3278		:- assert_must_succeed((bsets_clp:not_total_function([(int(1),int(7)), (int(2),int(1))],
3279		[int(1),int(2)],[int(7)],_WF))).
3280		:- assert_must_succeed((bsets_clp:not_total_function(X,[int(1),int(2)],[int(7),int(6)],_WF),
3281		X = [(int(2),int(7)),(int(2),int(6))])).
3282
3283		:- block not_total_function(-,?,?,?), not_total_function(?,-,?,?).
3284		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
3285		% we do not need the Range; this means we can match more closures (e.g., lambda)
3286		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
3287		not_equal_object_wf(FFDomain,Domain,WF).
3288		not_total_function(FF,Domain,Range,WF) :- nonvar(FF),
3289		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
3290		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
3291		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
3292		not_total_function(FF,Domain,Range,WF) :- nonvar(FF), FF=closure(P,T,Pred),
3293		% example: f = %t.(t : NATURAL|t + 100) & f /: NATURAL +-> NATURAL
3294		is_lambda_value_domain_closure(P,T,Pred, FFDomain,_Expr),
3295		get_range_id_expression(P,T,TRangeID),!,
3296		equality_objects_wf(FFDomain,Domain,SubRes,WF), % compare: subset_test for not_partial_function
3297		when(nonvar(SubRes),
3298		(SubRes=pred_false -> true % not equal -> it is not a total function over the domain
3299		; check_not_lambda_closure_range(P,T,Pred,TRangeID,Range,WF))).
3300		not_total_function(R,Domain,Range,WF) :-
3301		try_expand_and_convert_to_avl_with_check(R,ER,not_total_function_range),
3302		try_expand_and_convert_to_avl_unless_large_wf(Range,ERange,WF),
3303		not_total_function2(ER,Domain,ERange,WF).
3304
3305		% repeat block, in case Domain or R is a closure
3306		:- block not_total_function2(-,?,?,?), not_total_function2(?,-,?,?).
3307		not_total_function2(R,Domain,Range,WF) :-
3308		expand_and_convert_to_avl_set_warn(R,AER,not_total_function2,'ARG /: ? --> ?',WF),
3309		!,
3310		not_total_function_avl(AER,Domain,Range,WF).
3311		not_total_function2(R,Domain,ERange,WF) :-
3312		expand_custom_set_to_list_wf(R,ER,_,not_total_function2,WF),
3313		try_expand_and_convert_to_avl_with_check(Domain,EDomain,keep_intervals(1000),not_total_function_domain),
3314		not_tf(ER,[],EDomain,ERange,WF).
3315
3316		not_total_function_avl(_AER,Domain,_Range,_WF) :- is_infinite_explicit_set(Domain),!,
3317		true. % a finite AVL set cannot be a total function over an infinite domain
3318		not_total_function_avl(AER,Domain,Range,WF) :-
3319		expand_and_convert_to_avl_set_warn(Domain,ADom,not_total_function2,'? /: ARG --> ?',WF),
3320		!,
3321		(is_avl_total_function_over_domain(AER,ADom)
3322		->
3323		is_not_avl_relation_over_range(AER,Range,WF)
3324		; true
3325		).
3326		not_total_function_avl(AER,EDomain,ERange,WF) :-
3327		expand_custom_set_to_list_wf(avl_set(AER),ER,_,not_total_function_avl,WF),
3328		not_tf(ER,[],EDomain,ERange,WF).
3329
3330
3331		:- use_module(kernel_equality,[membership_test_wf_with_force/4]).
3332
3333		:- block not_tf(-,?,?,?,?).
3334		not_tf([],_,Domain,_,WF) :- not_empty_set_wf(Domain,WF).
3335		not_tf([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf_with_force(SoFar,X,MemRes,WF),
3336		not_tf2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3337
3338		:- block not_tf2(-,?,?,?, ?,?,?,?). %, not_tf2(?,?,?,?, -,?,?), not_tf2(?,?,?,?, ?,-,?).
3339		not_tf2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).% :- check_element_of_lazy(X,SoFar,WF).
3340		not_tf2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3341		%not_element_of_wf(X,SoFar,WF),
3342		membership_test_wf_with_force(Dom,X,MemRes,WF),
3343		not_tf3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3344
3345		:- block not_tf3(-, ?,?,?,?, ?,?,?).
3346		not_tf3(pred_false,_X,_Y,_T,_SoFar,_Dom,_Ran,_WF).
3347		not_tf3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3348		remove_element_wf(X,Dom,Dom2,WF),
3349		membership_test_wf_with_force(Ran,Y,MemRes,WF),
3350		not_tf4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3351
3352		:- block not_tf4(-, ?,?,?,?, ?,?,?).
3353		not_tf4(pred_false,_X,_Y,_T,_SoFar,_Dom2,_Ran,_WF).
3354		not_tf4(pred_true,X,_Y,T,SoFar,Dom2,Ran,WF) :-
3355		%check_element_of_wf(Y,Ran,WF), %DO WE NEED THIS ????
3356		add_new_element_wf(X,SoFar,SoFar2,WF),
3357		not_tf(T,SoFar2,Dom2,Ran,WF).
3358
3359
3360
3361		:- 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)).
3362		:- 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)).
3363		:- 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)).
3364		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1)],[int(7)],_WF),
3365		X = [(int(1),int(7))])).
3366		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3367		X = [(int(2),int(7)),(int(1),int(6))])).
3368		:- assert_must_fail((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3369		X = [(int(1),int(7)),(int(2),int(6))])).
3370		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(3)],_WF),
3371		X = [(int(1),int(3)),(int(2),int(3))])).
3372		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3373		X = [(int(2),int(7)),(int(1),int(7))])).
3374		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1)],[int(7),int(8)],_WF),
3375		X = [(int(1),int(7))])).
3376		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7)],_WF),
3377		X = [(int(2),int(7))])).
3378		:- assert_must_succeed((bsets_clp:not_total_bijection([],[int(1)],[int(7)],_WF))).
3379		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(7),int(7))],[int(1)],[int(7)],_WF))).
3380		:- assert_must_succeed((bsets_clp:not_total_bijection([(int(1),int(7)), (int(2),int(1))],
3381		[int(1),int(2)],[int(7)],_WF))).
3382		:- assert_must_succeed((bsets_clp:not_total_bijection(X,[int(1),int(2)],[int(7),int(6)],_WF),
3383		X = [(int(2),int(7)),(int(2),int(6))])).
3384
3385		:- block not_total_bijection(-,?,?,?), not_total_bijection(?,-,?,?).
3386		not_total_bijection(FF,Domain,Range,WF) :-
3387		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
3388		not_equal_object_wf((FFDomain,FFRange),(Domain,Range),WF).
3389		not_total_bijection(avl_set(_),Domain,_Range,_WF) :-
3390	?	is_infinite_explicit_set(Domain),!.
3391		% a finite set cannot be a total bijection over an infinite domain, see test 1641
3392		not_total_bijection(R,Domain,Range,WF) :-
3393		try_expand_custom_set_wf(R,ER,not_total_bijection,WF),
3394		not_tot_bij(ER,[],Domain,Range,WF).
3395
3396		:- block not_tot_bij(-,?,?,?,?).
3397		not_tot_bij([],_,Domain,Range,WF) :- empty_not_tot_bij(Domain,Range,WF).
3398		not_tot_bij([(X,Y)|T],SoFar,Dom,Ran,WF) :- membership_test_wf(SoFar,X,MemRes,WF),
3399		not_tot_bij2(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3400
3401		:- use_module(kernel_equality,[empty_set_test_wf/3]).
3402		:- block empty_not_tot_bij(-,?,?).
3403		empty_not_tot_bij(Domain,Range,WF) :-
3404		empty_set_test_wf(Domain,EqRes,WF),
3405		empty_not_tot_bij2(EqRes,Range,WF).
3406		:- block empty_not_tot_bij2(-,?,?).
3407		empty_not_tot_bij2(pred_false,_,_).
3408		empty_not_tot_bij2(pred_true,Range,WF) :- not_empty_set_wf(Range,WF).
3409
3410		:- block not_tot_bij2(-,?,?,?,?,?,?,?).
3411		not_tot_bij2(pred_true,_X,_,_T,_SoFar,_Dom,_Ran,_WF).
3412		not_tot_bij2(pred_false,X,Y,T,SoFar,Dom,Ran,WF) :-
3413		membership_test_wf(Dom,X,MemRes,WF),
3414		not_tot_bij3(MemRes,X,Y,T,SoFar,Dom,Ran,WF).
3415
3416		:- block not_tot_bij3(-,?,?,?,?,?,?,?).
3417		not_tot_bij3(pred_false,_X,_,_T,_SoFar,_Dom,_Ran,_WF). % X not a member of domain
3418		not_tot_bij3(pred_true,X,Y,T,SoFar,Dom,Ran,WF) :-
3419		remove_element_wf(X,Dom,Dom2,WF),
3420		membership_test_wf(Ran,Y,MemRes,WF),
3421		not_tot_bij4(MemRes,X,Y,T,SoFar,Dom2,Ran,WF).
3422
3423		:- block not_tot_bij4(-,?,?,?,?,?,?,?).
3424		not_tot_bij4(pred_false,_X,_,_T,_SoFar,_Dom2,_Ran,_WF). % Y not a member of range
3425		not_tot_bij4(pred_true,X,Y,T,SoFar,Dom2,Ran,WF) :-
3426		remove_element_wf(Y,Ran,Ran2,WF),
3427		add_element_wf(X,SoFar,SoFar2,WF),
3428		not_tot_bij(T,SoFar2,Dom2,Ran2,WF).
3429
3430
3431
3432		:- 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)).
3433		:- 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)).
3434		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_restriction_wf([],[int(2),int(3)],[],WF),WF)).
3435		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[int(1)],[],_WF))).
3436		:- assert_must_succeed((bsets_clp:range_restriction_wf([],[],[],_WF))).
3437		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(1)],[],_WF))).
3438		:- assert_must_succeed((bsets_clp:range_restriction_wf([(int(1),int(2))],[int(2)],[(int(1),int(2))],_WF))).
3439		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3440		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3441		kernel_objects:equal_object(X,R))).
3442		:- assert_must_succeed((bsets_clp:range_restriction_wf(X,Y,R,_WF),
3443		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],Y=global_set('Name'),
3444		kernel_objects:equal_object(X,R))).
3445		:- assert_must_fail((bsets_clp:range_restriction_wf(X,[fd(3,'Name')],R,_WF),
3446		X = [(int(1),fd(3,'Name')),(int(2),fd(1,'Name'))],
3447		kernel_objects:equal_object(X,R))).
3448
3449		:- block range_restriction_wf(-,?,?,?),range_restriction_wf(?,-,-,?).
3450
3451		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3452		ok_to_try_restriction_explicit_set(S,R,Res),
3453		range_restriction_explicit_set_wf(R,S,SR,WF),!,
3454		equal_object_wf(SR,Res,range_restriction,WF).
3455		range_restriction_wf(R,S,Res,WF) :- /* R |> S */
3456		expand_custom_set_to_list_wf(R,ER,_,range_restriction,WF),
3457	?	relation_restriction_wf(ER,S,Res,pred_true,range,WF).
3458
3459		% heuristic: should we try restriction_explicit_set or
3460		% is relation_restriction with its stronger constraint propagation better
3461		ok_to_try_restriction_explicit_set(S,R,Res) :-
3462		nonvar(S),
3463		(var(Res) -> true
3464		; S=avl_set(_),
3465		nonvar(R), R=avl_set(_) % otherwise constraint propagation from normal relation_restriction better
3466		).
3467
3468		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:range_subtraction_wf([],[int(2)],[],WF),WF)).
3469		:- 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)).
3470		:- 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)).
3471		:- 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)).
3472
3473		:- block range_subtraction_wf(-,?,?,?),range_subtraction_wf(?,-,-,?).
3474		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3475		S==[],!,
3476		equal_object_wf(R,Res,range_subtraction1,WF).
3477		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3478		ok_to_try_restriction_explicit_set(S,R,Res),
3479		range_subtraction_explicit_set_wf(R,S,SR,WF),!,
3480		equal_object_wf(SR,Res,range_subtraction2,WF).
3481		range_subtraction_wf(R,S,Res,WF) :- /* R |>> S */
3482		expand_custom_set_to_list_wf(R,ER,_,range_subtraction,WF),
3483	?	relation_restriction_wf(ER,S,Res,pred_false,range,WF).
3484
3485		:- 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)).
3486		:- 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)).
3487
3488		:- block in_range_restriction_wf(-,-,-,?).
3489		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3490	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3491		; preference(convert_comprehension_sets_into_closures,true)),
3492		!,
3493		Rel \== [], % avoid setting up check_element_of for X then
3494		% x |-> y : Rel |>> Set <=> x|->y : Rel & y: Set
3495		check_element_of_wf(Pair,Rel,WF),
3496		Pair = (_,P2),
3497		check_element_of_wf(P2,Set,WF).
3498		in_range_restriction_wf(Pair,Rel,Set,WF) :-
3499		range_restriction_wf(Rel,Set,Res,WF),
3500		check_element_of_wf(Pair,Res,WF).
3501
3502		:- 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)).
3503		:- 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)).
3504
3505		:- block not_in_range_restriction_wf(-,-,-,?).
3506		not_in_range_restriction_wf(Pair,Rel,Set,WF) :-
3507		range_restriction_wf(Rel,Set,Res,WF),
3508		not_element_of_wf(Pair,Res,WF).
3509
3510		:- 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)).
3511		:- 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)).
3512
3513		:- block in_range_subtraction_wf(-,-,-,?).
3514		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3515	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3516		; preference(convert_comprehension_sets_into_closures,true)),
3517		!,
3518		Rel \== [], % avoid setting up check_element_of for X then
3519		% x |-> y : Rel |>> Set <=> x|->y : Rel & y/: Set
3520		check_element_of_wf(Pair,Rel,WF),
3521		Pair = (_,P2),
3522		not_element_of_wf(P2,Set,WF).
3523		in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3524		range_subtraction_wf(Rel,Set,Res,WF),
3525		check_element_of_wf(Pair,Res,WF).
3526
3527		:- 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)).
3528		:- 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)).
3529
3530		:- block not_in_range_subtraction_wf(-,-,-,?).
3531		not_in_range_subtraction_wf(Pair,Rel,Set,WF) :-
3532		range_subtraction_wf(Rel,Set,Res,WF),
3533		not_element_of_wf(Pair,Res,WF).
3534
3535
3536
3537		:- 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)).
3538		:- 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)).
3539
3540		:- block in_domain_restriction_wf(-,-,-,?).
3541		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3542	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3543		; preference(convert_comprehension_sets_into_closures,true)),
3544		!,
3545		Rel \== [], % avoid setting up check_element_of for X then
3546		% x |-> y : Set <| Rel <=> x|->y : Rel & x: Set
3547		check_element_of_wf(Pair,Rel,WF),
3548		Pair = (P1,_),
3549		check_element_of_wf(P1,Set,WF).
3550		in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3551		domain_restriction_wf(Set,Rel,Res,WF),
3552		check_element_of_wf(Pair,Res,WF).
3553
3554		:- 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)).
3555		:- 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)).
3556
3557		:- block not_in_domain_restriction_wf(-,-,-,?).
3558		not_in_domain_restriction_wf(Pair,Set,Rel,WF) :-
3559		domain_restriction_wf(Set,Rel,Res,WF),
3560		not_element_of_wf(Pair,Res,WF).
3561
3562		:- 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)).
3563		:- 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)).
3564		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_restriction_wf([int(2),int(3)],[],[],WF),WF)).
3565		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],[],_WF))).
3566		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[],R,_WF), R==[])).
3567		:- assert_must_fail((bsets_clp:domain_restriction_wf(_,[],R,_WF), R=[int(_)|_])).
3568		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2)],[(int(1),int(2))],[],_WF))).
3569		:- assert_must_succeed((bsets_clp:domain_restriction_wf([],[(int(1),int(2))],[],_WF))).
3570		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3571		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(1)],[(int(1),int(2)),(int(2),_)],_,_WF))).
3572		:- assert_must_succeed((bsets_clp:domain_restriction_wf([int(2),int(1)],X,R,_WF),
3573		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3574		kernel_objects:equal_object(X,R))).
3575
3576
3577		:- block domain_restriction_wf(?,-,?,?),domain_restriction_wf(-,?,-,?).
3578		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3579		ok_to_try_restriction_explicit_set(S,R,Res),
3580		domain_restriction_explicit_set_wf(S,R,SR,WF),!,
3581		equal_object_wf(SR,Res,domain_restriction,WF).
3582		domain_restriction_wf(S,R,Res,WF) :- /* S <| R */
3583		expand_custom_set_to_list_wf(R,ER,_,domain_restriction,WF),
3584	?	relation_restriction_wf(ER,S,Res,pred_true,domain,WF).
3585
3586		% a predicate to compute domain/range restriction/subtraction
3587		:- block relation_restriction_wf(?,-,- ,?,?,?),
3588		relation_restriction_wf(-,?,? ,?,?,?).
3589		relation_restriction_wf([],_S,Res,_AddWhen,_DomOrRange,WF) :-
3590	?	empty_set_wf(Res,WF).
3591		relation_restriction_wf([(X,Y)|T],S,Res,AddWhen,DomOrRange,WF) :-
3592		(DomOrRange=domain
3593		-> membership_test_wf(S,X,MemRes,WF) % TO DO: pass WF !
3594		; membership_test_wf(S,Y,MemRes,WF)),
3595		(nonvar(MemRes)
3596		%MemRes==AddWhen % MemRes already set; we will ensure that (X,Y) in Res below; this slows down Alstom Compilation Regle !
3597		% 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 ?
3598		-> true %,(MemRes==AddWhen -> true ; print_term_summary(relation_restriction([(X,Y)|T],S,Res,AddWhen,DomOrRange)),nl)
3599		; (AddWhen=pred_true -> InResult=MemRes
3600		; negate(InResult,MemRes)), % from bool_pred
3601	?	membership_test_wf(Res,(X,Y),InResult,WF)
3602		% TO DO: same for explicit version; gets called e.g. if S = 1..n (1..n <| [1,2,3] = [1,2])
3603		% can now solve e.g. {x|x <| [1,2,3] = [1,2] & card(x)=2} = {{1,2}}
3604		% or x <| s = [1,2,3] \/ {29|->29} & x <: 1..100 & s = %i.(i:1..50|i)
3605		),
3606	?	relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF).
3607		:- block relation_restriction_aux(-,?,?,?,?,?, ?,?,?).
3608		relation_restriction_aux(MemRes,X,Y,T,S,Res,AddWhen,DomOrRange,WF) :-
3609		MemRes==AddWhen,!, % (X,Y) should be added to result
3610		% TO DO: collect result until we delay ? and then do equal_object ?
3611	?	equal_cons(Res,(X,Y),RT), % was : equal_object([(X,Y)|RT],Res),
3612		%equal_cons_wf(Res,(X,Y),RT,WF), % makes tests 982, 1302, 1303 fail; TO DO: investigate
3613		%when(nonvar(RT), % causes problem for test 982
3614	?	relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3615		relation_restriction_aux(_MemRes,_X,_,T,S,RT,AddWhen,DomOrRange,WF) :-
3616		% the couple is filtered out
3617	?	relation_restriction_wf(T,S,RT,AddWhen,DomOrRange,WF).
3618
3619
3620		:- 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)).
3621		:- 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)).
3622		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([int(1)],[],[],WF),WF)).
3623		:- assert_must_succeed(exhaustive_kernel_check_wf(bsets_clp:domain_subtraction_wf([],[(int(11),int(21))],[(int(11),int(21))],WF),WF)).
3624		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2))],[],_WF))).
3625		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(3)],[(int(1),int(2))],[(int(1),int(2))],_WF))).
3626		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(1)],[(int(1),int(2)),(int(2),int(X))],R,_WF),
3627		R=[(int(2),int(YY))], YY==X)).
3628		:- assert_must_succeed((bsets_clp:domain_subtraction_wf([int(5),int(3)],X,R,_WF),
3629		X = [(int(1),fd(3,'Name')),(int(2),fd(3,'Name'))],
3630		kernel_objects:equal_object(X,R))).
3631		:- block domain_subtraction_wf(?,-,?,?),domain_subtraction_wf(-,?,-,?).
3632		domain_subtraction_wf(S,R,Res,WF) :- S==[],!,
3633		equal_object_wf(R,Res,domain_subtraction1,WF).
3634		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3635		ok_to_try_restriction_explicit_set(S,R,Res),
3636		domain_subtraction_explicit_set_wf(S,R,SR,WF),!,
3637		equal_object_wf(SR,Res,domain_subtraction2,WF).
3638		domain_subtraction_wf(S,R,Res,WF) :- /* S <<| R */
3639		expand_custom_set_to_list_wf(R,ER,_,domain_subtraction,WF),
3640		try_expand_and_convert_to_avl_with_check(S,AS,keep_intervals(500),domain_subtraction),
3641		% (ground(ER) -> domain_subtraction_acc(ER,AS,[],Res) ;
3642	?	relation_restriction_wf(ER,AS,Res,pred_false,domain,WF)
3643		% )
3644		.
3645
3646
3647
3648		:- 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)).
3649		:- 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)).
3650
3651		:- block in_domain_subtraction_wf(-,-,-,?).
3652
3653		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3654	?	(treat_arg_symbolically(Set) ; treat_arg_symbolically(Rel)
3655		; preference(convert_comprehension_sets_into_closures,true)),
3656		!,
3657		Rel \== [], % avoid setting up check_element_of for X then
3658		% x |-> y : Set <<| Rel <=> x|->y : Rel & x/: Set
3659		check_element_of_wf(Pair,Rel,WF),
3660		Pair = (P1,_),
3661		not_element_of_wf(P1,Set,WF).
3662		in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3663		domain_subtraction_wf(Set,Rel,Res,WF),
3664		check_element_of_wf(Pair,Res,WF).
3665
3666
3667		:- 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)).
3668		:- 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)).
3669
3670		:- block not_in_domain_subtraction_wf(-,-,-,?).
3671		not_in_domain_subtraction_wf(Pair,Set,Rel,WF) :-
3672		domain_subtraction_wf(Set,Rel,Res,WF),
3673		not_element_of_wf(Pair,Res,WF).
3674
3675		% similar to kernel_objects, but adds case for [_|_]
3676		treat_arg_symbolically(X) :- var(X),!.
3677		treat_arg_symbolically([H|T]) :- \+ ground(H) ; treat_arg_symbolically(T).
3678		treat_arg_symbolically(global_set(_)).
3679		treat_arg_symbolically(freetype(_)).
3680		treat_arg_symbolically(closure(P,T,B)) :- \+ kernel_objects:small_interval(P,T,B).
3681
3682		:- 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)).
3683		:- 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)).
3684		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(1),int(3))],X,_WF),
3685		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(3))]))).
3686		:- assert_must_succeed((bsets_clp:override_relation([(int(1),int(2)),(int(2),int(4))],[(int(3),int(6))],X,_WF),
3687		kernel_objects:equal_object(X,[(int(2),int(4)),(int(1),int(2)),(int(3),int(6))]))).
3688
3689		:- block override_relation(-,-,?,?). % overwrite AST node
3690		override_relation(R,S,Res,WF) :- R==[],!, equal_object_wf(S,Res,override_relation1,WF).
3691		override_relation(R,S,Res,WF) :- S==[],!, equal_object_wf(R,Res,override_relation2,WF).
3692		override_relation(R,S,Res,WF) :- Res==[],!, empty_set_wf(S,WF), empty_set_wf(R,WF).
3693		override_relation(R,S,Res,WF) :- /* R <+ S */
3694		override_custom_explicit_set_wf(R,S,ORes,WF),!,
3695		equal_object_wf(ORes,Res,override_relation3,WF).
3696		override_relation(R,S,Res,WF) :- /* R <+ S */
3697		domain_wf(S,DS,WF),
3698		domain_subtraction_wf(DS,R,DSR,WF),
3699		union_wf(DSR,S,Res,WF). % in principle we could call disjoint_union_wf, but fails 1112, 1751
3700
3701		:- 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)).
3702		:- 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)).
3703		:- 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)).
3704		:- 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)).
3705
3706		:- block in_override_relation_wf(-,-,-,?).
3707		in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3708		check_element_of_wf(Pair,Rel1,WF).
3709		in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3710		check_element_of_wf(Pair,S,WF).
3711		in_override_relation_wf((X,Y),Rel1,S,WF) :-
3712	?	(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3713		; preference(convert_comprehension_sets_into_closures,true)),
3714		!,
3715		domain_wf(S,DS,WF),
3716		membership_test_wf(DS,X,MemRes,WF),
3717		in_override_aux(MemRes,X,Y,Rel1,S,WF).
3718		in_override_relation_wf(Pair,Rel1,S,WF) :-
3719		override_relation(Rel1,S,Res,WF),
3720		check_element_of_wf(Pair,Res,WF).
3721
3722		:- block in_override_aux(-,?,?,?,?,?).
3723		in_override_aux(pred_true,X,Y,_R,S,WF) :-
3724		check_element_of_wf((X,Y),S,WF).
3725		in_override_aux(pred_false,X,Y,R,_S,WF) :-
3726		check_element_of_wf((X,Y),R,WF).
3727
3728		:- 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)).
3729		:- 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)).
3730
3731		:- block not_in_override_relation_wf(-,-,-,?).
3732		not_in_override_relation_wf(Pair,Rel1,S,WF) :- S==[],!, % Pair: Rel1 <+ S
3733		not_element_of_wf(Pair,Rel1,WF).
3734		not_in_override_relation_wf(Pair,Rel1,S,WF) :- Rel1==[],!,
3735		not_element_of_wf(Pair,S,WF).
3736		not_in_override_relation_wf((X,Y),Rel1,S,WF) :-
3737	?	(treat_arg_symbolically(S) ; treat_arg_symbolically(Rel1)
3738		; preference(convert_comprehension_sets_into_closures,true)),
3739		!,
3740		domain_wf(S,DS,WF),
3741		membership_test_wf(DS,X,MemRes,WF),
3742		not_in_override_aux(MemRes,X,Y,Rel1,S,WF).
3743		not_in_override_relation_wf(Pair,Rel1,S,WF) :-
3744		override_relation(Rel1,S,Res,WF),
3745		not_element_of_wf(Pair,Res,WF).
3746
3747		:- block not_in_override_aux(-,?,?,?,?,?).
3748		not_in_override_aux(pred_true,X,Y,_R,S,WF) :-
3749		not_element_of_wf((X,Y),S,WF).
3750		not_in_override_aux(pred_false,X,Y,R,_S,WF) :-
3751		not_element_of_wf((X,Y),R,WF).
3752
3753		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:override([],int(1),int(3),[(int(1),int(3))],WF),WF)).
3754		:- 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)).
3755		:- 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)).
3756
3757		% override for a single pair
3758		:- block override(-,?,?,?,?), override(?,-,?,?,?),
3759		override(?,?,-,?,?). % also wait on Y; try to generate avl if possible; can only be used in substitution anyway
3760		/* R <+ {X |-> Y} as used by substitution R(X) := Y */
3761		override(R,X,Y,Res,WF) :-
3762		override_pair_explicit_set(R,X,Y,ORes),!,
3763		equal_object_wf(ORes,Res,override1,WF).
3764		override(R,X,Y,Res,WF) :-
3765		if(try_expand_custom_set_to_list(R,ER,_,override),
3766		(
3767		override2(ER,X,Y,[(X,Y)],ORes,WF),
3768	?	equal_object_wf(ORes,Res,override2,WF)),
3769		( %print_term_summary(exception(R)), % Virtual Timeout exception occured
3770		override_relation(R,[(X,Y)],Res,WF)
3771		)).
3772
3773		:- block override2(-,?,?,?,?,?).
3774		override2([],_X,_Y,Remainder,Res,WF) :- equal_object_optimized_wf(Remainder,Res,override2,WF). %equal_object(Remainder,Res).
3775		override2([(V,W)|T],X,Y,Remainder,Res,WF) :-
3776		equality_objects_wf(V,X,EqRes,WF),
3777		override2c(EqRes,V,W,T,X,Y,Remainder,Res,WF).
3778
3779		:- block override2c(-, ?,?,?, ?,?,?,?,?).
3780		override2c(pred_true,_V,_W,T,X,Y,_Remainder,Res,WF) :-
3781		equal_cons_wf(Res,(X,Y),T2,WF),
3782		override2(T,X,Y,[],T2,WF). /* set remainder to [], we have already added (X,Y) */
3783		override2c(pred_false,V,W,T,X,Y,Remainder,Res,WF) :-
3784		equal_cons_wf(Res,(V,W),T2,WF),
3785		override2(T,X,Y,Remainder,T2,WF).
3786
3787
3788
3789		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],WF),WF)).
3790		:- 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)).
3791		:- 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)).
3792		:- 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)).
3793		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(2)],_WF)).
3794		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(2)],[],_WF)).
3795		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2))],[int(3)],[],_WF)).
3796		:- assert_must_succeed((bsets_clp:image_wf([(int(1),int(2)),(int(1),int(3))],
3797		[int(X)],R,_WF), X=1, kernel_objects:equal_object(R,[int(2),int(3)]))).
3798		:- assert_must_succeed((bsets_clp:image_wf([([int(1),int(2)],int(6)),
3799		([int(1),int(2),int(3)],int(7)),
3800		([int(2),int(1)],int(8))],
3801		[[int(X),int(1)]],R,_WF), X=2,
3802		kernel_objects:equal_object(R,[int(6),int(8)]))).
3803		:- assert_must_succeed(bsets_clp:image_wf([(int(1),int(2)),(int(2),int(2))],[int(1),int(2)],[int(2)],_WF)).
3804		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[int(1)],_WF)).
3805		:- assert_must_fail(bsets_clp:image_wf([(int(1),int(2))],[int(1)],[],_WF)).
3806
3807
3808		:- block image_wf(-,?,?,?).
3809		image_wf(Rel,_,Res,WF) :- Rel==[],!,empty_set_wf(Res,WF).
3810		image_wf(Rel,S,Res,WF) :-
3811		image_for_id_closure(Rel,S,Img),!, % we don't require S to be known here
3812		equal_object_wf(Img,Res,image_wf_id_closure,WF).
3813		image_wf(Rel,S,Res,WF) :-
3814		image_wf0(Rel,S,Res,WF).
3815
3816		:- block image_wf0(?,-,?,?).
3817		image_wf0(Rel,S,Res,WF) :- /* Res = Rel[S] */
3818		(S==[] -> empty_set_wf(Res,WF)
3819		; opt_push_wait_flag_call_stack_info(WF,b_operator_call(image,[Rel,S],unknown),WF2),
3820		image1(Rel,S,Res,WF2) ).
3821
3822		keep_symbolic(R) :- var(R),!,fail.
3823		keep_symbolic(closure(_,_,_)) :- preferences:get_preference(convert_comprehension_sets_into_closures,true),!.
3824	?	keep_symbolic(R) :- dont_expand_this_explicit_set(R).
3825
3826		:- block image1(-,?,?,?).
3827		image1(Rel,S,Res,WF) :-
3828		image_for_explicit_set(Rel,S,Img,WF),!,
3829		equal_object_wf(Img,Res,image1_1,WF),
3830		quick_propagate_subset_range(Res,Rel,WF).
3831		%image1(Rel,S,Res,WF) :- expand_custom_set_to_list(S,ES),!, image_of_set(ES,Rel,Res,WF).
3832		image1(Rel,Set,Res,WF) :-
3833	?	keep_symbolic(Rel),
3834		(preferences:get_preference(convert_comprehension_sets_into_closures,true), % in this case keep_symbolic is always true
3835		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
3836		-> debug_println(9,infinite_for_image1(Set)),
3837		fail
3838		; true),
3839		( dom_for_specific_closure(Rel,Domain,function(_),WF)
3840		-> !, expand_custom_set_to_list_wf(Set,ESet,_,image1,WF), % TO DO: what if keep_symbolic(Set)
3841		image_for_inf_fun(ESet,Domain,Rel,[],Res,WF)
3842		; get_relation_types(Rel,DomType,RangeType),!,
3843		expand_custom_set_to_list_wf(Set,ESet,_,image1_2,WF),
3844	?	(is_symbolic_closure(Rel)
3845		-> Symbolic=symbolic_try_expand, ground_value_check((Rel,ESet),GRel) % also wait for ESet to be ground so that we can catch enumeration warning exceptions, cf. test 2428 when theorem and foralls not expanded
3846		; Symbolic=expand, ground_value_check(Rel,GRel)
3847		),
3848		when(nonvar(GRel), image_for_large_relation(ESet,Rel,Symbolic,DomType,RangeType,[],Res,WF))
3849		/* ; get_relation_types(Rel,DomType,RangeType),!,
3850		% Rel should not be expanded, compute closure by calculating {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3851		print_term_summary(image_for_large_relation(DomType,RangeType,Set,Rel)),nl,
3852		image_closure(Set,Rel,DomType,RangeType,Closure ),
3853		translate:print_bvalue(Closure),nl,
3854		expand_custom_set(Closure,CRes), print(expanded),nl,
3855		equal_object(CRes,Res,image1_2) */
3856		).
3857		image1(Rel,S,Res,WF) :-
3858		expand_custom_set_to_list_wf(Rel,Relation,_,image1_2,WF), % bad if Rel is a big closure !
3859		% image_for_list_relation(Relation,S,Res).
3860		propagate_singleton_image(Relation,S,Res,WF),
3861		% TO DO: we could propagate cardinality constraints about Relation,S and Res
3862		% we could also try to infer all_different constraints in case card(S)=card(Res) and f is a function
3863		image_for_list_relation(Relation,S,[],Res,WF). %,print_term_summary(image2_res(Relation,S,Res)).
3864
3865		% propagate that f[{x}] = {r1,...,rk} => x|->ri : f (or {x}*{r1,...,rk} <: f); see test 1532
3866		propagate_singleton_image(R,S,Res,_) :-
3867		(var(S) ; var(Res) ; nonvar(R), is_custom_explicit_set(R,psi)), !.
3868		propagate_singleton_image(Relation,S,avl_set(Res),WF) :-
3869		custom_explicit_sets:singleton_set(S,El), % we have the image by a singleton set {El}
3870		expand_custom_set_to_list_wf(avl_set(Res),LR,_,prop_singleton,WF),
3871		!,
3872		l_check_element_of(LR, El, Relation, WF). % propagate x|->ri : f (will force membership)
3873		propagate_singleton_image(_,_,_,_).
3874
3875		l_check_element_of([],_,_,_).
3876		l_check_element_of([H|T],El,Relation,WF) :-
3877		check_element_of_wf((El,H),Relation,WF),
3878		l_check_element_of(T,El,Relation,WF).
3879
3880		% quick_propagate_in_range(Set, Relation,WF) : propagate that Set <: ran(Relation)
3881		:- block quick_propagate_subset_range(-,?,?).
3882		quick_propagate_subset_range(avl_set(_),_,_) :- !.
3883		quick_propagate_subset_range([],_,_) :- !.
3884		quick_propagate_subset_range([H|T],Relation,WF) :- is_custom_explicit_set(Relation,range_wf1),
3885		range_of_explicit_set_wf(Relation,Range,WF), !,
3886		quick_propagation_element_information(Range,H,WF,NewRange),
3887		quick_propagate_subset_range2(T,NewRange,WF).
3888		quick_propagate_subset_range(_,_,_).
3889
3890		:- block quick_propagate_subset_range2(-,?,?).
3891		quick_propagate_subset_range2([H|T],NewRange,WF) :- !,
3892		quick_propagation_element_information(NewRange,H,WF,NewRange1),
3893		quick_propagate_subset_range2(T,NewRange1,WF).
3894		quick_propagate_subset_range2(_,_,_).
3895
3896		:- use_module(btypechecker, [unify_types_strict/2]).
3897		get_relation_types(Value,Domain,Range) :-
3898		kernel_objects:infer_value_type(Value,VT),
3899		unify_types_strict(VT,set(couple(Domain,Range))). % deal also with seq types
3900		% VT=set(couple(Domain,Range)).
3901
3902		:- block image_for_large_relation(-,?,?,?,?,?,?,?), image_for_large_relation(?,?,?,?,?,-,?,?).
3903	?	image_for_large_relation([],_,_,_,_,Acc,Res,WF) :- equal_object_wf(Acc,Res,WF).
3904		image_for_large_relation([XX|T],Rel,Symbolic,DomType,RangeType,Acc,Res,WF) :-
3905		get_image_singleton_closure(XX,DomType,RangeType,Rel, Par,TPara,Body),
3906		expand_closure_direct_if_possible(Symbolic,Par,TPara,Body,ImagesForXX,WF),
3907		union_wf(Acc,ImagesForXX,NewAcc,WF),
3908		(T == [] -> equal_object_wf(NewAcc,Res,WF)
3909		; image_for_large_relation(T,Rel,Symbolic,DomType,RangeType,NewAcc,Res,WF)).
3910
3911		get_image_singleton_closure(XX,DomType,RangeType,Rel, [yy], [RangeType], Body) :-
3912		Body = b(member(b(couple(b(value(XX),DomType,[]),
3913		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3914		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]).
3915		% TO DO: simplify above if we have Rel = closure(P,T,B); which we usually will
3916
3917		expand_closure_direct_if_possible(symbolic_try_expand,Par,Types,Body,Result,WF) :- !,
3918		catch_enumeration_warning_exceptions(
3919		custom_explicit_sets:expand_normal_closure_direct(Par,Types,Body,Result,_Done,WF),
3920		(mark_bexpr_as_symbolic(Body,SBody),
3921		Result = closure(Par,Types,SBody) % TODO: we could set definitely_symbolic for next iteration
3922		),
3923		false,
3924		ignore(image_for_large_relation)).
3925		expand_closure_direct_if_possible(definitely_symbolic,Par,Types,Body,Result,_WF) :- !,
3926		mark_bexpr_as_symbolic(Body,SBody),
3927		Result = closure(Par,Types,SBody).
3928		expand_closure_direct_if_possible(_,Par,Types,Body,Result,WF) :-
3929		% do not memoize this (many different values):
3930		custom_explicit_sets:expand_normal_closure_direct(Par,Types,Body,Result,_Done,WF).
3931
3932
3933		/* no longer used
3934		% construct a closure for {yy|#(xx).(xx:Set & xx|->yy:Rel)}
3935		image_closure(Set,Rel,DomType,RangeType,Closure ) :- custom_explicit_sets:singleton_set(Set,XX),!,
3936		% do not set up existential quantifier if Set is singleton set
3937		Closure = closure([yy],[RangeType],Body),
3938		Body = b(member(b(couple(b(value(XX),DomType,[]),
3939		b(identifier(yy),RangeType,[])),couple(DomType,RangeType),[]),
3940		b(value(Rel),set(couple(DomType,RangeType)),[])),pred,[]).
3941		image_closure(Set,Rel,DomType,RangeType,Closure ) :-
3942		Closure = closure([yy],[RangeType],Body),
3943		couple_member_pred(xx,DomType,yy,RangeType,Rel, Predxxyy),
3944		Body = b(exists([b(identifier(xx),DomType,[])],
3945		b(conjunct(
3946		b(member(b(identifier(xx),DomType,[]),b(value(Set),set(DomType),[])),pred,[]), % TO DO : force evaluation !
3947		Predxxyy),
3948		pred,[])),pred,[used_ids([yy])]).
3949		*/
3950
3951		% very similar to rel_compose_with_inf_fun, indeed f[S] = ran((id(S);f))
3952		:- block image_for_inf_fun(-,?,?,?,?,?).
3953		image_for_inf_fun([],_Dom,_Rel2,Acc,Comp,WF) :- equal_object_wf(Acc,Comp,WF).
3954		image_for_inf_fun([X|T],Dom,Fun,Acc,CompRes,WF) :-
3955		membership_test_wf(Dom,X,MemRes,WF),
3956		image_for_inf_fun_aux(MemRes,X,T,Dom,Fun,Acc,CompRes,WF).
3957
3958		:- block image_for_inf_fun_aux(-,?,?, ?,?,?,?,?).
3959		image_for_inf_fun_aux(pred_true,X,T,Dom,Fun,Acc,CompRes,WF) :-
3960		apply_to(Fun,X,FX,WF), % TO DO: generalize to image so that we can apply it also to infinite relations ?
3961		add_element_wf(FX,Acc,NewAcc,WF), % will block until Acc Known !!
3962		% TO DO USE: equal_cons_wf(CompRes,FX,CT,WF) + accumulator !,
3963		image_for_inf_fun(T,Dom,Fun,NewAcc,CompRes,WF).
3964		image_for_inf_fun_aux(pred_false,_X,T,Dom,Fun,Acc,Comp,WF) :-
3965		image_for_inf_fun(T,Dom,Fun,Acc,Comp,WF).
3966
3967
3968		/*
3969		:- block image_of_set(-,?,?,?,?), image_of_set(?,?,-,?,?).
3970		image_of_set([],Rel,ImageSoFar,Res,WF) :- equal_object(ImageSoFar,Res).
3971		image_of_set([H|T],Rel,ImageSoFar,Res,WF) :-
3972		image_of_element(Rel,H,ImageSoFar,SF2,WF),
3973		image_of_set(T,Rel,SF2,Res,WF).
3974
3975		image_of_element([],_,Acc,Res,WF) :- equal_object(Acc,Res).
3976		image_of_element([(A,B)|T],H,Acc,Res,WF) :- equality....
3977		image_of_element(avl_set(),H,Acc,Res,WF) :- ....
3978		image_of_element(closure(),....
3979		*/
3980
3981		% Computing the image of a relation which is stored as a list: traverse the relation
3982		:- block image_for_list_relation(-,?,?,?,?).
3983	?	image_for_list_relation([],_,_,Res,WF) :- empty_set_wf(Res,WF).
3984		image_for_list_relation([(X,Y)|T],S,ImageSoFar,Res,WF) :-
3985		((T==[], definitely_not_empty(Res))
3986		-> MemRes=pred_true, % we need at least one more element for Res
3987		check_element_of_wf(X,S,WF)
3988		; (Res==[],ImageSoFar==[]) -> MemRes=pred_false, not_element_of_wf(X,S,WF) % Result empty: X cannot be in S
3989		; membership_test_wf(S,X,MemRes,WF)
3990		),
3991	?	image4(MemRes,Y,T,S,ImageSoFar,Res,WF).
3992
3993		definitely_not_empty(Set) :- nonvar(Set), Set \== [], \+ functor(Set,closure,3). % Set \= closure(_,_,_).
3994
3995		:- block image4(-, ?,?,?, ?,?,?).
3996		image4(pred_true, Y,T,S, ImageSoFar,Res,WF) :-
3997		(Res==[]
3998		-> MemRes=pred_true, check_element_of_wf(Y,ImageSoFar,WF)
3999		; membership_test_wf(ImageSoFar,Y,MemRes,WF)
4000		),
4001	?	image5(MemRes,Y,T,S,ImageSoFar,Res,WF).
4002		image4(pred_false, _Y,T,S, ImageSoFar,Res,WF) :-
4003	?	image_for_list_relation(T,S,ImageSoFar,Res,WF).
4004
4005		:- block image5(-, ?,?,? ,?,?,?).
4006		image5(pred_true,_Y,T,S,ImageSoFar,Res,WF) :- /* we have already added Y to the image */
4007	?	image_for_list_relation(T,S,ImageSoFar,Res,WF).
4008		image5(pred_false,Y,T,S,ImageSoFar,Res,WF) :-
4009		add_element_wf(Y,ImageSoFar,ImageSoFar2,WF),
4010		kernel_objects:mark_as_non_free(Y,image), % Y has been added to image, no longer freely choosable
4011		equal_cons_wf(Res,Y,Res2,WF),
4012	?	image_for_list_relation(T,S,ImageSoFar2,Res2,WF).
4013
4014
4015
4016		:- 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)).
4017		:- 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)).
4018		:- 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)).
4019		% version for computing closure1(Rel)[S]
4020		:- block image_for_closure1_wf(-,?,?,?),image_for_closure1_wf(?,-,?,?).
4021		image_for_closure1_wf(Rel,S,Res,WF) :- (Rel==[] ; S==[]),!,empty_set_wf(Res,WF).
4022		image_for_closure1_wf(Rel,Set,Res,WF) :-
4023		try_expand_and_convert_to_avl_unless_large_wf(Set,ESet,WF),
4024	?	image_for_closure1_wf_aux(Rel,ESet,Res,WF).
4025
4026		:- use_module(library(avl),[avl_height/2]).
4027		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
4028		((nonvar(S),S=avl_set(_))
4029		-> closure1_for_explicit_set_from(Rel,S,Closure1Rel),!,
4030		% if S is known: start from S (currently only deals with Rel=avl_set(_)
4031		range_wf(Closure1Rel,Res,WF)
4032		; Rel=avl_set(AR), avl_height(AR,AR_Height),
4033		((set_smaller_than(S,4),AR_Height>4)
4034		-> !, % TO DO: we could do the same for small S if Rel is large
4035		when(ground(S), (expand_and_convert_to_avl_set(S,ES,image_for_closure1_wf_aux,'closure1(ARG)[?]') ->
4036		closure1_for_explicit_set_from(Rel,avl_set(ES),Closure1Rel),
4037		range_wf(Closure1Rel,Res,WF)
4038		; image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S))
4039		))
4040		; % Don't do this if avl_height too large; then it is probably better to compute the image for S only
4041		AR_Height < 13, % how big should we make this magic constant; or should we time-out ? 2^14=16384
4042		closure1_for_explicit_set(Rel,Closure1Rel),!, % we can compute it effiently; don't use code below
4043		image_wf(Closure1Rel,S,Res,WF)
4044		)
4045		).
4046		image_for_closure1_wf_aux(Rel,S,Res,WF) :-
4047	?	propagate_result_in_range(Rel,S,Res,WF),
4048	?	image_for_closure1_iterate(Rel,S,[],Res,WF,first_iteration(S)).
4049
4050		% no need to treat avl_sets; already covered as special case above
4051		set_smaller_than([],_).
4052		set_smaller_than([_|T],N) :- N>1, nonvar(T), N1 is N-1, set_smaller_than(T,N1).
4053
4054		image_for_closure1_iterate(Rel,S,Acc,Res,WF,FIRST) :-
4055		image_wf0(Rel,S,Res1,WF),
4056		ground_value_check(Res1,RV),
4057	?	image_for_closure1_check_fix(RV,Rel,Acc,Res1,Res,WF,FIRST).
4058
4059		:- block image_for_closure1_check_fix(-,?,?,?,?,?,?).
4060		image_for_closure1_check_fix(_,Rel,Acc,Res1,Res,WF,FIRST) :-
4061		%try_expand_and_convert_to_avl_unless_large_wf(Res1,ERes1,WF),
4062		difference_set(Res1,Acc,New),
4063		try_expand_and_convert_to_avl(New,ENew), % we compute difference_set below; we most definitely will need an explicit finite representation
4064		(not_empty_set_wf(ENew,WF),
4065		union(ENew,Acc,Acc1), % Note: we do not call union_wf - should we do this
4066		% upon first iteration remove also S from New -> New2 and pass New2 to image_for_closure1_iterate
4067		% TO DO: investigate whether this also makes sense for further iterations; always remove S
4068		(FIRST=first_iteration(S) -> difference_set(ENew,S,New2) ; New2=ENew),
4069	?	image_for_closure1_iterate(Rel,New2,Acc1,Res,WF,not_first)
4070		;
4071	?	empty_set_wf(ENew,WF),equal_object_optimized_wf(Acc,Res,image_for_closure1_check_fix,WF)).
4072
4073		% propagate information that if closure1(Rel)[.] = Res => Res <: range(Rel)
4074		% x: 1..n --> 1..n & closure1(x)[{1}] = {} & n=100
4075		:- block propagate_result_in_range(?,?,-,?).
4076		propagate_result_in_range(Rel,_S,_Res,_WF) :-
4077		ground_value(Rel),!. % no propagation required
4078		propagate_result_in_range(Rel,S,[],WF) :- !,
4079		domain_wf(Rel,Domain,WF),
4080		not_subset_of_wf(S,Domain,WF).
4081		propagate_result_in_range(Rel,_,Res,WF) :-
4082		range_wf(Rel,Range,WF),
4083	?	check_subset_of_wf(Res,Range,WF).
4084
4085		:- use_module(probsrc(avl_tools),[avl_height_less_than/2]).
4086
4087		% version for computing iterate(K,Rel)[S]
4088		% iteration
4089		:- block image_for_iterate_wf(?,-,?,?,?,?), image_for_iterate_wf(?,?,-,?,?,?).
4090		image_for_iterate_wf(_Rel,_K,S,Res,_,WF) :- S==[],!,empty_set_wf(Res,WF).
4091		image_for_iterate_wf(Rel,int(K),S,Res,Type,WF) :-
4092		image_for_iterate_k(K,Rel,S,Res,Type,WF).
4093
4094		:- block image_for_iterate_k(-,?,?,?,?,?).
4095		image_for_iterate_k(K,Rel,S,Res,Type,WF) :-
4096		nonvar(Rel),
4097		Rel=avl_set(AVL),
4098		(var(S) -> avl_height_less_than(AVL,11) ; avl_height_less_than(AVL,3)),
4099		!, % compute the iteration once; possibly better constraint propagation and performance if S enumerated
4100		% e.g. x:{1,10,20} & iterate({1|->10,20|->1,10|->20},2)(x) = 20
4101		rel_iterate_wf(Rel,int(K),RelIterated,Type,WF),
4102		image_wf(RelIterated,S,Res,WF).
4103		image_for_iterate_k(K,Rel,S,Res,_,WF) :-
4104		image_for_iterate_k_loop(K,Rel,S,Res,WF).
4105
4106		:- block image_for_iterate_k_loop(?,?,-,?,?).
4107		image_for_iterate_k_loop(0,_Rel,Acc,Result,WF) :- !,
4108		equal_object_optimized_wf(Acc,Result,image_for_iterate_k,WF).
4109		image_for_iterate_k_loop(K,Rel,Acc,Result,WF) :-
4110		image_wf0(Rel,Acc,Acc1,WF), % we could try and detect fix point if K> some limit or time for iteration is measurable
4111		if((K>10, K mod 10 =:= 0, % check for fixpoint every 10 iterations
4112		nonvar(Acc1), Acc1=avl_set(_), quick_custom_explicit_set_approximate_size(Acc1,Size1),
4113		quick_custom_explicit_set_approximate_size(Acc,Size0),
4114		Size0=Size1, % only check for equality if approximate sizes match
4115		equal_explicit_sets_wf(Acc,Acc1,WF)),
4116		K1=0, % fixpoint found, no need to continue iterating
4117		K1 is K-1),
4118		image_for_iterate_k_loop(K1,Rel,Acc1,Result,WF).
4119
4120		special_operator_for_image(b(Rel,Type,_),Kind,Args) :- special_image_aux(Rel,Type,Kind,Args).
4121		special_image_aux(closure(Rel),_,closure,[Rel]). % we have closure1(Rel)[Set] -> avoid computing full closure
4122		special_image_aux(iteration(Rel,K),Type,iteration(Type),[Rel,K]).
4123		% TODO: reflexive closure, id_closure (this will probably be more natural as special case for a value)
4124
4125	?	image_for_special_operator(closure,[Rel],S,Res,WF) :- image_for_closure1_wf(Rel,S,Res,WF).
4126		image_for_special_operator(iteration(Type),[Rel,K],S,Res,WF) :-
4127		image_for_iterate_wf(Rel,K,S,Res,Type,WF).
4128
4129		:- use_module(kernel_objects,[singleton_set_element/4]).
4130		apply_fun_for_special_operator(Kind,EArgs,FunArg,Res,WF,Span) :-
4131		InitialSet = [FunArg], % TODO: try convert to AVL, note: closure1 not really useful in fun. application context
4132		image_for_special_operator(Kind,EArgs,InitialSet,SetRes,WF),
4133		singleton_set_element(SetRes,Res,Span,WF).
4134
4135		% iterate(%x.(x:NATURAL|x+2),2000)(20) much faster this way, 15 ms vs 4 seconds
4136		% iterate(%x.(x:NATURAL|x+2),2000)[{20}]: ditto
4137
4138
4139		% -----------------------------------
4140
4141		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:apply_to([(int(2),int(22))],int(2),int(22),WF),WF)).
4142		:- 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)
4143		:- 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)
4144		:- assert_must_succeed(bsets_clp:apply_to([(int(1),int(2))],int(1),int(2),_WF)).
4145		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),int(2),_WF),F=[(int(3),int(2)),(int(2),int(1))])).
4146		:- assert_must_succeed((bsets_clp:apply_to(F,X,int(1),_WF),F=[(int(3),int(2)),(int(2),int(1))],X=int(2))).
4147		:- assert_must_succeed((bsets_clp:apply_to(F,int(3),_,_WF),F=[(int(3),[int(2),int(3)]),(int(2),[])])).
4148
4149		:- assert_must_fail(bsets_clp:apply_to([(int(1),int(2)),(int(1),int(3))],int(1),int(3),_WF)).
4150		/* input not a function */
4151		apply_to(R,X,Y,WF) :- apply_to(R,X,Y,unknown,unknown,WF).
4152	?	apply_to(R,X,Y,Span,WF) :- apply_to(R,X,Y,unknown,Span,WF).
4153
4154		% comment in to perform profiling at function call level; can lead to big slowdowns
4155		%:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4156		%:- use_module(source_profiler,[opt_add_source_location_hits/2]).
4157		%apply_to(_R,_X,_Y,_FunctionType,Span,_WF) :- opt_add_source_location_hits(Span,1),fail.
4158
4159		:- block apply_to(-,-,-,?,?,?).
4160		apply_to(R,X,Y,_FunctionType,Span,WF) :-
4161		% we could check if WD condition discharged in Span
4162		(\+ preferences:preference(find_abort_values,false) ; preference(data_validation_mode,true)),
4163		!,
4164		apply_to_var_block_abort(R,X,Y,R,Span,WF). % we have to know R before we can do anything
4165		apply_to(R,X,Y,FunctionType,Span,WF) :-
4166		(var(R),var(X) -> force_in_domain_wf(X,R,WF) ; true),
4167	?	apply_to1(R,X,Y,R,FunctionType,Span,WF).
4168
4169
4170
4171		:- use_module(preferences,[preference/2]).
4172		:- use_module(clpfd_tables,[can_translate_function_to_element_constraint/2,check_apply_with_element_constraint/5]).
4173		:- block apply_to1(-,-,?,?,?,?,?).
4174		apply_to1(R,X,Y,InitialRel,FunctionType,Span,WF) :-
4175		(var(R) -> apply_to_var(R,X,Y,InitialRel,Span,WF)
4176		; R\=[], can_translate_function_to_element_constraint(R,FunctionType) ->
4177		check_apply_with_element_constraint(R,X,Y,FunctionType,WF)
4178	?	; apply_to_nonvar(R,X,Y,InitialRel,Span,WF),
4179		propagate_range_membership(R,Y)
4180		).
4181		:- block apply_to2(-,-,?,?,?,?).
4182		apply_to2(R,X,Y,InitialRel,Span,WF) :-
4183		(var(R)
4184		-> apply_to_var(R,X,Y,InitialRel,Span,WF)
4185	?	; apply_to_nonvar(R,X,Y,InitialRel,Span,WF)
4186		).
4187
4188		:- use_module(clpfd_lists,[get_finite_fdset_information/2,combine_fdset_information/3,
4189		assert_fdset_information/2,get_fdset_information/2]).
4190		% tested in test 1478; initially slows down NQueens
4191		%:- block propagate_range_membership(-,?). % not necessary
4192		propagate_range_membership([(_,RanEl)|T],X) :- nonvar(RanEl),
4193		preferences:preference(use_clpfd_solver,true),
4194		preferences:preference(find_abort_values,false),
4195		get_finite_fdset_information(RanEl,Info), % TO DO: try and detect if we can apply element/3 from clpfd
4196		\+ ground(X),
4197		get_fdset_information(X,InfoX),
4198		Info \= InfoX, % avoids NQueens slowdown; TO DO: check if more precise than InfoX; otherwise no use in collecting info
4199		!,
4200		propagate_range_membership(T,Info,X).
4201		propagate_range_membership(_,_).
4202		:- block propagate_range_membership(-,?,?).
4203		propagate_range_membership([],Info,El) :- !,
4204		% 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 ??
4205		assert_fdset_information(Info,El).
4206		propagate_range_membership([(_,RanEl)|T],Acc,X) :-
4207		nonvar(RanEl), % otherwise we have no info: we may just as well stop
4208		get_finite_fdset_information(RanEl,RInfo),
4209		combine_fdset_information(Acc,RInfo,NewAcc),
4210		NewAcc \= no_fdset_info,
4211		!,
4212		propagate_range_membership(T,NewAcc,X).
4213		propagate_range_membership(_,_,_).
4214
4215
4216		apply_to_var(R,X,Y,InitialRel,Span,WF) :-
4217		mark_var_set_as_non_empty(R),
4218		get_wait_flag(1.0,apply_to_var,WF,WF1), % see tests 1393, 1562??
4219		% was: get_wait_flag0(WF,WF1), but see test 1706 (in conjunction for improvement for test 2033)
4220		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 ??)
4221		(var(R) ->
4222		R=[(X,Y)|Tail],
4223		optional_functionality_check(Tail,X,WF)
4224		; apply_to_nonvar(R,X,Y,InitialRel,Span,WF))).
4225
4226		:- block apply_to_var_block_abort(-,?,?,?,?,?).
4227		apply_to_var_block_abort(R,X,Y,InitialRel,Span,WF) :-
4228		apply_to_nonvar(R,X,Y,InitialRel,Span,WF).
4229
4230		optional_functionality_check(Tail,X,WF) :-
4231		preferences:preference(disprover_mode,true),!,
4232		not_in_domain_wf(X,Tail,WF). % we assert that R is a function ; when disproving we can assume well-definedness
4233		% 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)
4234		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
4235
4236
4237		:- use_module(closures,[is_recursive_closure/3]).
4238		:- use_module(memoization,[is_memoization_closure/4,apply_to_memoize/8]).
4239		:- load_files(library(system), [when(compile_time), imports([environ/2])]).
4240		:- if(\+ environ(no_wd_checking,true)).
4241		apply_to_nonvar([],X,_Y,InitialRel,Span,WF) :-
4242		\+ preferences:preference(find_abort_values,false),
4243		add_wd_error_span('function applied outside of domain (#2): ', '@fun'(X,InitialRel),Span,WF).
4244		:- endif.
4245		apply_to_nonvar([(X2,Y2)|T],X,Y,InitialRel,Span,WF) :-
4246		equality_objects_wf(X2,X,EqRes,WF),
4247		% this check on Y2 below is important if both Y and Y2 are instantiated but X,X2 not yet
4248		% example: aload_R07_cbc.mch (Savary) or cbc_sequence check for R08_ByteArray for aload_R07 event (test 1349)
4249		% however: slows down test 583 !
4250		(var(EqRes) -> equality_objects_wf(Y2,Y,EqResY,WF),
4251		prop_apply_eqxy(EqResY,EqRes) % propagate: if Y/=Y2 => X/=X2
4252		; EqResY=not_called),
4253	?	apply_to4(EqRes,EqResY,Y2,T,X,Y,InitialRel,Span,WF).
4254		apply_to_nonvar(avl_set(A),X,Y,_InitialRel,Span,WF) :-
4255		apply_to_avl_set(A,X,Y,Span,WF).
4256		apply_to_nonvar(closure(P,T,B),X,Y,_InitialRel,Span,WF) :-
4257		%is_custom_explicit_set(Closure,apply), % should also work for avl_set,...
4258	?	(is_memoization_closure(P,T,B,MemoID)
4259		% Function application with memoization; currently enabled by add /*@desc memo */ pragma to abstract constant
4260		-> apply_to_memoize(MemoID,P,T,B,X,Y,Span,WF)
4261		; 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 ??)
4262		-> % print_term_summary(apply_recursive_closure(X,P,T,B)),
4263		%hit_profiler:add_profile_hit(rec_apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4264		ground_value_check(X,XV), block_apply_closure_to_nonvar_groundx(XV,X,Y,P,T,B,Span,WF)
4265		; %hit_profiler:add_profile_hit(apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)),
4266		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF)).
4267
4268
4269		:- block block_apply_closure_to_nonvar_groundx(-,?,?, ?,?,?, ?,?).
4270		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).
4271
4272		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4273		kernel_tools:ground_bexpr(B),
4274		!, % then if the element of function succeeds there is no need to check WD
4275		if(check_element_of_function_closure(X,Y,P,T,B,WF),
4276		true, % No need to check for well-definedness; no pending choice points
4277		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
4278		).
4279		apply_closure_to_nonvar_groundx(X,Y,P,T,B,Span,WF) :-
4280		apply_closure_to_nonvar(X,Y,P,T,B,Span,WF).
4281
4282		% if we first check preferences:preference(find_abort_values,false) to avoid a choice
4283		% point, we get a big slow-down on Alstom models; e.g., vesg_Mar12
4284		% WARNING: This choice point can be set up in WF0 !
4285		apply_closure_to_nonvar(X,Y,P,T,B,_,WF) :-
4286		(preferences:preference(find_abort_values,true) -> true ; !), % slow down ???!
4287		check_element_of_function_closure(X,Y,P,T,B,WF) .
4288		apply_closure_to_nonvar(X,_,P,T,B,Span,WF) :- % removing this clause doubles runtime of COMPUTE_GRADIENT_CHANGE
4289		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF).
4290
4291		apply_closure_to_nonvar_wd_check(X,P,T,B,Span,WF) :-
4292		\+ preferences:preference(find_abort_values,false),
4293		not_in_domain_wf(X,closure(P,T,B),WF),
4294		when((ground(X),ground(closure(P,T,B))),
4295		add_wd_error_span('function applied outside of domain (#3): ', '@fun'(X,closure(P,T,B)),Span,WF)).
4296
4297
4298		% propagate equality_objects between range and domain elements for function application:
4299		:- block prop_apply_eqxy(-,-).
4300		prop_apply_eqxy(Eqy,Eqx) :- var(Eqy),!, (Eqx = pred_true -> Eqy = pred_true ; true).
4301		prop_apply_eqxy(pred_false,pred_false).
4302		prop_apply_eqxy(pred_true,_).
4303
4304		:- block apply_to4(-,?,?, -,?,?,?,?,?).
4305		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4306		var(EqResX),!, % Tail bound
4307		(Tail == []
4308		-> (preferences:preference(find_abort_values,false)
4309		-> EqResX = pred_true,
4310		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4311		; apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4312		)
4313		; 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,...)
4314		; Tail = closure(_,_,_) -> apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)
4315		; Tail \= [_|_] -> add_internal_error('Illegal Tail: ',apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF)),fail
4316		; 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
4317	?	apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3)
4318		).
4319		apply_to4(pred_true,EqResY,Y2, Tail,X,Y,_InitialRel,_,WF) :-
4320	?	(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to4,WF) ; EqResY = pred_true),
4321		optional_functionality_check(Tail,X,WF).
4322	?	apply_to4(pred_false,_EqResY,_Y2,T,X,Y,InitialRel,Span,WF) :- apply_to2(T,X,Y,InitialRel,Span,WF).
4323
4324		% we delay setting up equality_objects until X3 is at least partially known, see test 1715 Alstom_essai2_boucle1
4325		% TO DO: we could check if X3==X above
4326		:- block apply_to4_call5(-,?,?, ?,?,?,?,?,?, -,?,?).
4327		apply_to4_call5(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF, _X3,_Y3,_T3) :- nonvar(EqResX),!,
4328		apply_to4(EqResX,EqResY,Y2,Tail,X,Y,InitialRel,Span,WF).
4329		apply_to4_call5(EqResX,EqResY,Y2, _Tail,X,Y,InitialRel,Span,WF, X3,Y3,T3) :- % X3 must now be bound
4330		equality_objects_wf(X3,X,EqRes3,WF),
4331	?	apply_to5(EqResX,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, InitialRel,Span,WF).
4332
4333		% version which wait suntil first argument known
4334		:- block apply_to4_block(-,?,?, ?,?,?,?,?,?).
4335		apply_to4_block(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF) :-
4336		apply_to4(EqResX,EqResY,Y2, Tail,X,Y,InitialRel,Span,WF).
4337
4338
4339		% apply_to5: implements a watched-literal style treatment of function application
4340		% we watch whether X unifies with two elements of the function, if only one element left we can force equality
4341		% TEST:
4342		% 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
4343		:- block apply_to5(-,?,-, ?,?,?,?, ?,?, ?,?,?),apply_to5(-,?,?, ?,?,?,-, ?,?, ?,?,?).
4344		apply_to5(EqRes,EqResY,EqRes3, Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4345		var(EqRes),!,
4346		% EqRes3 and T3 must be known; TO DO: improve predicate so that we have to wait on T3 only when EqRes3=pred_false
4347		(EqRes3 = pred_false -> % we cannot match next element, move tail one forward
4348		(T3 = [] -> EqRes=pred_true ; true),
4349		apply_to4(EqRes,EqResY,Y2,T3,X,Y,InitialRel,Span,WF)
4350		; /* EqRes3 = pred_true */
4351		% we match the next entry in the list; discard Y2 and jump to (X3,Y3) and return as solution
4352	?	equal_object_wf(Y3,Y,apply_to6,WF), optional_functionality_check(T3,X,WF),
4353		% TO DO: we could also do equality_objects if necessary between Y and Y3, as in apply_to4 for Y and Y2
4354		opt_force_false(EqRes)
4355		).
4356		apply_to5(pred_true,EqResY,EqRes3, Y2,X3,Y3,T3, X,Y, _InitialRel,_Span,WF) :-
4357		(EqResY==not_called -> equal_object_wf(Y2,Y,apply_to5,WF) ; EqResY = pred_true),
4358		opt_force_false(EqRes3),
4359		optional_functionality_check([(X3,Y3)|T3],X,WF).
4360		apply_to5(pred_false,_EqResY,EqRes3, _Y2,_X3,Y3,T3, X,Y, InitialRel,Span,WF) :-
4361		(var(EqRes3) -> % it can be that EqRes3 is about to be triggered
4362		equality_objects_wf(Y3,Y,EqResY3,WF),
4363		prop_apply_eqxy(EqResY3,EqRes3) % propagate: if Y/=Y3 => X/=X3
4364		; EqResY3=not_called),
4365		apply_to4(EqRes3,EqResY3,Y3, T3,X,Y,InitialRel,Span,WF).
4366
4367		opt_force_false(EqRes) :-
4368		(preference(find_abort_values,false) -> EqRes=pred_false
4369		; true). % TO DO: if EqRes becomes pred_true: raise abort_error as the relation was not a function
4370
4371
4372
4373		/********************************************/
4374		/* surjection_relation(R,Domain,Range) */
4375		/* R : Domain <->> Range */
4376		/********************************************/
4377		:- 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)
4378		:- 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)).
4379
4380		surjection_relation_wf(R,Domain,Range,WF) :-
4381		is_surjective(R,Range,WF),
4382		% TODO: is not optimal since ran(R)<:Range is already implied by is_surjective and
4383		% checked a second time by relation_over_wf/4
4384	?	relation_over_wf(R,Domain,Range,WF).
4385
4386		:- 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)).
4387
4388		not_surjection_relation_wf(R,Domain,Range,WF) :-
4389		expand_custom_set_to_list_wf(R,ER,Done,not_surjection_relation_wf,WF),
4390		not_tot_surj_rel(ER,Done,[],Domain,Range,Range,WF).
4391
4392		/*********************************************/
4393		/* total_surjection_relation(R,Domain,Range) */
4394		/* R : Domain <<->> Range */
4395		/*********************************************/
4396		:- 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)).
4397		:- 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)).
4398
4399
4400		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1)],[int(11),int(12)]),L),
4401		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
4402		length(SSL,1))).
4403		%:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11),int(12)]),L), length(L,7))).
4404		% the new domain predicate also instantiates from result; meaning that duplicate solutions are now generated
4405		:- 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))).
4406		:- assert_must_succeed((findall(R,bsets_clp:total_surjection_relation(R,[int(1),int(2)],[int(11)]),L),
4407		length(L,1))).
4408
4409		total_surjection_relation(R,Domain,Range) :- init_wait_flags(WF,[total_surjection_relation]),
4410	?	total_surjection_relation_wf(R,Domain,Range,WF), ground_wait_flags(WF).
4411
4412		total_surjection_relation_wf(R,Domain,Range,WF) :-
4413	?	relation_over_wf(R,Domain,Range,WF),
4414		check_relation_is_total(R,Domain,WF), % calls domain which now instantiates R if Domain known
4415		check_relation_is_surjective(R,Range,WF).
4416
4417
4418		:- 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)).
4419		:- 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)).
4420		:- 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)).
4421
4422		not_total_surjection_relation_wf(R,Domain,Range,WF) :-
4423		expand_custom_set_to_list_wf(R,ER,Done,not_total_surjection_relation_wf,WF),
4424	?	not_tot_surj_rel(ER,Done,Domain,Domain,Range,Range,WF).
4425
4426
4427		/********************************************/
4428		/* partial_surjection(R,DomType,RangeType) */
4429		/* R : DomType +->> RangeType */
4430		/********************************************/
4431
4432		:- 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)
4433		:- 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)).
4434		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4435		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4436		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],global_set('Name')),
4437		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4438		:- assert_must_succeed((bsets_clp:partial_surjection_wf(X,[int(1),int(2),int(3)],global_set('Name'),_WF),
4439		kernel_objects:equal_object(X,[(int(2),fd(1,'Name')),(int(1),fd(2,'Name')),(int(3),fd(3,'Name'))]))).
4440		:- assert_must_succeed((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4441		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4442		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4443		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))]))). /* mult. */
4444		:- assert_must_succeed((X=[(int(2),int(7)),(int(1),int(6)),(int(3),int(6))],
4445		bsets_clp:partial_surjection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]))).
4446		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4447		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4448		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4449		X = [(int(2),int(7)),(int(1),int(7))])).
4450		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4451		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4452		:- assert_must_succeed_multiple((bsets_clp:partial_surjection(_X,
4453		[int(1),int(2),int(3),int(4),int(5),int(6),int(7)],[int(2),int(3),int(4)]) )).
4454		:- assert_must_fail((bsets_clp:partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4455		X = [(int(2),int(7)),(int(2),int(6))])).
4456
4457		partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[partial_surjection]),
4458		partial_surjection_wf(R,Domain,Range,WF),
4459	?	ground_wait_flags(WF).
4460
4461		:- block partial_surjection_wf(-,-,?,?).
4462		partial_surjection_wf(R,Domain,Range,WF) :-
4463		check_card_greater_equal(Domain,geq,Range,CardDom,CardRange),
4464		(surjection_has_to_be_total_injection(CardDom,CardRange)
4465		% LAW: card(setX) = card(setY) => ff: setX +->> setY <=> ff: setX >-> setY
4466	?	-> total_function_wf(R,Domain,Range,WF),
4467		injective(R,WF)
4468		; is_surjective(R,Range,WF),
4469		partial_function_wf(R,Domain,Range,WF)
4470		).
4471
4472
4473		% check_card_greater_equal(A,B) : quick check that card(A) >= card(B); also works with infinite cardinality
4474		% TO DO: replace by a better constraint propagating predicate (also working for partially instantiated lists,...)
4475		% compared with computing card and setting up < constraint: will only compute card if it can be done efficiently + deals with inf
4476		% check_card_greater_equal(SetA,EQ,SetB) ; EQ=eq or geq
4477		:- block check_card_greater_equal(-,?,?,?,?).
4478		check_card_greater_equal([],_,R,0,0) :- !, empty_set(R).
4479		check_card_greater_equal(A,EQ,B,CA,CB) :- check_card_greater_equal2(A,EQ,B,CA,CB).
4480
4481		:- use_module(inf_arith,[block_inf_greater_equal/2]).
4482		:- block check_card_greater_equal2(?,?,-,?,?).
4483		check_card_greater_equal2(A,EQ,B,CardA,CardB) :-
4484		efficient_card_for_set(A,CardA,CodeA),
4485		efficient_card_for_set(B,CardB,CodeB),!,
4486		call(CodeA), call(CodeB),
4487		(EQ=eq -> CardA=CardB ; block_inf_greater_equal(CardA,CardB)).
4488		check_card_greater_equal2(_A,_,_B,'?','?').
4489
4490
4491		:- block is_surjective(-,-,?).
4492		is_surjective(R,Range,WF) :-
4493		(var(R) -> setup_surj_range(Range,R,WF)
4494		; range_wf(R,Range,WF)).
4495
4496		setup_surj_range(Range,R,WF) :-
4497		setup_range(Range,Res,DONE,WF),
4498		equal_when_done(Res,R,DONE).
4499		:- block equal_when_done(?,?,-).
4500	?	equal_when_done(Res,R,_DONE) :- equal_object(Res,R).
4501
4502
4503		:- block setup_range(-,?,?,?).
4504		setup_range(global_set(G),Res,DONE,WF) :-
4505		expand_custom_set_wf(global_set(G),ES,setup_range,WF),
4506		setup_range(ES,Res,DONE,WF).
4507		setup_range(freetype(ID),Res,DONE,WF) :-
4508		expand_custom_set_wf(freetype(ID),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4509		setup_range(avl_set(S),Res,DONE,WF) :-
4510		expand_custom_set_wf(avl_set(S),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4511		setup_range(closure(P,T,B),Res,DONE,WF) :-
4512		expand_custom_set_wf(closure(P,T,B),ES,setup_range,WF), setup_range(ES,Res,DONE,WF).
4513		setup_range([],_,done,_WF).
4514		setup_range([H|T],[(_,H)|ST],DONE,WF) :- setup_range(T,ST,DONE,WF).
4515
4516
4517
4518		:- assert_must_succeed(exhaustive_kernel_fail_check_wf(bsets_clp:not_partial_surjection_wf([(int(1),int(6)),(int(2),int(7))],
4519		[int(1),int(2)],[int(7),int(6)],WF),WF)).
4520		:- 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)],
4521		[int(7),int(6),int(2)],WF),WF)).
4522		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4523		X = [(int(2),int(7)),(int(1),int(6))])).
4524		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4525		X = [(int(2),int(7)),(int(2),int(6))])).
4526		:- assert_must_fail((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4527		X = [(int(2),int(7)),(int(1),int(6))])).
4528		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4529		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4530		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4531		X = [(int(2),int(7)),(int(1),int(7))])).
4532		:- assert_must_succeed((bsets_clp:not_partial_surjection(X,[int(1),int(2),int(3)],[int(7),int(6)]),
4533		X = [(int(2),int(7)),(int(1),int(6)),(int(3),int(8))])).
4534
4535
4536
4537		/* /: Domain +->> Range */
4538		not_partial_surjection(R,Domain,Range) :- init_wait_flags(WF,[not_partial_surjection]),
4539		not_partial_surjection_wf(R,Domain,Range,WF),
4540		ground_wait_flags(WF).
4541
4542		:- block not_partial_surjection_wf(-,?,?,?).
4543		not_partial_surjection_wf(R,DomType,RangeType,WF) :-
4544	?	partial_surjection_test_wf(R,DomType,RangeType,pred_false,WF).
4545
4546
4547		%not_surjective_relation_wf(R,DomType,RType,WF) :-
4548		% invert_relation_wf(R,IR,WF),
4549		% not_total_relation_wf(IR,RType,DomType,WF).
4550
4551
4552		:- 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)).
4553		:- 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)).
4554
4555		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF) :-
4556		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4557		(IsPF==pred_false -> PredRes=pred_false
4558		; range_wf(R,RelRan,WF),
4559	?	conjoin_test(IsPF,IsSurjective,PredRes,WF),
4560	?	subset_test(RangeType,RelRan,IsSurjective,WF)
4561		).
4562
4563
4564		:- 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)).
4565		:- 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)).
4566
4567		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4568		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4569		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4570		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4571		:- assert_must_succeed((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4572		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(7))]))).
4573		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4574		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4575		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4576		kernel_objects:equal_object(X,[(int(2),int(7))]))).
4577		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4578		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6)),(int(1),int(8))]))).
4579		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4580		kernel_objects:equal_object(X,[(int(2),int(7)),(int(3),int(6)),(int(1),int(7))]))).
4581		:- assert_must_fail((bsets_clp:total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4582		kernel_objects:equal_object(X,[]))).
4583
4584		/****************************************/
4585		/* total_relation_wf(R,Domain,Range,WF) */
4586		/* R : Domain <<-> Range */
4587		/****************************************/
4588
4589	?	total_relation_wf(R,Domain,Range,WF) :- relation_over_wf(R,Domain,Range,WF),
4590		check_relation_is_total(R,Domain,WF).
4591
4592		% this predicates assume that the relation's range and domain have already been checked
4593		check_relation_is_total(Relation,Domain,WF) :- domain_wf(Relation,Domain,WF).
4594		check_relation_is_surjective(Relation,Range,WF) :-
4595		range_wf(Relation,Range,WF). % we could also call is_surjective (which does setup_surj_range) ?
4596
4597		:- 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)).
4598		:- 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)).
4599		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4600		X = [(int(2),int(7)),(int(1),int(6))])).
4601		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4602		X = [(int(2),int(7)),(int(1),int(7))])).
4603		:- assert_must_fail((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4604		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(7))])).
4605		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4606		X = [(int(2),int(7)),(int(2),int(6))])).
4607		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4608		X = [(int(2),int(7))])).
4609		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4610		X = [(int(2),int(7)),(int(1),int(6)),(int(1),int(8))])).
4611		:- assert_must_succeed((bsets_clp:not_total_relation_wf(X,[int(1),int(2)],[int(7),int(6)],_WF),
4612		X = [(int(2),int(7)),(int(3),int(6)),(int(1),int(7))])).
4613
4614		:- block not_total_relation_wf(-,?,?,?).
4615		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),custom_explicit_sets:is_definitely_maximal_set(Range),
4616		% we do not need the Range; this means we can match more closures (e.g., lambda)
4617		custom_explicit_sets:dom_for_specific_closure(FF,FFDomain,function(_),WF),!,
4618		not_equal_object_wf(FFDomain,Domain,WF).
4619		not_total_relation_wf(FF,Domain,Range,WF) :- nonvar(FF),
4620		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(_),WF),!,
4621		equality_objects_wf(FFDomain,Domain,Result,WF), % not yet implemented ! % TODO ! -> sub_set,equal,super_set
4622		when(nonvar(Result),(Result=pred_false -> true ; not_subset_of_wf(FFRange,Range,WF))).
4623		not_total_relation_wf(R,Domain,Range,WF) :-
4624		expand_custom_set_to_list_wf(R,ER,Done,not_total_relation_wf,WF),
4625	?	not_tot_surj_rel(ER,Done,Domain,Domain,[],Range,WF). % empty DelRange means we don't do surjective test
4626
4627		% can be used to check not total, not surj, not total surj relation
4628		:- block not_tot_surj_rel(-,?,?,?,?,?,?).
4629		not_tot_surj_rel([],_,DelDomain,_,DelRange,_,WF) :-
4630	?	at_least_one_set_not_empty(DelDomain,DelRange,WF).
4631		not_tot_surj_rel([_|_],Done,DelDom,Dom,_DelRan,_Ran,_WF) :- nonvar(Done),
4632		Done \= no_check_to_be_done,
4633		nonvar(DelDom),DelDom \= [],
4634		nonvar(Dom),is_infinite_explicit_set(Dom),
4635		!. % a finite expanded list can never be a total relation over an infinite domain
4636		not_tot_surj_rel([(X,Y)|T],_Done,DelDom,Dom,DelRan,Ran,WF) :-
4637		membership_test_wf(Dom,X,MemRes,WF),
4638	?	not_tr2(MemRes,X,Y,T,DelDom,Dom,DelRan,Ran,WF).
4639
4640		% check if one of the two sets is non-empty
4641		at_least_one_set_not_empty(Set1,Set2,_) :- (Set=Set1 ; Set=Set2),
4642		nonvar(Set),
4643		(Set=avl_set(_) ; Set=[_|_]), % we can avoid leaving choice point
4644		!.
4645		at_least_one_set_not_empty(Set1,_,WF) :- not_empty_set_wf(Set1,WF).
4646		at_least_one_set_not_empty(Set1,Set2,WF) :- empty_set_wf(Set1,WF),not_empty_set_wf(Set2,WF).
4647
4648		:- block not_tr2(-,?,?,?,?,?,?,?,?).
4649		not_tr2(pred_false,_X,_Y,_T,_DelDom,_Dom,_DelRan,_Ran,_WF).
4650		not_tr2(pred_true,X,Y,T,DelDom,Dom,DelRan,Ran,WF) :-
4651		delete_element_wf(X,DelDom,DelDom2,WF), % set DelDom initially to [] to avoid totality check
4652		membership_test_wf(Ran,Y,MemRes,WF),
4653	?	not_tr3(MemRes,Y,T,DelDom2,Dom,DelRan,Ran,WF).
4654
4655		:- block not_tr3(-,?,?,?,?,?,?,?).
4656		not_tr3(pred_false,_Y,_T,_DelDom2,_Dom,_DelRan,_Ran,_WF).
4657		not_tr3(pred_true,Y,T,DelDom2,Dom,DelRan,Ran,WF) :-
4658		delete_element_wf(Y,DelRan,DelRan2,WF), % set DelRan initially to [] to avoid surjection check
4659	?	not_tot_surj_rel(T,no_check_to_be_done,DelDom2,Dom,DelRan2,Ran,WF).
4660
4661		/******************************************/
4662		/* total_surjection(R,DomType,RangeType) */
4663		/* R : DomType -->> RangeType */
4664		/******************************************/
4665
4666		:- 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)
4667		:- 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
4668		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1)],[int(7)]),
4669		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4670		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4671		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4672		:- assert_must_succeed((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7)]),
4673		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4674		:- assert_must_fail((bsets_clp:total_surjection([],[int(1)],[int(7)]))).
4675		:- assert_must_fail((bsets_clp:total_surjection([(int(7),int(7))],[int(1)],[int(7)]))).
4676		:- assert_must_fail((bsets_clp:total_surjection([(int(1),int(7)), (int(2),int(1))],
4677		[int(1),int(2)],[int(7)]))).
4678		:- assert_must_fail((bsets_clp:total_surjection(X,[int(1),int(2)],[int(7),int(6)]),
4679		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4680
4681
4682		total_surjection(R,Domain,Range) :- init_wait_flags(WF),
4683		total_surjection_wf(R,Domain,Range,WF),
4684	?	ground_wait_flags(WF).
4685
4686		:- block total_surjection_wf(-,-,?,?).
4687		total_surjection_wf(R,DomType,RangeType,WF) :-
4688		check_card_greater_equal(DomType,geq,RangeType,CardDom,CardRange),
4689	?	total_function_wf(R,DomType,RangeType,WF),
4690		% setup_surj_range(RangeType,R,WF).
4691		(surjection_has_to_be_total_injection(CardDom,CardRange)
4692		% LAW: card(setX) = card(setY) => ff: setX -->> setY <=> ff: setX >-> setY
4693		-> 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
4694		; check_relation_is_surjective(R,RangeType,WF)).
4695		% invert_relation_wf(R,IR,WF), total_relation_wf(IR,RangeType,DomType,WF).
4696
4697		surjection_has_to_be_total_injection(CardDom,CardRange) :- number(CardDom), CardDom=CardRange.
4698		% 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)
4699
4700		:- 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)).
4701		:- 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)).
4702		:- 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)).
4703		:- 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)).
4704		:- 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)).
4705		:- 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)).
4706
4707		:- block not_total_surjection_wf(-,?,?,?), not_total_surjection_wf(?,-,-,?).
4708		not_total_surjection_wf(R,DomType,RangeType,WF) :-
4709		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4710		not_total_surjection2(PredRes,R,DomType,RangeType,WF).
4711		:- block not_total_surjection2(-,?,?,?,?).
4712		not_total_surjection2(pred_false,_R,_DomType,_RangeType,_WF).
4713		not_total_surjection2(pred_true,R,_DomType,RangeType,WF) :-
4714		range_wf(R,RelRange,WF),
4715		opt_push_wait_flag_call_stack_info(WF,b_operator_call(not_subset,
4716		[RangeType,b_operator(range,[R])],unknown),WF2),
4717		not_subset_of_wf(RangeType,RelRange,WF2).
4718		%not_surjective_relation_wf(R,DomType,RangeType,WF).
4719
4720		:- 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)).
4721		:- 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)
4722		:- 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)).
4723
4724		% reified total function check:
4725		total_function_test_wf(R,DomType,RangeType,PredRes,WF) :-
4726		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4727		(IsPF==pred_false -> PredRes=pred_false
4728		; domain_wf(R,RelDom,WF),
4729	?	conjoin_test(IsPF,IsTotal,PredRes,WF),
4730		opt_push_wait_flag_call_stack_info(WF,b_operator_call(subset,
4731		[DomType,b_operator(domain,[R])],unknown),WF2),
4732	?	subset_test(DomType,RelDom,IsTotal,WF2)
4733		).
4734
4735		/*******************************************/
4736		/* partial_injection(R,DomType,RangeType) */
4737		/* R : DomType >+> RangeType */
4738		/*******************************************/
4739
4740		:- 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)).
4741		:- 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)).
4742		:- 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)).
4743		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1)],[int(7)]),
4744		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4745		:- assert_must_succeed((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4746		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4747		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7)]),
4748		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4749		:- assert_must_succeed((bsets_clp:partial_injection([],[int(1)],[int(7)]))).
4750		:- assert_must_fail((bsets_clp:partial_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4751		:- assert_must_fail((bsets_clp:partial_injection([(int(1),int(7)), (int(2),int(1))],
4752		[int(1),int(2)],[int(7)]))).
4753		:- assert_must_fail((bsets_clp:partial_injection(X,[int(1),int(2)],[int(7),int(6)]),
4754		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4755
4756
4757		partial_injection(R,Domain,Range) :- init_wait_flags(WF),
4758		partial_injection_wf(R,Domain,Range,WF),
4759	?	ground_wait_flags(WF).
4760
4761		:- block partial_injection_wf(-,-,?,?).
4762		partial_injection_wf(FF,Domain,Range,WF) :- nonvar(FF),
4763		custom_explicit_sets:dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection),WF),!,
4764		check_domain_subset_for_closure_wf(FF,FFDomain,Domain,WF),
4765		check_range_subset_for_closure_wf(FF,FFRange,Range,WF).
4766		partial_injection_wf(R,DomType,RangeType,WF) :-
4767		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF), % should we use very_large?
4768		partial_function_wf(ER,DomType,RangeType,WF),
4769		injective(ER,WF).
4770		% invert_relation_wf(R,IR,WF),
4771		% partial_function_wf(IR,RangeType,DomType,WF).
4772
4773		:- assert_must_succeed(exhaustive_kernel_check_wfdet(bsets_clp:injective([(int(1),int(6)),(int(4),int(7)),(int(2),int(8))],WF),WF)).
4774		:- 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)).
4775
4776		:- block injective(-,?).
4777		injective(FF,WF) :-
4778		custom_explicit_sets:dom_range_for_specific_closure(FF,_FFDomain,_FFRange,function(bijection),WF),!.
4779		injective(avl_set(AVL),_WF) :- !,
4780		is_injective_avl_relation(AVL,_Range). % seems slightly faster than injective/3 code below
4781		injective(closure(P,T,B),WF) :- !,
4782		symbolic_injectivity_check(closure(P,T,B),WF).
4783		injective(Rel,WF) :- expand_custom_set_to_list_wf(Rel,ERel,_,injective,WF),
4784		injective(ERel,[],WF).
4785
4786		%:- use_module(library(lists),[maplist/3]).
4787		% for FD-sets we could setup all_different constraint
4788		:- block injective(-,?,?).
4789		injective([],_SoFar,_).
4790		% (maplist(get_fd_val,SoFar,FDL) -> clpfd:all_distinct(FDL) ; true). %clpfd_interface:clpfd_alldifferent(FDL) ; true).
4791		%get_fd_val(int(H),H).
4792		injective([(_From,To)|T],SoFar,WF) :-
4793		not_element_of_wf(To,SoFar,WF), /* check that it is injective */
4794		add_new_element_wf(To,SoFar,SoFar2,WF), %SoFar2=[To|SoFar], could also work and be faster ?
4795		injective(T,SoFar2,WF).
4796		% no case for global_set: it cannot be a relation; two cases below not required because of expand_custom_set_to_list
4797		%injective(avl_set(S),SoFar,WF) :- expand_custom_set_wf(avl_set(S),ES,inj,WF), injective(ES,SoFar,WF).
4798		%injective(closure(P,T,B),SoFar,WF) :- expand_custom_set_wf(closure(P,T,B),ES,inj,WF), injective(ES,SoFar,WF).
4799
4800
4801
4802		/* /: Dom >+> R */
4803
4804		:- 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)).
4805		:- 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)).
4806		:- 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)).
4807		:- 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)).
4808
4809		:- block not_partial_injection(-,?,?,?).
4810		not_partial_injection(R,DomType,RangeType,WF) :-
4811		partial_function_test_wf(R,DomType,RangeType,IsPF,WF),
4812		not_partial_injection2(IsPF,R,DomType,RangeType,WF).
4813
4814		:- block not_partial_injection2(-,?,?,?,?).
4815		not_partial_injection2(pred_false,_R,_DomType,_RType,_WF).
4816		not_partial_injection2(pred_true,R,DomType,RType,WF) :-
4817		not_injection_wf(R,DomType,RType,WF).
4818
4819		not_injection_wf(R,DomType,RType,WF) :-
4820		invert_relation_wf(R,IR,WF),
4821		not_partial_function(IR,RType,DomType,WF).
4822
4823		/*****************************************/
4824		/* total_injection(R,DomType,RangeType) */
4825		/* R : DomType >-> RangeType */
4826		/*****************************************/
4827
4828		:- 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)).
4829		:- 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)).
4830		:- 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)).
4831		:- 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)).
4832		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1)],[int(7)]),
4833		kernel_objects:equal_object(X,[(int(1),int(7))]))).
4834		:- assert_must_succeed((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4835		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(6))]))).
4836		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7)]),
4837		kernel_objects:equal_object(X,[(int(2),int(7)),(int(1),int(7))]))).
4838		:- assert_must_fail((bsets_clp:total_injection([],[int(1)],[int(7)]))).
4839		:- assert_must_fail((bsets_clp:total_injection([(int(7),int(7))],[int(1)],[int(7)]))).
4840		:- assert_must_fail((bsets_clp:total_injection([(int(1),int(7)), (int(2),int(1))],
4841		[int(1),int(2)],[int(7)]))).
4842		:- assert_must_fail((bsets_clp:total_injection(X,[int(1),int(2)],[int(7),int(6)]),
4843		kernel_objects:equal_object(X,[(int(2),int(7)),(int(2),int(6))]))).
4844
4845
4846		total_injection(R,Domain,Range) :- init_wait_flags(WF),
4847		total_injection_wf(R,Domain,Range,WF),
4848	?	ground_wait_flags(WF).
4849
4850		:- block total_injection_wf(-,-,?,?). % with just ?,-,?,? we may wait too long to start injective check
4851		% Note: no need to check: dom_range_for_specific_closure(FF,FFDomain,FFRange,function(bijection)),
4852		total_injection_wf(R,DomType,RangeType,WF) :-
4853		check_card_greater_equal(RangeType,geq,DomType,_,_), % there must be more Range elements than domain elements; pigeonhole principle
4854	?	total_injection_wf2(R,DomType,RangeType,WF).
4855		total_injection_wf2(R,DomType,RangeType,WF) :-
4856		try_expand_and_convert_to_avl_unless_large_wf(R,ER,WF),
4857	?	total_function_wf(ER,DomType,RangeType,WF),
4858		injective(ER,WF).
4859
4860
4861		:- block not_total_injection(-,?,?,?), not_total_injection(?,-,-,?).
4862		not_total_injection(R,DomType,RangeType,WF) :-
4863		total_function_test_wf(R,DomType,RangeType,PredRes,WF),
4864		not_total_injection2(PredRes,R,DomType,RangeType,WF).
4865
4866		:- block not_total_injection2(-,?,?,?,?).
4867		not_total_injection2(pred_false,_R,_Dom,_Ran,_WF).
4868		not_total_injection2(pred_true,R,DomType,RangeType,WF) :-
4869		% TO DO: replace DomType and RangeType by full Type
4870		not_injection_wf(R,DomType,RangeType,WF).
4871
4872		/***********************************/
4873		/* partial_bijection(R,DomType,RangeType) */
4874		/* R : DomType >+>> RangeType */
4875		/***********************************/
4876
4877		:- 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)
4878		:- 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)).
4879		:- assert_must_succeed((partial_bijection(X,[int(1),int(2)],[int(7),int(6)]),
4880		kernel_objects:equal_object(X,[(int(1),int(6)),(int(2),int(7))]))).
4881		:- assert_must_succeed((partial_bijection(X,[int(1),int(2),int(3),int(4)],[int(7),int(6)]),
4882		X = [(int(2),int(7)),(int(3),int(6))])).
4883		:- assert_must_fail((partial_bijection(X,[int(1),int(2)],[int(7),int(6),int(5)]),
4884		X = [(int(2),int(7)),(int(1),int(6))])).
4885
4886		partial_bijection(R,Domain,Range) :- init_wait_flags(WF),
4887		partial_bijection_wf(R,Domain,Range,WF),
4888	?	ground_wait_flags(WF).
4889
4890		partial_bijection_wf(R,DomType,RangeType,WF) :-
4891		partial_injection_wf(R,DomType,RangeType,WF),
4892	?	partial_surjection_wf(R,DomType,RangeType,WF).
4893
4894		:- 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)).
4895		:- 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)).
4896
4897		:- 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)).
4898
4899		:- 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)).
4900		:- 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)).
4901
4902
4903		:- block not_partial_bijection(-,?,?,?), not_partial_bijection(?,-,-,?).
4904		not_partial_bijection(R,DomType,RangeType,WF) :-
4905		% >+>> = +->> + injective
4906		partial_surjection_test_wf(R,DomType,RangeType,PredRes,WF),
4907		not_partial_bijection2(PredRes,R,DomType,RangeType,WF).
4908
4909		:- block not_partial_bijection2(-,?,?,?,?).
4910		not_partial_bijection2(pred_false,_R,_DomType,_RangeType,_WF).
4911		not_partial_bijection2(pred_true,R,DomType,RangeType,WF) :-
4912		not_injection_wf(R,DomType,RangeType,WF).
4913
4914
4915
4916		/* The transitive (not reflexive) closure of a relation (closure1) */
4917
4918		:- 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))]))).
4919		:- 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))]))).
4920		:- 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))]))).
4921		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4))],X),
4922		kernel_objects:equal_object(X,[(int(1),int(4))]))).
4923		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2))],X),
4924		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),
4925		(int(1),int(2))]))).
4926		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(1),int(4)),(int(4),int(2)),(int(2),int(3))],X),
4927		kernel_objects:equal_object(X,[(int(1),int(4)),(int(4),int(2)),(int(2),int(3)),
4928		(int(4),int(3)),(int(1),int(2)),(int(1),int(3))]))).
4929		:- assert_must_succeed((bsets_clp:relational_trans_closure([(int(A),int(2)),(int(2),int(6))],
4930		[(int(1),int(2)),(int(1),int(6)),(int(2),int(6))]),A=1)).
4931
4932		relational_trans_closure(Rel,Res) :- relational_trans_closure_wf(Rel,Res,no_wf_available).
4933
4934		% transitive closure for relations (closure1)
4935		:- block relational_trans_closure_wf(-,?,?).
4936		relational_trans_closure_wf(Relation,Result,WF) :-
4937		try_expand_and_convert_to_avl_with_check(Relation,ARelation,relational_trans_closure_wf),
4938	?	relational_trans_closure2(ARelation,Result,WF).
4939		:- block relational_trans_closure2(-,?,?).
4940		relational_trans_closure2(ARelation,Result,WF) :-
4941		(closure1_for_explicit_set(ARelation,Res)
4942		-> kernel_objects:equal_object_wf(Res,Result,relational_trans_closure_wf,WF)
4943		; expand_custom_set_to_list_wf(ARelation,ERelation,_,relational_trans_closure2,WF),
4944		is_full_relation(ERelation,WaitVar), % still required??
4945		% we could do a check_subset_of_wf(ERelation,Resul,WF) if Result is nonvar and ERelation not ground
4946	?	compute_trans_closure(ERelation,Result,WaitVar,WF)
4947		).
4948
4949		:- block compute_trans_closure(?,?,-,?).
4950		compute_trans_closure(Relation,Result,_,WF) :-
4951	?	compute_trans_closure2(Relation,1,Result,WF).
4952
4953		compute_trans_closure2(Relation,Cnt,Result,WF) :-
4954		one_closure_iteration(Relation,Relation,Relation,Result1,Added,Done,WF),
4955	?	compute_trans_closure3(Relation,Cnt,Result1,Added,Done,Result,WF).
4956
4957		:- block compute_trans_closure3(?,?,?,?,-,?,?).
4958		compute_trans_closure3(Relation,Cnt,Result1,Added,_Done,Result,WF) :-
4959		( equal_object_wf(Result1,Relation,relational_trans_closure_wf,WF), % should we do equality_objects here?
4960		equal_object_optimized_wf(Result,Result1,compute_trans_closure,WF)
4961		;
4962		Added==possibly_added,
4963		not_equal_object_wf(Result1,Relation,WF), % not a fixpoint; continue
4964		IterCnt is Cnt+1,
4965	?	compute_trans_closure2(Result1,IterCnt,Result,WF)
4966		).
4967
4968		:- block one_closure_iteration(?,?,-,?,?,?,?).
4969		one_closure_iteration([],_,IterRes,OutRel,Added,Done,WF) :-
4970		equal_object_wf(IterRes,OutRel,one_closure_iteration,WF),
4971		(var(Added) -> Added=not_added ; true),
4972		Done=done.
4973		one_closure_iteration([(X,Y)|T],ExpandedPreviousRel,PreviousRel,OutRel,Added,Done,WF) :-
4974		add_tuples(ExpandedPreviousRel,X,Y,PreviousRel,IntRel,Added,DoneTuples,WF),
4975	?	one_closure_iteration_block(DoneTuples,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
4976
4977		:- block one_closure_iteration_block(-,?,?,?,?,?,?,?).
4978		one_closure_iteration_block(_,T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF) :-
4979	?	one_closure_iteration(T,ExpandedPreviousRel,IntRel,OutRel,Added,Done,WF).
4980
4981		add_tuples([],_,_,OutRel,OutRel,_Added,done,_).
4982		add_tuples([(X,Y)|T],OX,OY,InRel,OutRel,Added,Done,WF) :-
4983		% add tuple (X,OY) if we have Y=OX
4984		equality_objects_wf(Y,OX,EqRes,WF),
4985	?	add_tuples_aux(EqRes,X,T,OX,OY,InRel,OutRel,Added,Done,WF).
4986
4987		:- block add_tuples_aux(-,?,?,?,?,?,?,?,?,?).
4988		add_tuples_aux(pred_true,X,T,OX,OY,InRel,OutRel,possibly_added,Done,WF) :-
4989		add_element_wf((X,OY),InRel,IntRel,WF), % add transitive couple X -> OY
4990	?	add_tuples(T,OX,OY,IntRel,OutRel,_,Done,WF).
4991		add_tuples_aux(pred_false,_X,T,OX,OY,InRel,OutRel,Added,Done,WF) :- % no transitive couple needed
4992	?	add_tuples(T,OX,OY,InRel,OutRel,Added,Done,WF).
4993
4994
4995		:- assert_must_succeed((is_full_relation(X,R),var(R),X=[],R==true)).
4996		:- 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)).
4997		:- block is_full_relation(-,?).
4998		is_full_relation([],R) :- !,R=true.
4999	?	is_full_relation([H|T],W) :- !, is_full_relation_aux(H,T,W).
5000		is_full_relation(X,R) :-
5001		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation(X,R)),fail.
5002
5003		:- block is_full_relation_aux(-,?,?).
5004	?	is_full_relation_aux((X,Y),T,W) :- !, is_full_relation_aux2(X,Y,T,W).
5005		is_full_relation_aux(X,T,W) :-
5006		add_internal_error('Illegal Set for is_full_relation: ',is_full_relation_aux(X,T,W)),fail.
5007		:- block is_full_relation_aux2(-,?,?,?), is_full_relation_aux2(?,-,?,?).
5008	?	is_full_relation_aux2(_X,_Y,T,W) :- is_full_relation(T,W).
5009
5010		/* ------------------ */
5011
5012		:- 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)
5013
5014		in_closure1_wf(Pair,Relation,WF) :- %Pair = (_A,B),
5015		%in_domain_wf_lazy(A,Relation,WF), % done below
5016		%check_element_of_wf((_,B),Relation,WF), % multiple solutions for _, see test 634, 637
5017	?	in_closure1_membership_test_wf(Pair,Relation,pred_true,WF).
5018
5019
5020		:- 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)).
5021
5022		not_in_closure1_wf(Pair,Relation,WF) :-
5023	?	in_closure1_membership_test_wf(Pair,Relation,pred_false,WF).
5024
5025		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[],Res,_WF),Res==pred_false)).
5026		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(2))],Res,_WF),Res==pred_true)).
5027		:- assert_must_succeed((bsets_clp:in_closure1_membership_test_wf((int(1),int(2)),[(int(1),int(3))],Res,_WF),Res==pred_false)).
5028		:- 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)).
5029		:- 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)
5030		:- 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)).
5031		:- 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)).
5032		:- 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)).
5033		:- 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)
5034		:- 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)).
5035
5036		:- block force_in_domain(-,?,?,?).
5037		force_in_domain(pred_false,_A,_Relation,_WF).
5038		force_in_domain(pred_true,A,Relation,WF) :- % force A to be in domain, avoid enumeration warnings,...
5039		% maybe only for non-ground A
5040		in_domain_wf_lazy(A,Relation,WF). % slowdown Loop.mch (tests 634, 637) if we use in_domain_wf ?
5041
5042		% (x,y) : closure1(Rel)
5043		:- block in_closure1_membership_test_wf(?,-,?,?).
5044		in_closure1_membership_test_wf((A,B),CSRelation,MemRes,WF) :-
5045		is_custom_explicit_set(CSRelation,in_closure1),
5046		!,
5047	?	image_for_closure1_wf(CSRelation,[A],Image,WF),
5048		force_in_domain(MemRes,A,CSRelation,WF),
5049		membership_test_wf(Image,B,MemRes,WF).
5050		in_closure1_membership_test_wf((X,Y),Relation,MemRes,WF) :-
5051		expand_custom_set_to_list_wf(Relation,ERelation,_,in_closure1_membership_test_wf,WF),
5052		Discarded = [], % pairs discarded in current iteration
5053		force_in_domain(MemRes,X,Relation,WF),
5054		in_closure1_membership_test_wf2(ERelation,X,Y,Discarded,MemRes,WF).
5055
5056		:- block in_closure1_membership_test_wf2(-,?,?,?,?,?).
5057		in_closure1_membership_test_wf2([],_X,_Y,_,MemRes,_WF) :- MemRes=pred_false.
5058		in_closure1_membership_test_wf2([(V,W)|Rest],X,Y,Discarded,MemRes,WF) :- % TO DO: Rest==[] -->
5059		equality_objects_wf(V,X,VXResult,WF),
5060		in_closure1_membership_test_wf3(VXResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5061
5062		:- block in_closure1_membership_test_wf3(-,?,?,?,?,?,?,?,?).
5063		in_closure1_membership_test_wf3(pred_false,V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5064		in_closure1_membership_test_wf2(Rest,X,Y,[(V,W)|Discarded],MemRes,WF).
5065		in_closure1_membership_test_wf3(pred_true,V,W,Rest,X,Y,Discarded,MemRes,WF) :- % V=X
5066		propagate_false(MemRes,WYResult),
5067		% TODO: Res=[],Discarded=[] -> MemRes=WYResult
5068		equality_objects_wf(W,Y,WYResult,WF), % MemRes = pred_false => WYResult = pred_false
5069		in_closure1_membership_test_wf4(WYResult,V,W,Rest,X,Y,Discarded,MemRes,WF).
5070
5071		:- block in_closure1_membership_test_wf4(-,?,?,?,?,?,?,?,?).
5072		in_closure1_membership_test_wf4(pred_false,_V,W,Rest,X,Y,Discarded,MemRes,WF) :-
5073		append(Discarded,Rest,Restart),
5074		in_closure1_membership_test_wf2(Restart,W,Y,[],MemRes1,WF),
5075		propagate_false(MemRes,MemRes1), % MemRes = pred_false -> MemRes1=pred_false
5076		when(nonvar(MemRes1),
5077		(MemRes1=pred_true -> MemRes=pred_true
5078		; in_closure1_membership_test_wf2(Rest,X,Y,Discarded,MemRes,WF) % (V,W) not in Discarded: was not useful
5079		)).
5080		in_closure1_membership_test_wf4(pred_true,_V,_W,_Rest,_X,_Y,_Discarded,MemRes,_WF) :- % W=Y
5081		MemRes = pred_true.
5082		/* ------------------ */
5083
5084		:- block propagate_false(-,?).
5085		propagate_false(pred_false,pred_false).
5086		propagate_false(pred_true,_).
5087