166.37/63.23	YES
166.37/63.24	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
166.37/63.24	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
166.37/63.24	
166.37/63.24	
166.37/63.24	Quasi decreasingness of the given CTRS could be proven:
166.37/63.24	
166.37/63.24	(0) CTRS
166.37/63.24	(1) CTRSToQTRSProof [SOUND, 0 ms]
166.37/63.24	(2) QTRS
166.37/63.24	(3) AAECC Innermost [EQUIVALENT, 0 ms]
166.37/63.24	(4) QTRS
166.37/63.24	(5) DependencyPairsProof [EQUIVALENT, 5 ms]
166.37/63.24	(6) QDP
166.37/63.24	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	(8) AND
166.37/63.24	    (9) QDP
166.37/63.24	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (11) QDP
166.37/63.24	        (12) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (13) QDP
166.37/63.24	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (15) YES
166.37/63.24	    (16) QDP
166.37/63.24	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (18) QDP
166.37/63.24	        (19) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (20) QDP
166.37/63.24	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (22) YES
166.37/63.24	    (23) QDP
166.37/63.24	        (24) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (25) QDP
166.37/63.24	        (26) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (27) QDP
166.37/63.24	        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (29) YES
166.37/63.24	    (30) QDP
166.37/63.24	        (31) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (32) QDP
166.37/63.24	        (33) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (34) QDP
166.37/63.24	        (35) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (36) YES
166.37/63.24	    (37) QDP
166.37/63.24	        (38) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (39) QDP
166.37/63.24	        (40) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (41) QDP
166.37/63.24	        (42) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (43) YES
166.37/63.24	    (44) QDP
166.37/63.24	        (45) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (46) QDP
166.37/63.24	        (47) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (48) QDP
166.37/63.24	        (49) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	        (50) YES
166.37/63.24	    (51) QDP
166.37/63.24	        (52) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	        (53) QDP
166.37/63.24	        (54) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	        (55) QDP
166.37/63.24	        (56) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	        (57) QDP
166.37/63.24	        (58) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	        (59) QDP
166.37/63.24	        (60) QDPQMonotonicMRRProof [EQUIVALENT, 179 ms]
166.37/63.24	        (61) QDP
166.37/63.24	        (62) NonInfProof [EQUIVALENT, 396 ms]
166.37/63.24	        (63) QDP
166.37/63.24	        (64) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	        (65) QDP
166.37/63.24	        (66) QDPOrderProof [EQUIVALENT, 623 ms]
166.37/63.24	        (67) QDP
166.37/63.24	        (68) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	        (69) QDP
166.37/63.24	        (70) NonInfProof [EQUIVALENT, 394 ms]
166.37/63.24	        (71) AND
166.37/63.24	            (72) QDP
166.37/63.24	                (73) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (74) QDP
166.37/63.24	                (75) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (76) QDP
166.37/63.24	                (77) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (78) QDP
166.37/63.24	                (79) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (80) QDP
166.37/63.24	                (81) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (82) QDP
166.37/63.24	                (83) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (84) QDP
166.37/63.24	                (85) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (86) QDP
166.37/63.24	                (87) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (88) QDP
166.37/63.24	                (89) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (90) QDP
166.37/63.24	                (91) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (92) QDP
166.37/63.24	                (93) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (94) QDP
166.37/63.24	                (95) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (96) QDP
166.37/63.24	                (97) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (98) QDP
166.37/63.24	                (99) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (100) QDP
166.37/63.24	                (101) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (102) QDP
166.37/63.24	                (103) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (104) QDP
166.37/63.24	                (105) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (106) QDP
166.37/63.24	                (107) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (108) QDP
166.37/63.24	                (109) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (110) QDP
166.37/63.24	                (111) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (112) QDP
166.37/63.24	                (113) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (114) QDP
166.37/63.24	                (115) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (116) QDP
166.37/63.24	                (117) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (118) QDP
166.37/63.24	                (119) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (120) QDP
166.37/63.24	                (121) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (122) QDP
166.37/63.24	                (123) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	                (124) QDP
166.37/63.24	                (125) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	                (126) QDP
166.37/63.24	                (127) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (128) QDP
166.37/63.24	                (129) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (130) QDP
166.37/63.24	                (131) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	                (132) QDP
166.37/63.24	                (133) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	                (134) QDP
166.37/63.24	                (135) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (136) QDP
166.37/63.24	                (137) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (138) QDP
166.37/63.24	                (139) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (140) QDP
166.37/63.24	                (141) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (142) QDP
166.37/63.24	                (143) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (144) QDP
166.37/63.24	                (145) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (146) QDP
166.37/63.24	                (147) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (148) QDP
166.37/63.24	                (149) QDPSizeChangeProof [EQUIVALENT, 0 ms]
166.37/63.24	                (150) YES
166.37/63.24	            (151) QDP
166.37/63.24	                (152) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (153) QDP
166.37/63.24	                (154) QDPQMonotonicMRRProof [EQUIVALENT, 24 ms]
166.37/63.24	                (155) QDP
166.37/63.24	                (156) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                (157) QDP
166.37/63.24	                (158) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                (159) QDP
166.37/63.24	                (160) NonInfProof [EQUIVALENT, 50 ms]
166.37/63.24	                (161) AND
166.37/63.24	                    (162) QDP
166.37/63.24	                        (163) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (164) QDP
166.37/63.24	                        (165) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (166) QDP
166.37/63.24	                        (167) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (168) QDP
166.37/63.24	                        (169) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (170) QDP
166.37/63.24	                        (171) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (172) QDP
166.37/63.24	                        (173) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (174) QDP
166.37/63.24	                        (175) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (176) QDP
166.37/63.24	                        (177) UsableRulesProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (178) QDP
166.37/63.24	                        (179) QReductionProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (180) QDP
166.37/63.24	                        (181) TransformationProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (182) QDP
166.37/63.24	                        (183) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (184) TRUE
166.37/63.24	                    (185) QDP
166.37/63.24	                        (186) DependencyGraphProof [EQUIVALENT, 0 ms]
166.37/63.24	                        (187) TRUE
166.37/63.24	
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(0)
166.37/63.24	Obligation:
166.37/63.24	Conditional term rewrite system:
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	The conditional TRS C consists of the following conditional rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) <= empty(fstsplit(m, st_1)) -> false
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) <= leq(m, length(st_2)) -> true, empty(fstsplit(m, st_2)) -> false
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) <= leq(m, length(st_2)) -> false, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))) -> false
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m) <= empty(map_f(two, head(in_2))) -> true
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) <= leq(m, length(st_3)) -> true, empty(fstsplit(m, st_3)) -> false
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) <= leq(m, length(st_3)) -> false, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))) -> false
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m) <= empty(map_f(three, head(in_3))) -> true
166.37/63.24	
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(1) CTRSToQTRSProof (SOUND)
166.37/63.24	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(2)
166.37/63.24	Obligation:
166.37/63.24	Q restricted rewrite system:
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	Q is empty.
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(3) AAECC Innermost (EQUIVALENT)
166.37/63.24	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	The TRS R 2 is 
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	
166.37/63.24	The signature Sigma is {ring_6, U1_7, U2_7, U4_7, U3_7, U5_7, U6_7, U7_7, U8_7, U9_7}
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(4)
166.37/63.24	Obligation:
166.37/63.24	Q restricted rewrite system:
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	The set Q consists of the following terms:
166.37/63.24	
166.37/63.24	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.24	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   fstsplit(0, x0)
166.37/63.24	   fstsplit(s(x0), nil)
166.37/63.24	   fstsplit(s(x0), cons(x1, x2))
166.37/63.24	   sndsplit(0, x0)
166.37/63.24	   sndsplit(s(x0), nil)
166.37/63.24	   sndsplit(s(x0), cons(x1, x2))
166.37/63.24	   empty(nil)
166.37/63.24	   empty(cons(x0, x1))
166.37/63.24	   leq(0, x0)
166.37/63.24	   leq(s(x0), 0)
166.37/63.24	   leq(s(x0), s(x1))
166.37/63.24	   length(nil)
166.37/63.24	   length(cons(x0, x1))
166.37/63.24	   app(nil, x0)
166.37/63.24	   app(cons(x0, x1), x2)
166.37/63.24	   map_f(x0, nil)
166.37/63.24	   map_f(x0, cons(x1, x2))
166.37/63.24	   head(cons(x0, x1))
166.37/63.24	   tail(cons(x0, x1))
166.37/63.24	
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(5) DependencyPairsProof (EQUIVALENT)
166.37/63.24	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(6)
166.37/63.24	Obligation:
166.37/63.24	Q DP problem:
166.37/63.24	The TRS P consists of the following rules:
166.37/63.24	
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(fstsplit(m, st_1))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, st_1)
166.37/63.24	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> SNDSPLIT(m, st_1)
166.37/63.24	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, st_1)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> LEQ(m, length(st_2))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH(st_2)
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(fstsplit(m, app(map_f(two, head(in_2)), st_2)))
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, app(map_f(two, head(in_2)), st_2))
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> APP(map_f(two, head(in_2)), st_2)
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(two, head(in_2))
166.37/63.24	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_2)
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> TAIL(in_2)
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> SNDSPLIT(m, app(map_f(two, head(in_2)), st_2))
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> APP(map_f(two, head(in_2)), st_2)
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(two, head(in_2))
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_2)
166.37/63.24	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, app(map_f(two, head(in_2)), st_2))
166.37/63.24	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(fstsplit(m, st_2))
166.37/63.24	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, st_2)
166.37/63.24	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> SNDSPLIT(m, st_2)
166.37/63.24	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, st_2)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U5^1(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(map_f(two, head(in_2)))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(two, head(in_2))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_2)
166.37/63.24	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> TAIL(in_2)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> LEQ(m, length(st_3))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH(st_3)
166.37/63.24	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(fstsplit(m, st_3))
166.37/63.24	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, st_3)
166.37/63.24	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> SNDSPLIT(m, st_3)
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(fstsplit(m, app(map_f(three, head(in_3)), st_3)))
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT(m, app(map_f(three, head(in_3)), st_3))
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> APP(map_f(three, head(in_3)), st_3)
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(three, head(in_3))
166.37/63.24	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_3)
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> TAIL(in_3)
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> SNDSPLIT(m, app(map_f(three, head(in_3)), st_3))
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> APP(map_f(three, head(in_3)), st_3)
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(three, head(in_3))
166.37/63.24	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_3)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY(map_f(three, head(in_3)))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F(three, head(in_3))
166.37/63.24	   RING(st_1, in_2, st_2, in_3, st_3, m) -> HEAD(in_3)
166.37/63.24	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> TAIL(in_3)
166.37/63.24	   FSTSPLIT(s(n), cons(h, t)) -> FSTSPLIT(n, t)
166.37/63.24	   SNDSPLIT(s(n), cons(h, t)) -> SNDSPLIT(n, t)
166.37/63.24	   LEQ(s(n), s(m)) -> LEQ(n, m)
166.37/63.24	   LENGTH(cons(h, t)) -> LENGTH(t)
166.37/63.24	   APP(cons(h, t), x) -> APP(t, x)
166.37/63.24	   MAP_F(pid, cons(h, t)) -> APP(f(pid, h), map_f(pid, t))
166.37/63.24	   MAP_F(pid, cons(h, t)) -> MAP_F(pid, t)
166.37/63.24	
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	The set Q consists of the following terms:
166.37/63.24	
166.37/63.24	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.24	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   fstsplit(0, x0)
166.37/63.24	   fstsplit(s(x0), nil)
166.37/63.24	   fstsplit(s(x0), cons(x1, x2))
166.37/63.24	   sndsplit(0, x0)
166.37/63.24	   sndsplit(s(x0), nil)
166.37/63.24	   sndsplit(s(x0), cons(x1, x2))
166.37/63.24	   empty(nil)
166.37/63.24	   empty(cons(x0, x1))
166.37/63.24	   leq(0, x0)
166.37/63.24	   leq(s(x0), 0)
166.37/63.24	   leq(s(x0), s(x1))
166.37/63.24	   length(nil)
166.37/63.24	   length(cons(x0, x1))
166.37/63.24	   app(nil, x0)
166.37/63.24	   app(cons(x0, x1), x2)
166.37/63.24	   map_f(x0, nil)
166.37/63.24	   map_f(x0, cons(x1, x2))
166.37/63.24	   head(cons(x0, x1))
166.37/63.24	   tail(cons(x0, x1))
166.37/63.24	
166.37/63.24	We have to consider all minimal (P,Q,R)-chains.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(7) DependencyGraphProof (EQUIVALENT)
166.37/63.24	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 45 less nodes.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(8)
166.37/63.24	Complex Obligation (AND)
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(9)
166.37/63.24	Obligation:
166.37/63.24	Q DP problem:
166.37/63.24	The TRS P consists of the following rules:
166.37/63.24	
166.37/63.24	   APP(cons(h, t), x) -> APP(t, x)
166.37/63.24	
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.24	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.24	   fstsplit(0, x) -> nil
166.37/63.24	   fstsplit(s(n), nil) -> nil
166.37/63.24	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.24	   sndsplit(0, x) -> x
166.37/63.24	   sndsplit(s(n), nil) -> nil
166.37/63.24	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.24	   empty(nil) -> true
166.37/63.24	   empty(cons(h, t)) -> false
166.37/63.24	   leq(0, m) -> true
166.37/63.24	   leq(s(n), 0) -> false
166.37/63.24	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.24	   length(nil) -> 0
166.37/63.24	   length(cons(h, t)) -> s(length(t))
166.37/63.24	   app(nil, x) -> x
166.37/63.24	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.24	   map_f(pid, nil) -> nil
166.37/63.24	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.24	   head(cons(h, t)) -> h
166.37/63.24	   tail(cons(h, t)) -> t
166.37/63.24	
166.37/63.24	The set Q consists of the following terms:
166.37/63.24	
166.37/63.24	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.24	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   fstsplit(0, x0)
166.37/63.24	   fstsplit(s(x0), nil)
166.37/63.24	   fstsplit(s(x0), cons(x1, x2))
166.37/63.24	   sndsplit(0, x0)
166.37/63.24	   sndsplit(s(x0), nil)
166.37/63.24	   sndsplit(s(x0), cons(x1, x2))
166.37/63.24	   empty(nil)
166.37/63.24	   empty(cons(x0, x1))
166.37/63.24	   leq(0, x0)
166.37/63.24	   leq(s(x0), 0)
166.37/63.24	   leq(s(x0), s(x1))
166.37/63.24	   length(nil)
166.37/63.24	   length(cons(x0, x1))
166.37/63.24	   app(nil, x0)
166.37/63.24	   app(cons(x0, x1), x2)
166.37/63.24	   map_f(x0, nil)
166.37/63.24	   map_f(x0, cons(x1, x2))
166.37/63.24	   head(cons(x0, x1))
166.37/63.24	   tail(cons(x0, x1))
166.37/63.24	
166.37/63.24	We have to consider all minimal (P,Q,R)-chains.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(10) UsableRulesProof (EQUIVALENT)
166.37/63.24	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(11)
166.37/63.24	Obligation:
166.37/63.24	Q DP problem:
166.37/63.24	The TRS P consists of the following rules:
166.37/63.24	
166.37/63.24	   APP(cons(h, t), x) -> APP(t, x)
166.37/63.24	
166.37/63.24	R is empty.
166.37/63.24	The set Q consists of the following terms:
166.37/63.24	
166.37/63.24	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.24	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   fstsplit(0, x0)
166.37/63.24	   fstsplit(s(x0), nil)
166.37/63.24	   fstsplit(s(x0), cons(x1, x2))
166.37/63.24	   sndsplit(0, x0)
166.37/63.24	   sndsplit(s(x0), nil)
166.37/63.24	   sndsplit(s(x0), cons(x1, x2))
166.37/63.24	   empty(nil)
166.37/63.24	   empty(cons(x0, x1))
166.37/63.24	   leq(0, x0)
166.37/63.24	   leq(s(x0), 0)
166.37/63.24	   leq(s(x0), s(x1))
166.37/63.24	   length(nil)
166.37/63.24	   length(cons(x0, x1))
166.37/63.24	   app(nil, x0)
166.37/63.24	   app(cons(x0, x1), x2)
166.37/63.24	   map_f(x0, nil)
166.37/63.24	   map_f(x0, cons(x1, x2))
166.37/63.24	   head(cons(x0, x1))
166.37/63.24	   tail(cons(x0, x1))
166.37/63.24	
166.37/63.24	We have to consider all minimal (P,Q,R)-chains.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(12) QReductionProof (EQUIVALENT)
166.37/63.24	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.24	
166.37/63.24	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.24	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.24	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.24	   fstsplit(0, x0)
166.37/63.24	   fstsplit(s(x0), nil)
166.37/63.24	   fstsplit(s(x0), cons(x1, x2))
166.37/63.24	   sndsplit(0, x0)
166.37/63.24	   sndsplit(s(x0), nil)
166.37/63.24	   sndsplit(s(x0), cons(x1, x2))
166.37/63.24	   empty(nil)
166.37/63.24	   empty(cons(x0, x1))
166.37/63.24	   leq(0, x0)
166.37/63.24	   leq(s(x0), 0)
166.37/63.24	   leq(s(x0), s(x1))
166.37/63.24	   length(nil)
166.37/63.24	   length(cons(x0, x1))
166.37/63.24	   app(nil, x0)
166.37/63.24	   app(cons(x0, x1), x2)
166.37/63.24	   map_f(x0, nil)
166.37/63.24	   map_f(x0, cons(x1, x2))
166.37/63.24	   head(cons(x0, x1))
166.37/63.24	   tail(cons(x0, x1))
166.37/63.24	
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(13)
166.37/63.24	Obligation:
166.37/63.24	Q DP problem:
166.37/63.24	The TRS P consists of the following rules:
166.37/63.24	
166.37/63.24	   APP(cons(h, t), x) -> APP(t, x)
166.37/63.24	
166.37/63.24	R is empty.
166.37/63.24	Q is empty.
166.37/63.24	We have to consider all minimal (P,Q,R)-chains.
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(14) QDPSizeChangeProof (EQUIVALENT)
166.37/63.24	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.24	
166.37/63.24	From the DPs we obtained the following set of size-change graphs:
166.37/63.24	*APP(cons(h, t), x) -> APP(t, x)
166.37/63.24	The graph contains the following edges 1 > 1, 2 >= 2
166.37/63.24	
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(15)
166.37/63.24	YES
166.37/63.24	
166.37/63.24	----------------------------------------
166.37/63.24	
166.37/63.24	(16)
166.37/63.24	Obligation:
166.37/63.24	Q DP problem:
166.37/63.24	The TRS P consists of the following rules:
166.37/63.24	
166.37/63.24	   MAP_F(pid, cons(h, t)) -> MAP_F(pid, t)
166.37/63.24	
166.37/63.24	The TRS R consists of the following rules:
166.37/63.24	
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.24	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.24	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.24	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(17) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(18)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   MAP_F(pid, cons(h, t)) -> MAP_F(pid, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(19) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(20)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   MAP_F(pid, cons(h, t)) -> MAP_F(pid, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	Q is empty.
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(21) QDPSizeChangeProof (EQUIVALENT)
166.37/63.25	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.25	
166.37/63.25	From the DPs we obtained the following set of size-change graphs:
166.37/63.25	*MAP_F(pid, cons(h, t)) -> MAP_F(pid, t)
166.37/63.25	The graph contains the following edges 1 >= 1, 2 > 2
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(22)
166.37/63.25	YES
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(23)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LENGTH(cons(h, t)) -> LENGTH(t)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(24) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(25)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LENGTH(cons(h, t)) -> LENGTH(t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(26) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(27)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LENGTH(cons(h, t)) -> LENGTH(t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	Q is empty.
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(28) QDPSizeChangeProof (EQUIVALENT)
166.37/63.25	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.25	
166.37/63.25	From the DPs we obtained the following set of size-change graphs:
166.37/63.25	*LENGTH(cons(h, t)) -> LENGTH(t)
166.37/63.25	The graph contains the following edges 1 > 1
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(29)
166.37/63.25	YES
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(30)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LEQ(s(n), s(m)) -> LEQ(n, m)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(31) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(32)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LEQ(s(n), s(m)) -> LEQ(n, m)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(33) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(34)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   LEQ(s(n), s(m)) -> LEQ(n, m)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	Q is empty.
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(35) QDPSizeChangeProof (EQUIVALENT)
166.37/63.25	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.25	
166.37/63.25	From the DPs we obtained the following set of size-change graphs:
166.37/63.25	*LEQ(s(n), s(m)) -> LEQ(n, m)
166.37/63.25	The graph contains the following edges 1 > 1, 2 > 2
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(36)
166.37/63.25	YES
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(37)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   SNDSPLIT(s(n), cons(h, t)) -> SNDSPLIT(n, t)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(38) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(39)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   SNDSPLIT(s(n), cons(h, t)) -> SNDSPLIT(n, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(40) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(41)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   SNDSPLIT(s(n), cons(h, t)) -> SNDSPLIT(n, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	Q is empty.
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(42) QDPSizeChangeProof (EQUIVALENT)
166.37/63.25	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.25	
166.37/63.25	From the DPs we obtained the following set of size-change graphs:
166.37/63.25	*SNDSPLIT(s(n), cons(h, t)) -> SNDSPLIT(n, t)
166.37/63.25	The graph contains the following edges 1 > 1, 2 > 2
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(43)
166.37/63.25	YES
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(44)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   FSTSPLIT(s(n), cons(h, t)) -> FSTSPLIT(n, t)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(45) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(46)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   FSTSPLIT(s(n), cons(h, t)) -> FSTSPLIT(n, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(47) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(48)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   FSTSPLIT(s(n), cons(h, t)) -> FSTSPLIT(n, t)
166.37/63.25	
166.37/63.25	R is empty.
166.37/63.25	Q is empty.
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(49) QDPSizeChangeProof (EQUIVALENT)
166.37/63.25	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.37/63.25	
166.37/63.25	From the DPs we obtained the following set of size-change graphs:
166.37/63.25	*FSTSPLIT(s(n), cons(h, t)) -> FSTSPLIT(n, t)
166.37/63.25	The graph contains the following edges 1 > 1, 2 > 2
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(50)
166.37/63.25	YES
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(51)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U5^1(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U1(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U2(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2(false, st_1, in_2, st_2, in_3, st_3, m) -> U4(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   U2(true, st_1, in_2, st_2, in_3, st_3, m) -> U3(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U5(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U6(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6(true, st_1, in_2, st_2, in_3, st_3, m) -> U7(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   U6(false, st_1, in_2, st_2, in_3, st_3, m) -> U8(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8(false, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   ring(st_1, in_2, st_2, in_3, st_3, m) -> U9(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9(true, st_1, in_2, st_2, in_3, st_3, m) -> ring(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(52) UsableRulesProof (EQUIVALENT)
166.37/63.25	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(53)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U5^1(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(54) QReductionProof (EQUIVALENT)
166.37/63.25	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.25	
166.37/63.25	   ring(x0, x1, x2, x3, x4, x5)
166.37/63.25	   U1(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U4(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U2(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U3(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U5(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U7(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U6(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U8(false, x0, x1, x2, x3, x4, x5)
166.37/63.25	   U9(true, x0, x1, x2, x3, x4, x5)
166.37/63.25	
166.37/63.25	
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(55)
166.37/63.25	Obligation:
166.37/63.25	Q DP problem:
166.37/63.25	The TRS P consists of the following rules:
166.37/63.25	
166.37/63.25	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U5^1(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.25	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.25	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.25	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.25	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.25	
166.37/63.25	The TRS R consists of the following rules:
166.37/63.25	
166.37/63.25	   sndsplit(0, x) -> x
166.37/63.25	   sndsplit(s(n), nil) -> nil
166.37/63.25	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.25	   fstsplit(0, x) -> nil
166.37/63.25	   fstsplit(s(n), nil) -> nil
166.37/63.25	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.25	   empty(nil) -> true
166.37/63.25	   empty(cons(h, t)) -> false
166.37/63.25	   tail(cons(h, t)) -> t
166.37/63.25	   head(cons(h, t)) -> h
166.37/63.25	   map_f(pid, nil) -> nil
166.37/63.25	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.25	   app(nil, x) -> x
166.37/63.25	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.25	   length(nil) -> 0
166.37/63.25	   length(cons(h, t)) -> s(length(t))
166.37/63.25	   leq(0, m) -> true
166.37/63.25	   leq(s(n), 0) -> false
166.37/63.25	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.25	
166.37/63.25	The set Q consists of the following terms:
166.37/63.25	
166.37/63.25	   fstsplit(0, x0)
166.37/63.25	   fstsplit(s(x0), nil)
166.37/63.25	   fstsplit(s(x0), cons(x1, x2))
166.37/63.25	   sndsplit(0, x0)
166.37/63.25	   sndsplit(s(x0), nil)
166.37/63.25	   sndsplit(s(x0), cons(x1, x2))
166.37/63.25	   empty(nil)
166.37/63.25	   empty(cons(x0, x1))
166.37/63.25	   leq(0, x0)
166.37/63.25	   leq(s(x0), 0)
166.37/63.25	   leq(s(x0), s(x1))
166.37/63.25	   length(nil)
166.37/63.25	   length(cons(x0, x1))
166.37/63.25	   app(nil, x0)
166.37/63.25	   app(cons(x0, x1), x2)
166.37/63.25	   map_f(x0, nil)
166.37/63.25	   map_f(x0, cons(x1, x2))
166.37/63.25	   head(cons(x0, x1))
166.37/63.25	   tail(cons(x0, x1))
166.37/63.25	
166.37/63.25	We have to consider all minimal (P,Q,R)-chains.
166.37/63.25	----------------------------------------
166.37/63.25	
166.37/63.25	(56) TransformationProof (EQUIVALENT)
166.37/63.25	By narrowing [LPAR04] the rule RING(st_1, in_2, st_2, in_3, st_3, m) -> U5^1(empty(map_f(two, head(in_2))), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.27	
166.37/63.27	   (RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5),RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5))
166.37/63.27	
166.37/63.27	
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(57)
166.37/63.27	Obligation:
166.37/63.27	Q DP problem:
166.37/63.27	The TRS P consists of the following rules:
166.37/63.27	
166.37/63.27	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.27	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.27	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.27	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.27	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.27	
166.37/63.27	The TRS R consists of the following rules:
166.37/63.27	
166.37/63.27	   sndsplit(0, x) -> x
166.37/63.27	   sndsplit(s(n), nil) -> nil
166.37/63.27	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.27	   fstsplit(0, x) -> nil
166.37/63.27	   fstsplit(s(n), nil) -> nil
166.37/63.27	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.27	   empty(nil) -> true
166.37/63.27	   empty(cons(h, t)) -> false
166.37/63.27	   tail(cons(h, t)) -> t
166.37/63.27	   head(cons(h, t)) -> h
166.37/63.27	   map_f(pid, nil) -> nil
166.37/63.27	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.27	   app(nil, x) -> x
166.37/63.27	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.27	   length(nil) -> 0
166.37/63.27	   length(cons(h, t)) -> s(length(t))
166.37/63.27	   leq(0, m) -> true
166.37/63.27	   leq(s(n), 0) -> false
166.37/63.27	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.27	
166.37/63.27	The set Q consists of the following terms:
166.37/63.27	
166.37/63.27	   fstsplit(0, x0)
166.37/63.27	   fstsplit(s(x0), nil)
166.37/63.27	   fstsplit(s(x0), cons(x1, x2))
166.37/63.27	   sndsplit(0, x0)
166.37/63.27	   sndsplit(s(x0), nil)
166.37/63.27	   sndsplit(s(x0), cons(x1, x2))
166.37/63.27	   empty(nil)
166.37/63.27	   empty(cons(x0, x1))
166.37/63.27	   leq(0, x0)
166.37/63.27	   leq(s(x0), 0)
166.37/63.27	   leq(s(x0), s(x1))
166.37/63.27	   length(nil)
166.37/63.27	   length(cons(x0, x1))
166.37/63.27	   app(nil, x0)
166.37/63.27	   app(cons(x0, x1), x2)
166.37/63.27	   map_f(x0, nil)
166.37/63.27	   map_f(x0, cons(x1, x2))
166.37/63.27	   head(cons(x0, x1))
166.37/63.27	   tail(cons(x0, x1))
166.37/63.27	
166.37/63.27	We have to consider all minimal (P,Q,R)-chains.
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(58) TransformationProof (EQUIVALENT)
166.37/63.27	By narrowing [LPAR04] the rule RING(st_1, in_2, st_2, in_3, st_3, m) -> U9^1(empty(map_f(three, head(in_3))), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.27	
166.37/63.27	   (RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5),RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5))
166.37/63.27	
166.37/63.27	
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(59)
166.37/63.27	Obligation:
166.37/63.27	Q DP problem:
166.37/63.27	The TRS P consists of the following rules:
166.37/63.27	
166.37/63.27	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.27	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.27	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.27	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.27	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.27	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.27	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.27	
166.37/63.27	The TRS R consists of the following rules:
166.37/63.27	
166.37/63.27	   sndsplit(0, x) -> x
166.37/63.27	   sndsplit(s(n), nil) -> nil
166.37/63.27	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.27	   fstsplit(0, x) -> nil
166.37/63.27	   fstsplit(s(n), nil) -> nil
166.37/63.27	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.27	   empty(nil) -> true
166.37/63.27	   empty(cons(h, t)) -> false
166.37/63.27	   tail(cons(h, t)) -> t
166.37/63.27	   head(cons(h, t)) -> h
166.37/63.27	   map_f(pid, nil) -> nil
166.37/63.27	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.27	   app(nil, x) -> x
166.37/63.27	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.27	   length(nil) -> 0
166.37/63.27	   length(cons(h, t)) -> s(length(t))
166.37/63.27	   leq(0, m) -> true
166.37/63.27	   leq(s(n), 0) -> false
166.37/63.27	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.27	
166.37/63.27	The set Q consists of the following terms:
166.37/63.27	
166.37/63.27	   fstsplit(0, x0)
166.37/63.27	   fstsplit(s(x0), nil)
166.37/63.27	   fstsplit(s(x0), cons(x1, x2))
166.37/63.27	   sndsplit(0, x0)
166.37/63.27	   sndsplit(s(x0), nil)
166.37/63.27	   sndsplit(s(x0), cons(x1, x2))
166.37/63.27	   empty(nil)
166.37/63.27	   empty(cons(x0, x1))
166.37/63.27	   leq(0, x0)
166.37/63.27	   leq(s(x0), 0)
166.37/63.27	   leq(s(x0), s(x1))
166.37/63.27	   length(nil)
166.37/63.27	   length(cons(x0, x1))
166.37/63.27	   app(nil, x0)
166.37/63.27	   app(cons(x0, x1), x2)
166.37/63.27	   map_f(x0, nil)
166.37/63.27	   map_f(x0, cons(x1, x2))
166.37/63.27	   head(cons(x0, x1))
166.37/63.27	   tail(cons(x0, x1))
166.37/63.27	
166.37/63.27	We have to consider all minimal (P,Q,R)-chains.
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(60) QDPQMonotonicMRRProof (EQUIVALENT)
166.37/63.27	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
166.37/63.27	
166.37/63.27	
166.37/63.27	Strictly oriented rules of the TRS R:
166.37/63.27	
166.37/63.27	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
166.37/63.27	
166.37/63.27	Used ordering: Polynomial interpretation [POLO]:
166.37/63.27	
166.37/63.27	   POL(0) = 0
166.37/63.27	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + 2*x_3 + 2*x_5
166.37/63.27	   POL(U1^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U2^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U3^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U4^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U5^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(U9^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_4 + 2*x_6
166.37/63.27	   POL(app(x_1, x_2)) = x_1 + x_2
166.37/63.27	   POL(cons(x_1, x_2)) = 2 + x_1 + x_2
166.37/63.27	   POL(empty(x_1)) = 0
166.37/63.27	   POL(f(x_1, x_2)) = 0
166.37/63.27	   POL(false) = 0
166.37/63.27	   POL(fstsplit(x_1, x_2)) = 2*x_2
166.37/63.27	   POL(head(x_1)) = 2*x_1
166.37/63.27	   POL(length(x_1)) = 0
166.37/63.27	   POL(leq(x_1, x_2)) = 0
166.37/63.27	   POL(map_f(x_1, x_2)) = 0
166.37/63.27	   POL(nil) = 0
166.37/63.27	   POL(s(x_1)) = 0
166.37/63.27	   POL(sndsplit(x_1, x_2)) = x_2
166.37/63.27	   POL(tail(x_1)) = 2*x_1
166.37/63.27	   POL(three) = 0
166.37/63.27	   POL(true) = 0
166.37/63.27	   POL(two) = 0
166.37/63.27	
166.37/63.27	
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(61)
166.37/63.27	Obligation:
166.37/63.27	Q DP problem:
166.37/63.27	The TRS P consists of the following rules:
166.37/63.27	
166.37/63.27	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.27	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.27	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.27	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.27	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.27	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.27	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.27	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.27	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.27	
166.37/63.27	The TRS R consists of the following rules:
166.37/63.27	
166.37/63.27	   sndsplit(0, x) -> x
166.37/63.27	   sndsplit(s(n), nil) -> nil
166.37/63.27	   fstsplit(0, x) -> nil
166.37/63.27	   fstsplit(s(n), nil) -> nil
166.37/63.27	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.27	   empty(nil) -> true
166.37/63.27	   empty(cons(h, t)) -> false
166.37/63.27	   tail(cons(h, t)) -> t
166.37/63.27	   head(cons(h, t)) -> h
166.37/63.27	   map_f(pid, nil) -> nil
166.37/63.27	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.27	   app(nil, x) -> x
166.37/63.27	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.27	   length(nil) -> 0
166.37/63.27	   length(cons(h, t)) -> s(length(t))
166.37/63.27	   leq(0, m) -> true
166.37/63.27	   leq(s(n), 0) -> false
166.37/63.27	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.27	
166.37/63.27	The set Q consists of the following terms:
166.37/63.27	
166.37/63.27	   fstsplit(0, x0)
166.37/63.27	   fstsplit(s(x0), nil)
166.37/63.27	   fstsplit(s(x0), cons(x1, x2))
166.37/63.27	   sndsplit(0, x0)
166.37/63.27	   sndsplit(s(x0), nil)
166.37/63.27	   sndsplit(s(x0), cons(x1, x2))
166.37/63.27	   empty(nil)
166.37/63.27	   empty(cons(x0, x1))
166.37/63.27	   leq(0, x0)
166.37/63.27	   leq(s(x0), 0)
166.37/63.27	   leq(s(x0), s(x1))
166.37/63.27	   length(nil)
166.37/63.27	   length(cons(x0, x1))
166.37/63.27	   app(nil, x0)
166.37/63.27	   app(cons(x0, x1), x2)
166.37/63.27	   map_f(x0, nil)
166.37/63.27	   map_f(x0, cons(x1, x2))
166.37/63.27	   head(cons(x0, x1))
166.37/63.27	   tail(cons(x0, x1))
166.37/63.27	
166.37/63.27	We have to consider all minimal (P,Q,R)-chains.
166.37/63.27	----------------------------------------
166.37/63.27	
166.37/63.27	(62) NonInfProof (EQUIVALENT)
166.37/63.27	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
166.37/63.27	
166.37/63.27	Note that final constraints are written in bold face.
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x6, x7, x8, x9, x10, x11) -> U1^1(empty(fstsplit(x11, x6)), x6, x7, x8, x9, x10, x11), U1^1(false, x12, x13, x14, x15, x16, x17) -> RING(sndsplit(x17, x12), cons(fstsplit(x17, x12), x13), x14, x15, x16, x17) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U1^1(empty(fstsplit(x11, x6)), x6, x7, x8, x9, x10, x11)=U1^1(false, x12, x13, x14, x15, x16, x17)  ==>  U1^1(false, x12, x13, x14, x15, x16, x17)_>=_RING(sndsplit(x17, x12), cons(fstsplit(x17, x12), x13), x14, x15, x16, x17))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (fstsplit(x11, x6)=x1858 & empty(x1858)=false  ==>  U1^1(false, x6, x7, x8, x9, x10, x11)_>=_RING(sndsplit(x11, x6), cons(fstsplit(x11, x6), x7), x8, x9, x10, x11))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1858)=false which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (false=false & fstsplit(x11, x6)=cons(x1860, x1859)  ==>  U1^1(false, x6, x7, x8, x9, x10, x11)_>=_RING(sndsplit(x11, x6), cons(fstsplit(x11, x6), x7), x8, x9, x10, x11))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (fstsplit(x11, x6)=cons(x1860, x1859)  ==>  U1^1(false, x6, x7, x8, x9, x10, x11)_>=_RING(sndsplit(x11, x6), cons(fstsplit(x11, x6), x7), x8, x9, x10, x11))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x11, x6)=cons(x1860, x1859) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (cons(x1864, fstsplit(x1865, x1863))=cons(x1860, x1859) & (\/x1866,x1867,x1868,x1869,x1870,x1871:fstsplit(x1865, x1863)=cons(x1866, x1867)  ==>  U1^1(false, x1863, x1868, x1869, x1870, x1871, x1865)_>=_RING(sndsplit(x1865, x1863), cons(fstsplit(x1865, x1863), x1868), x1869, x1870, x1871, x1865))  ==>  U1^1(false, cons(x1864, x1863), x7, x8, x9, x10, s(x1865))_>=_RING(sndsplit(s(x1865), cons(x1864, x1863)), cons(fstsplit(s(x1865), cons(x1864, x1863)), x7), x8, x9, x10, s(x1865)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (U1^1(false, cons(x1864, x1863), x7, x8, x9, x10, s(x1865))_>=_RING(sndsplit(s(x1865), cons(x1864, x1863)), cons(fstsplit(s(x1865), cons(x1864, x1863)), x7), x8, x9, x10, s(x1865)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain U1^1(false, x104, x105, x106, x107, x108, x109) -> RING(sndsplit(x109, x104), cons(fstsplit(x109, x104), x105), x106, x107, x108, x109), RING(x110, x111, x112, x113, x114, x115) -> U1^1(empty(fstsplit(x115, x110)), x110, x111, x112, x113, x114, x115) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(sndsplit(x109, x104), cons(fstsplit(x109, x104), x105), x106, x107, x108, x109)=RING(x110, x111, x112, x113, x114, x115)  ==>  RING(x110, x111, x112, x113, x114, x115)_>=_U1^1(empty(fstsplit(x115, x110)), x110, x111, x112, x113, x114, x115))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x110, x111, x106, x107, x108, x109)_>=_U1^1(empty(fstsplit(x109, x110)), x110, x111, x106, x107, x108, x109))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U4^1(false, x134, x135, x136, x137, x138, x139) -> RING(x134, tail(x135), sndsplit(x139, app(map_f(two, head(x135)), x136)), cons(fstsplit(x139, app(map_f(two, head(x135)), x136)), x137), x138, x139), RING(x140, x141, x142, x143, x144, x145) -> U1^1(empty(fstsplit(x145, x140)), x140, x141, x142, x143, x144, x145) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x134, tail(x135), sndsplit(x139, app(map_f(two, head(x135)), x136)), cons(fstsplit(x139, app(map_f(two, head(x135)), x136)), x137), x138, x139)=RING(x140, x141, x142, x143, x144, x145)  ==>  RING(x140, x141, x142, x143, x144, x145)_>=_U1^1(empty(fstsplit(x145, x140)), x140, x141, x142, x143, x144, x145))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x134, x141, x142, x143, x138, x139)_>=_U1^1(empty(fstsplit(x139, x134)), x134, x141, x142, x143, x138, x139))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U5^1(true, x146, x147, x148, x149, x150, x151) -> RING(x146, tail(x147), x148, x149, x150, x151), RING(x152, x153, x154, x155, x156, x157) -> U1^1(empty(fstsplit(x157, x152)), x152, x153, x154, x155, x156, x157) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x146, tail(x147), x148, x149, x150, x151)=RING(x152, x153, x154, x155, x156, x157)  ==>  RING(x152, x153, x154, x155, x156, x157)_>=_U1^1(empty(fstsplit(x157, x152)), x152, x153, x154, x155, x156, x157))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x146, x153, x148, x149, x150, x151)_>=_U1^1(empty(fstsplit(x151, x146)), x146, x153, x148, x149, x150, x151))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U7^1(false, x170, x171, x172, x173, x174, x175) -> RING(x170, x171, x172, x173, sndsplit(x175, x174), x175), RING(x176, x177, x178, x179, x180, x181) -> U1^1(empty(fstsplit(x181, x176)), x176, x177, x178, x179, x180, x181) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x170, x171, x172, x173, sndsplit(x175, x174), x175)=RING(x176, x177, x178, x179, x180, x181)  ==>  RING(x176, x177, x178, x179, x180, x181)_>=_U1^1(empty(fstsplit(x181, x176)), x176, x177, x178, x179, x180, x181))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x170, x171, x172, x173, x180, x175)_>=_U1^1(empty(fstsplit(x175, x170)), x170, x171, x172, x173, x180, x175))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U9^1(true, x182, x183, x184, x185, x186, x187) -> RING(x182, x183, x184, tail(x185), x186, x187), RING(x188, x189, x190, x191, x192, x193) -> U1^1(empty(fstsplit(x193, x188)), x188, x189, x190, x191, x192, x193) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x182, x183, x184, tail(x185), x186, x187)=RING(x188, x189, x190, x191, x192, x193)  ==>  RING(x188, x189, x190, x191, x192, x193)_>=_U1^1(empty(fstsplit(x193, x188)), x188, x189, x190, x191, x192, x193))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x182, x183, x184, x191, x186, x187)_>=_U1^1(empty(fstsplit(x187, x182)), x182, x183, x184, x191, x186, x187))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U8^1(false, x200, x201, x202, x203, x204, x205) -> RING(x200, x201, x202, tail(x203), sndsplit(x205, app(map_f(three, head(x203)), x204)), x205), RING(x206, x207, x208, x209, x210, x211) -> U1^1(empty(fstsplit(x211, x206)), x206, x207, x208, x209, x210, x211) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x200, x201, x202, tail(x203), sndsplit(x205, app(map_f(three, head(x203)), x204)), x205)=RING(x206, x207, x208, x209, x210, x211)  ==>  RING(x206, x207, x208, x209, x210, x211)_>=_U1^1(empty(fstsplit(x211, x206)), x206, x207, x208, x209, x210, x211))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x200, x201, x202, x209, x210, x205)_>=_U1^1(empty(fstsplit(x205, x200)), x200, x201, x202, x209, x210, x205))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U3^1(false, x218, x219, x220, x221, x222, x223) -> RING(x218, x219, sndsplit(x223, x220), cons(fstsplit(x223, x220), x221), x222, x223), RING(x224, x225, x226, x227, x228, x229) -> U1^1(empty(fstsplit(x229, x224)), x224, x225, x226, x227, x228, x229) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x218, x219, sndsplit(x223, x220), cons(fstsplit(x223, x220), x221), x222, x223)=RING(x224, x225, x226, x227, x228, x229)  ==>  RING(x224, x225, x226, x227, x228, x229)_>=_U1^1(empty(fstsplit(x229, x224)), x224, x225, x226, x227, x228, x229))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x218, x219, x226, x227, x222, x223)_>=_U1^1(empty(fstsplit(x223, x218)), x218, x219, x226, x227, x222, x223))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain U1^1(false, x244, x245, x246, x247, x248, x249) -> RING(sndsplit(x249, x244), cons(fstsplit(x249, x244), x245), x246, x247, x248, x249), RING(x250, x251, x252, x253, x254, x255) -> U2^1(leq(x255, length(x252)), x250, x251, x252, x253, x254, x255) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(sndsplit(x249, x244), cons(fstsplit(x249, x244), x245), x246, x247, x248, x249)=RING(x250, x251, x252, x253, x254, x255)  ==>  RING(x250, x251, x252, x253, x254, x255)_>=_U2^1(leq(x255, length(x252)), x250, x251, x252, x253, x254, x255))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x250, x251, x246, x247, x248, x249)_>=_U2^1(leq(x249, length(x246)), x250, x251, x246, x247, x248, x249))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U4^1(false, x274, x275, x276, x277, x278, x279) -> RING(x274, tail(x275), sndsplit(x279, app(map_f(two, head(x275)), x276)), cons(fstsplit(x279, app(map_f(two, head(x275)), x276)), x277), x278, x279), RING(x280, x281, x282, x283, x284, x285) -> U2^1(leq(x285, length(x282)), x280, x281, x282, x283, x284, x285) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x274, tail(x275), sndsplit(x279, app(map_f(two, head(x275)), x276)), cons(fstsplit(x279, app(map_f(two, head(x275)), x276)), x277), x278, x279)=RING(x280, x281, x282, x283, x284, x285)  ==>  RING(x280, x281, x282, x283, x284, x285)_>=_U2^1(leq(x285, length(x282)), x280, x281, x282, x283, x284, x285))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x274, x281, x282, x283, x278, x279)_>=_U2^1(leq(x279, length(x282)), x274, x281, x282, x283, x278, x279))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U5^1(true, x286, x287, x288, x289, x290, x291) -> RING(x286, tail(x287), x288, x289, x290, x291), RING(x292, x293, x294, x295, x296, x297) -> U2^1(leq(x297, length(x294)), x292, x293, x294, x295, x296, x297) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x286, tail(x287), x288, x289, x290, x291)=RING(x292, x293, x294, x295, x296, x297)  ==>  RING(x292, x293, x294, x295, x296, x297)_>=_U2^1(leq(x297, length(x294)), x292, x293, x294, x295, x296, x297))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x286, x293, x288, x289, x290, x291)_>=_U2^1(leq(x291, length(x288)), x286, x293, x288, x289, x290, x291))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U7^1(false, x310, x311, x312, x313, x314, x315) -> RING(x310, x311, x312, x313, sndsplit(x315, x314), x315), RING(x316, x317, x318, x319, x320, x321) -> U2^1(leq(x321, length(x318)), x316, x317, x318, x319, x320, x321) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x310, x311, x312, x313, sndsplit(x315, x314), x315)=RING(x316, x317, x318, x319, x320, x321)  ==>  RING(x316, x317, x318, x319, x320, x321)_>=_U2^1(leq(x321, length(x318)), x316, x317, x318, x319, x320, x321))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x310, x311, x312, x313, x320, x315)_>=_U2^1(leq(x315, length(x312)), x310, x311, x312, x313, x320, x315))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U9^1(true, x322, x323, x324, x325, x326, x327) -> RING(x322, x323, x324, tail(x325), x326, x327), RING(x328, x329, x330, x331, x332, x333) -> U2^1(leq(x333, length(x330)), x328, x329, x330, x331, x332, x333) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x322, x323, x324, tail(x325), x326, x327)=RING(x328, x329, x330, x331, x332, x333)  ==>  RING(x328, x329, x330, x331, x332, x333)_>=_U2^1(leq(x333, length(x330)), x328, x329, x330, x331, x332, x333))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x322, x323, x324, x331, x326, x327)_>=_U2^1(leq(x327, length(x324)), x322, x323, x324, x331, x326, x327))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U8^1(false, x340, x341, x342, x343, x344, x345) -> RING(x340, x341, x342, tail(x343), sndsplit(x345, app(map_f(three, head(x343)), x344)), x345), RING(x346, x347, x348, x349, x350, x351) -> U2^1(leq(x351, length(x348)), x346, x347, x348, x349, x350, x351) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x340, x341, x342, tail(x343), sndsplit(x345, app(map_f(three, head(x343)), x344)), x345)=RING(x346, x347, x348, x349, x350, x351)  ==>  RING(x346, x347, x348, x349, x350, x351)_>=_U2^1(leq(x351, length(x348)), x346, x347, x348, x349, x350, x351))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x340, x341, x342, x349, x350, x345)_>=_U2^1(leq(x345, length(x342)), x340, x341, x342, x349, x350, x345))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U3^1(false, x358, x359, x360, x361, x362, x363) -> RING(x358, x359, sndsplit(x363, x360), cons(fstsplit(x363, x360), x361), x362, x363), RING(x364, x365, x366, x367, x368, x369) -> U2^1(leq(x369, length(x366)), x364, x365, x366, x367, x368, x369) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x358, x359, sndsplit(x363, x360), cons(fstsplit(x363, x360), x361), x362, x363)=RING(x364, x365, x366, x367, x368, x369)  ==>  RING(x364, x365, x366, x367, x368, x369)_>=_U2^1(leq(x369, length(x366)), x364, x365, x366, x367, x368, x369))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x358, x359, x366, x367, x362, x363)_>=_U2^1(leq(x363, length(x366)), x358, x359, x366, x367, x362, x363))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x396, x397, x398, x399, x400, x401) -> U2^1(leq(x401, length(x398)), x396, x397, x398, x399, x400, x401), U2^1(false, x402, x403, x404, x405, x406, x407) -> U4^1(empty(fstsplit(x407, app(map_f(two, head(x403)), x404))), x402, x403, x404, x405, x406, x407) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U2^1(leq(x401, length(x398)), x396, x397, x398, x399, x400, x401)=U2^1(false, x402, x403, x404, x405, x406, x407)  ==>  U2^1(false, x402, x403, x404, x405, x406, x407)_>=_U4^1(empty(fstsplit(x407, app(map_f(two, head(x403)), x404))), x402, x403, x404, x405, x406, x407))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (length(x398)=x1888 & leq(x401, x1888)=false  ==>  U2^1(false, x396, x397, x398, x399, x400, x401)_>=_U4^1(empty(fstsplit(x401, app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, x401))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x401, x1888)=false which results in the following new constraints:
166.37/63.27	
166.37/63.27	(3)    (false=false & length(x398)=0  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1890)))
166.37/63.27	
166.37/63.27	(4)    (leq(x1892, x1891)=false & length(x398)=s(x1891) & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892))  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1892)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (length(x398)=0  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1890)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x398)=s(x1891) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (s(length(x1900))=s(x1891) & leq(x1892, x1891)=false & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on length(x398)=0 which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (0=0  ==>  U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (length(x1900)=x1891 & leq(x1892, x1891)=false & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (9) using rule (VI) where we applied the induction hypothesis (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) with sigma = [x1893 / x1900, x1894 / x396, x1895 / x397, x1896 / x399, x1897 / x400] which results in the following new constraint:
166.37/63.27	
166.37/63.27	(10)    (U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (10) using rule (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(11)    (U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892)  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) the following chains were created:
166.37/63.27	*We consider the chain U2^1(false, x506, x507, x508, x509, x510, x511) -> U4^1(empty(fstsplit(x511, app(map_f(two, head(x507)), x508))), x506, x507, x508, x509, x510, x511), U4^1(false, x512, x513, x514, x515, x516, x517) -> RING(x512, tail(x513), sndsplit(x517, app(map_f(two, head(x513)), x514)), cons(fstsplit(x517, app(map_f(two, head(x513)), x514)), x515), x516, x517) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U4^1(empty(fstsplit(x511, app(map_f(two, head(x507)), x508))), x506, x507, x508, x509, x510, x511)=U4^1(false, x512, x513, x514, x515, x516, x517)  ==>  U4^1(false, x512, x513, x514, x515, x516, x517)_>=_RING(x512, tail(x513), sndsplit(x517, app(map_f(two, head(x513)), x514)), cons(fstsplit(x517, app(map_f(two, head(x513)), x514)), x515), x516, x517))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (fstsplit(x511, x1914)=x1913 & empty(x1913)=false  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1913)=false which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (false=false & fstsplit(x511, x1914)=cons(x1919, x1918)  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (fstsplit(x511, x1914)=cons(x1919, x1918)  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x511, x1914)=cons(x1919, x1918) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (cons(x1923, fstsplit(x1924, x1922))=cons(x1919, x1918) & (\/x1925,x1926,x1927,x1928,x1929,x1930,x1931:fstsplit(x1924, x1922)=cons(x1925, x1926)  ==>  U4^1(false, x1927, x1928, x1929, x1930, x1931, x1924)_>=_RING(x1927, tail(x1928), sndsplit(x1924, app(map_f(two, head(x1928)), x1929)), cons(fstsplit(x1924, app(map_f(two, head(x1928)), x1929)), x1930), x1931, x1924))  ==>  U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x676, cons(x677, x678), x679, x680, x681, x682) -> U5^1(empty(map_f(two, x677)), x676, cons(x677, x678), x679, x680, x681, x682), U5^1(true, x683, x684, x685, x686, x687, x688) -> RING(x683, tail(x684), x685, x686, x687, x688) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U5^1(empty(map_f(two, x677)), x676, cons(x677, x678), x679, x680, x681, x682)=U5^1(true, x683, x684, x685, x686, x687, x688)  ==>  U5^1(true, x683, x684, x685, x686, x687, x688)_>=_RING(x683, tail(x684), x685, x686, x687, x688))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (two=x1933 & map_f(x1933, x677)=x1932 & empty(x1932)=true  ==>  U5^1(true, x676, cons(x677, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(x677, x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1932)=true which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (true=true & two=x1933 & map_f(x1933, x677)=nil  ==>  U5^1(true, x676, cons(x677, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(x677, x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (two=x1933 & map_f(x1933, x677)=nil  ==>  U5^1(true, x676, cons(x677, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(x677, x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on map_f(x1933, x677)=nil which results in the following new constraints:
166.37/63.27	
166.37/63.27	(5)    (nil=nil & two=x1936  ==>  U5^1(true, x676, cons(nil, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(nil, x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	(6)    (app(f(x1939, x1938), map_f(x1939, x1937))=nil & two=x1939 & (\/x1940,x1941,x1942,x1943,x1944,x1945:map_f(x1939, x1937)=nil & two=x1939  ==>  U5^1(true, x1940, cons(x1937, x1941), x1942, x1943, x1944, x1945)_>=_RING(x1940, tail(cons(x1937, x1941)), x1942, x1943, x1944, x1945))  ==>  U5^1(true, x676, cons(cons(x1938, x1937), x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(cons(x1938, x1937), x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (U5^1(true, x676, cons(nil, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(nil, x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (f(x1939, x1938)=x1946 & map_f(x1939, x1937)=x1947 & app(x1946, x1947)=nil & two=x1939  ==>  U5^1(true, x676, cons(cons(x1938, x1937), x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(cons(x1938, x1937), x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on app(x1946, x1947)=nil which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (x1948=nil & f(x1939, x1938)=nil & map_f(x1939, x1937)=x1948 & two=x1939  ==>  U5^1(true, x676, cons(cons(x1938, x1937), x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(cons(x1938, x1937), x678)), x679, x680, x681, x682))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We solved constraint (9) using rules (I), (II).
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain U1^1(false, x696, x697, x698, x699, x700, x701) -> RING(sndsplit(x701, x696), cons(fstsplit(x701, x696), x697), x698, x699, x700, x701), RING(x702, x703, x704, x705, x706, x707) -> U6^1(leq(x707, length(x706)), x702, x703, x704, x705, x706, x707) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(sndsplit(x701, x696), cons(fstsplit(x701, x696), x697), x698, x699, x700, x701)=RING(x702, x703, x704, x705, x706, x707)  ==>  RING(x702, x703, x704, x705, x706, x707)_>=_U6^1(leq(x707, length(x706)), x702, x703, x704, x705, x706, x707))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x702, x703, x698, x699, x700, x701)_>=_U6^1(leq(x701, length(x700)), x702, x703, x698, x699, x700, x701))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U4^1(false, x726, x727, x728, x729, x730, x731) -> RING(x726, tail(x727), sndsplit(x731, app(map_f(two, head(x727)), x728)), cons(fstsplit(x731, app(map_f(two, head(x727)), x728)), x729), x730, x731), RING(x732, x733, x734, x735, x736, x737) -> U6^1(leq(x737, length(x736)), x732, x733, x734, x735, x736, x737) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x726, tail(x727), sndsplit(x731, app(map_f(two, head(x727)), x728)), cons(fstsplit(x731, app(map_f(two, head(x727)), x728)), x729), x730, x731)=RING(x732, x733, x734, x735, x736, x737)  ==>  RING(x732, x733, x734, x735, x736, x737)_>=_U6^1(leq(x737, length(x736)), x732, x733, x734, x735, x736, x737))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x726, x733, x734, x735, x730, x731)_>=_U6^1(leq(x731, length(x730)), x726, x733, x734, x735, x730, x731))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U5^1(true, x738, x739, x740, x741, x742, x743) -> RING(x738, tail(x739), x740, x741, x742, x743), RING(x744, x745, x746, x747, x748, x749) -> U6^1(leq(x749, length(x748)), x744, x745, x746, x747, x748, x749) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x738, tail(x739), x740, x741, x742, x743)=RING(x744, x745, x746, x747, x748, x749)  ==>  RING(x744, x745, x746, x747, x748, x749)_>=_U6^1(leq(x749, length(x748)), x744, x745, x746, x747, x748, x749))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x738, x745, x740, x741, x742, x743)_>=_U6^1(leq(x743, length(x742)), x738, x745, x740, x741, x742, x743))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U7^1(false, x762, x763, x764, x765, x766, x767) -> RING(x762, x763, x764, x765, sndsplit(x767, x766), x767), RING(x768, x769, x770, x771, x772, x773) -> U6^1(leq(x773, length(x772)), x768, x769, x770, x771, x772, x773) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x762, x763, x764, x765, sndsplit(x767, x766), x767)=RING(x768, x769, x770, x771, x772, x773)  ==>  RING(x768, x769, x770, x771, x772, x773)_>=_U6^1(leq(x773, length(x772)), x768, x769, x770, x771, x772, x773))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x762, x763, x764, x765, x772, x767)_>=_U6^1(leq(x767, length(x772)), x762, x763, x764, x765, x772, x767))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U9^1(true, x774, x775, x776, x777, x778, x779) -> RING(x774, x775, x776, tail(x777), x778, x779), RING(x780, x781, x782, x783, x784, x785) -> U6^1(leq(x785, length(x784)), x780, x781, x782, x783, x784, x785) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x774, x775, x776, tail(x777), x778, x779)=RING(x780, x781, x782, x783, x784, x785)  ==>  RING(x780, x781, x782, x783, x784, x785)_>=_U6^1(leq(x785, length(x784)), x780, x781, x782, x783, x784, x785))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x774, x775, x776, x783, x778, x779)_>=_U6^1(leq(x779, length(x778)), x774, x775, x776, x783, x778, x779))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U8^1(false, x792, x793, x794, x795, x796, x797) -> RING(x792, x793, x794, tail(x795), sndsplit(x797, app(map_f(three, head(x795)), x796)), x797), RING(x798, x799, x800, x801, x802, x803) -> U6^1(leq(x803, length(x802)), x798, x799, x800, x801, x802, x803) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x792, x793, x794, tail(x795), sndsplit(x797, app(map_f(three, head(x795)), x796)), x797)=RING(x798, x799, x800, x801, x802, x803)  ==>  RING(x798, x799, x800, x801, x802, x803)_>=_U6^1(leq(x803, length(x802)), x798, x799, x800, x801, x802, x803))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x792, x793, x794, x801, x802, x797)_>=_U6^1(leq(x797, length(x802)), x792, x793, x794, x801, x802, x797))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U3^1(false, x810, x811, x812, x813, x814, x815) -> RING(x810, x811, sndsplit(x815, x812), cons(fstsplit(x815, x812), x813), x814, x815), RING(x816, x817, x818, x819, x820, x821) -> U6^1(leq(x821, length(x820)), x816, x817, x818, x819, x820, x821) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x810, x811, sndsplit(x815, x812), cons(fstsplit(x815, x812), x813), x814, x815)=RING(x816, x817, x818, x819, x820, x821)  ==>  RING(x816, x817, x818, x819, x820, x821)_>=_U6^1(leq(x821, length(x820)), x816, x817, x818, x819, x820, x821))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x810, x811, x818, x819, x814, x815)_>=_U6^1(leq(x815, length(x814)), x810, x811, x818, x819, x814, x815))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x872, x873, x874, x875, x876, x877) -> U6^1(leq(x877, length(x876)), x872, x873, x874, x875, x876, x877), U6^1(true, x878, x879, x880, x881, x882, x883) -> U7^1(empty(fstsplit(x883, x882)), x878, x879, x880, x881, x882, x883) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U6^1(leq(x877, length(x876)), x872, x873, x874, x875, x876, x877)=U6^1(true, x878, x879, x880, x881, x882, x883)  ==>  U6^1(true, x878, x879, x880, x881, x882, x883)_>=_U7^1(empty(fstsplit(x883, x882)), x878, x879, x880, x881, x882, x883))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (length(x876)=x1960 & leq(x877, x1960)=true  ==>  U6^1(true, x872, x873, x874, x875, x876, x877)_>=_U7^1(empty(fstsplit(x877, x876)), x872, x873, x874, x875, x876, x877))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x877, x1960)=true which results in the following new constraints:
166.37/63.27	
166.37/63.27	(3)    (true=true & length(x876)=x1961  ==>  U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.27	
166.37/63.27	(4)    (leq(x1964, x1963)=true & length(x876)=s(x1963) & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964))  ==>  U6^1(true, x872, x873, x874, x875, x876, s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), x876)), x872, x873, x874, x875, x876, s(x1964)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x876)=s(x1963) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (s(length(x1970))=s(x1963) & leq(x1964, x1963)=true & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (length(x1970)=x1963 & leq(x1964, x1963)=true & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (7) using rule (VI) where we applied the induction hypothesis (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) with sigma = [x1965 / x1970, x1966 / x872, x1967 / x873, x1968 / x874, x1969 / x875] which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (8) using rule (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964)  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) the following chains were created:
166.37/63.27	*We consider the chain U6^1(true, x982, x983, x984, x985, x986, x987) -> U7^1(empty(fstsplit(x987, x986)), x982, x983, x984, x985, x986, x987), U7^1(false, x988, x989, x990, x991, x992, x993) -> RING(x988, x989, x990, x991, sndsplit(x993, x992), x993) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U7^1(empty(fstsplit(x987, x986)), x982, x983, x984, x985, x986, x987)=U7^1(false, x988, x989, x990, x991, x992, x993)  ==>  U7^1(false, x988, x989, x990, x991, x992, x993)_>=_RING(x988, x989, x990, x991, sndsplit(x993, x992), x993))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (fstsplit(x987, x986)=x1983 & empty(x1983)=false  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1983)=false which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (false=false & fstsplit(x987, x986)=cons(x1985, x1984)  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (fstsplit(x987, x986)=cons(x1985, x1984)  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x987, x986)=cons(x1985, x1984) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (cons(x1989, fstsplit(x1990, x1988))=cons(x1985, x1984) & (\/x1991,x1992,x1993,x1994,x1995,x1996:fstsplit(x1990, x1988)=cons(x1991, x1992)  ==>  U7^1(false, x1993, x1994, x1995, x1996, x1988, x1990)_>=_RING(x1993, x1994, x1995, x1996, sndsplit(x1990, x1988), x1990))  ==>  U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141) -> U9^1(empty(map_f(three, x1138)), x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141), U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147) -> RING(x1142, x1143, x1144, tail(x1145), x1146, x1147) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U9^1(empty(map_f(three, x1138)), x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)=U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147)  ==>  U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147)_>=_RING(x1142, x1143, x1144, tail(x1145), x1146, x1147))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (three=x1998 & map_f(x1998, x1138)=x1997 & empty(x1997)=true  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1997)=true which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (true=true & three=x1998 & map_f(x1998, x1138)=nil  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (three=x1998 & map_f(x1998, x1138)=nil  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on map_f(x1998, x1138)=nil which results in the following new constraints:
166.37/63.27	
166.37/63.27	(5)    (nil=nil & three=x2001  ==>  U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	(6)    (app(f(x2004, x2003), map_f(x2004, x2002))=nil & three=x2004 & (\/x2005,x2006,x2007,x2008,x2009,x2010:map_f(x2004, x2002)=nil & three=x2004  ==>  U9^1(true, x2005, x2006, x2007, cons(x2002, x2008), x2009, x2010)_>=_RING(x2005, x2006, x2007, tail(cons(x2002, x2008)), x2009, x2010))  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (f(x2004, x2003)=x2011 & map_f(x2004, x2002)=x2012 & app(x2011, x2012)=nil & three=x2004  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on app(x2011, x2012)=nil which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (x2013=nil & f(x2004, x2003)=nil & map_f(x2004, x2002)=x2013 & three=x2004  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We solved constraint (9) using rules (I), (II).
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x1184, x1185, x1186, x1187, x1188, x1189) -> U6^1(leq(x1189, length(x1188)), x1184, x1185, x1186, x1187, x1188, x1189), U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195) -> U8^1(empty(fstsplit(x1195, app(map_f(three, head(x1193)), x1194))), x1190, x1191, x1192, x1193, x1194, x1195) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U6^1(leq(x1189, length(x1188)), x1184, x1185, x1186, x1187, x1188, x1189)=U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195)  ==>  U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195)_>=_U8^1(empty(fstsplit(x1195, app(map_f(three, head(x1193)), x1194))), x1190, x1191, x1192, x1193, x1194, x1195))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (length(x1188)=x2017 & leq(x1189, x2017)=false  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, x1189)_>=_U8^1(empty(fstsplit(x1189, app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, x1189))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x1189, x2017)=false which results in the following new constraints:
166.37/63.27	
166.37/63.27	(3)    (false=false & length(x1188)=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2019)))
166.37/63.27	
166.37/63.27	(4)    (leq(x2021, x2020)=false & length(x1188)=s(x2020) & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021))  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2021)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (length(x1188)=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2019)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x1188)=s(x2020) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (s(length(x2029))=s(x2020) & leq(x2021, x2020)=false & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on length(x1188)=0 which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (0=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (length(x2029)=x2020 & leq(x2021, x2020)=false & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (9) using rule (VI) where we applied the induction hypothesis (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) with sigma = [x2022 / x2029, x2023 / x1184, x2024 / x1185, x2025 / x1186, x2026 / x1187] which results in the following new constraint:
166.37/63.27	
166.37/63.27	(10)    (U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (10) using rule (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(11)    (U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021)  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) the following chains were created:
166.37/63.27	*We consider the chain U6^1(false, x1312, x1313, x1314, x1315, x1316, x1317) -> U8^1(empty(fstsplit(x1317, app(map_f(three, head(x1315)), x1316))), x1312, x1313, x1314, x1315, x1316, x1317), U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323) -> RING(x1318, x1319, x1320, tail(x1321), sndsplit(x1323, app(map_f(three, head(x1321)), x1322)), x1323) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U8^1(empty(fstsplit(x1317, app(map_f(three, head(x1315)), x1316))), x1312, x1313, x1314, x1315, x1316, x1317)=U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323)  ==>  U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323)_>=_RING(x1318, x1319, x1320, tail(x1321), sndsplit(x1323, app(map_f(three, head(x1321)), x1322)), x1323))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (fstsplit(x1317, x2043)=x2042 & empty(x2042)=false  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x2042)=false which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (false=false & fstsplit(x1317, x2043)=cons(x2048, x2047)  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (fstsplit(x1317, x2043)=cons(x2048, x2047)  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x1317, x2043)=cons(x2048, x2047) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (cons(x2052, fstsplit(x2053, x2051))=cons(x2048, x2047) & (\/x2054,x2055,x2056,x2057,x2058,x2059,x2060:fstsplit(x2053, x2051)=cons(x2054, x2055)  ==>  U8^1(false, x2056, x2057, x2058, x2059, x2060, x2053)_>=_RING(x2056, x2057, x2058, tail(x2059), sndsplit(x2053, app(map_f(three, head(x2059)), x2060)), x2053))  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.27	*We consider the chain RING(x1368, x1369, x1370, x1371, x1372, x1373) -> U2^1(leq(x1373, length(x1370)), x1368, x1369, x1370, x1371, x1372, x1373), U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379) -> U3^1(empty(fstsplit(x1379, x1376)), x1374, x1375, x1376, x1377, x1378, x1379) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U2^1(leq(x1373, length(x1370)), x1368, x1369, x1370, x1371, x1372, x1373)=U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379)  ==>  U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379)_>=_U3^1(empty(fstsplit(x1379, x1376)), x1374, x1375, x1376, x1377, x1378, x1379))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (length(x1370)=x2061 & leq(x1373, x2061)=true  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, x1373)_>=_U3^1(empty(fstsplit(x1373, x1370)), x1368, x1369, x1370, x1371, x1372, x1373))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x1373, x2061)=true which results in the following new constraints:
166.37/63.27	
166.37/63.27	(3)    (true=true & length(x1370)=x2062  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.27	
166.37/63.27	(4)    (leq(x2065, x2064)=true & length(x1370)=s(x2064) & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065))  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), x1370)), x1368, x1369, x1370, x1371, x1372, s(x2065)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x1370)=s(x2064) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (s(length(x2071))=s(x2064) & leq(x2065, x2064)=true & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(7)    (length(x2071)=x2064 & leq(x2065, x2064)=true & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (7) using rule (VI) where we applied the induction hypothesis (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) with sigma = [x2066 / x2071, x2067 / x1368, x2068 / x1369, x2069 / x1371, x2070 / x1372] which results in the following new constraint:
166.37/63.27	
166.37/63.27	(8)    (U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (8) using rule (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(9)    (U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065)  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) the following chains were created:
166.37/63.27	*We consider the chain U2^1(true, x1532, x1533, x1534, x1535, x1536, x1537) -> U3^1(empty(fstsplit(x1537, x1534)), x1532, x1533, x1534, x1535, x1536, x1537), U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543) -> RING(x1538, x1539, sndsplit(x1543, x1540), cons(fstsplit(x1543, x1540), x1541), x1542, x1543) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (U3^1(empty(fstsplit(x1537, x1534)), x1532, x1533, x1534, x1535, x1536, x1537)=U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543)  ==>  U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543)_>=_RING(x1538, x1539, sndsplit(x1543, x1540), cons(fstsplit(x1543, x1540), x1541), x1542, x1543))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (fstsplit(x1537, x1534)=x2084 & empty(x2084)=false  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x2084)=false which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (false=false & fstsplit(x1537, x1534)=cons(x2086, x2085)  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (fstsplit(x1537, x1534)=cons(x2086, x2085)  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x1537, x1534)=cons(x2086, x2085) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(5)    (cons(x2090, fstsplit(x2091, x2089))=cons(x2086, x2085) & (\/x2092,x2093,x2094,x2095,x2096,x2097:fstsplit(x2091, x2089)=cons(x2092, x2093)  ==>  U3^1(false, x2094, x2095, x2089, x2096, x2097, x2091)_>=_RING(x2094, x2095, sndsplit(x2091, x2089), cons(fstsplit(x2091, x2089), x2096), x2097, x2091))  ==>  U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(6)    (U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5) the following chains were created:
166.37/63.27	*We consider the chain U1^1(false, x1564, x1565, x1566, x1567, x1568, x1569) -> RING(sndsplit(x1569, x1564), cons(fstsplit(x1569, x1564), x1565), x1566, x1567, x1568, x1569), RING(x1570, cons(x1571, x1572), x1573, x1574, x1575, x1576) -> U5^1(empty(map_f(two, x1571)), x1570, cons(x1571, x1572), x1573, x1574, x1575, x1576) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(sndsplit(x1569, x1564), cons(fstsplit(x1569, x1564), x1565), x1566, x1567, x1568, x1569)=RING(x1570, cons(x1571, x1572), x1573, x1574, x1575, x1576)  ==>  RING(x1570, cons(x1571, x1572), x1573, x1574, x1575, x1576)_>=_U5^1(empty(map_f(two, x1571)), x1570, cons(x1571, x1572), x1573, x1574, x1575, x1576))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1570, cons(x1571, x1565), x1566, x1567, x1568, x1569)_>=_U5^1(empty(map_f(two, x1571)), x1570, cons(x1571, x1565), x1566, x1567, x1568, x1569))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U4^1(false, x1595, x1596, x1597, x1598, x1599, x1600) -> RING(x1595, tail(x1596), sndsplit(x1600, app(map_f(two, head(x1596)), x1597)), cons(fstsplit(x1600, app(map_f(two, head(x1596)), x1597)), x1598), x1599, x1600), RING(x1601, cons(x1602, x1603), x1604, x1605, x1606, x1607) -> U5^1(empty(map_f(two, x1602)), x1601, cons(x1602, x1603), x1604, x1605, x1606, x1607) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1595, tail(x1596), sndsplit(x1600, app(map_f(two, head(x1596)), x1597)), cons(fstsplit(x1600, app(map_f(two, head(x1596)), x1597)), x1598), x1599, x1600)=RING(x1601, cons(x1602, x1603), x1604, x1605, x1606, x1607)  ==>  RING(x1601, cons(x1602, x1603), x1604, x1605, x1606, x1607)_>=_U5^1(empty(map_f(two, x1602)), x1601, cons(x1602, x1603), x1604, x1605, x1606, x1607))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (tail(x1596)=cons(x1602, x1603) & app(x2099, x1597)=x2098 & sndsplit(x1600, x2098)=x1604 & cons(fstsplit(x1600, app(map_f(two, head(x1596)), x1597)), x1598)=x1605  ==>  RING(x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600)_>=_U5^1(empty(map_f(two, x1602)), x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1596)=cons(x1602, x1603) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (x2102=cons(x1602, x1603) & app(x2099, x1597)=x2098 & sndsplit(x1600, x2098)=x1604 & cons(fstsplit(x1600, app(map_f(two, head(cons(x2103, x2102))), x1597)), x1598)=x1605  ==>  RING(x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600)_>=_U5^1(empty(map_f(two, x1602)), x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rules (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (RING(x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600)_>=_U5^1(empty(map_f(two, x1602)), x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U5^1(true, x1608, x1609, x1610, x1611, x1612, x1613) -> RING(x1608, tail(x1609), x1610, x1611, x1612, x1613), RING(x1614, cons(x1615, x1616), x1617, x1618, x1619, x1620) -> U5^1(empty(map_f(two, x1615)), x1614, cons(x1615, x1616), x1617, x1618, x1619, x1620) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1608, tail(x1609), x1610, x1611, x1612, x1613)=RING(x1614, cons(x1615, x1616), x1617, x1618, x1619, x1620)  ==>  RING(x1614, cons(x1615, x1616), x1617, x1618, x1619, x1620)_>=_U5^1(empty(map_f(two, x1615)), x1614, cons(x1615, x1616), x1617, x1618, x1619, x1620))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (tail(x1609)=cons(x1615, x1616)  ==>  RING(x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613)_>=_U5^1(empty(map_f(two, x1615)), x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1609)=cons(x1615, x1616) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(3)    (x2104=cons(x1615, x1616)  ==>  RING(x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613)_>=_U5^1(empty(map_f(two, x1615)), x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (3) using rule (III) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(4)    (RING(x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613)_>=_U5^1(empty(map_f(two, x1615)), x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U7^1(false, x1633, x1634, x1635, x1636, x1637, x1638) -> RING(x1633, x1634, x1635, x1636, sndsplit(x1638, x1637), x1638), RING(x1639, cons(x1640, x1641), x1642, x1643, x1644, x1645) -> U5^1(empty(map_f(two, x1640)), x1639, cons(x1640, x1641), x1642, x1643, x1644, x1645) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1633, x1634, x1635, x1636, sndsplit(x1638, x1637), x1638)=RING(x1639, cons(x1640, x1641), x1642, x1643, x1644, x1645)  ==>  RING(x1639, cons(x1640, x1641), x1642, x1643, x1644, x1645)_>=_U5^1(empty(map_f(two, x1640)), x1639, cons(x1640, x1641), x1642, x1643, x1644, x1645))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1633, cons(x1640, x1641), x1635, x1636, x1644, x1638)_>=_U5^1(empty(map_f(two, x1640)), x1633, cons(x1640, x1641), x1635, x1636, x1644, x1638))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U9^1(true, x1646, x1647, x1648, x1649, x1650, x1651) -> RING(x1646, x1647, x1648, tail(x1649), x1650, x1651), RING(x1652, cons(x1653, x1654), x1655, x1656, x1657, x1658) -> U5^1(empty(map_f(two, x1653)), x1652, cons(x1653, x1654), x1655, x1656, x1657, x1658) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1646, x1647, x1648, tail(x1649), x1650, x1651)=RING(x1652, cons(x1653, x1654), x1655, x1656, x1657, x1658)  ==>  RING(x1652, cons(x1653, x1654), x1655, x1656, x1657, x1658)_>=_U5^1(empty(map_f(two, x1653)), x1652, cons(x1653, x1654), x1655, x1656, x1657, x1658))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1646, cons(x1653, x1654), x1648, x1656, x1650, x1651)_>=_U5^1(empty(map_f(two, x1653)), x1646, cons(x1653, x1654), x1648, x1656, x1650, x1651))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U8^1(false, x1665, x1666, x1667, x1668, x1669, x1670) -> RING(x1665, x1666, x1667, tail(x1668), sndsplit(x1670, app(map_f(three, head(x1668)), x1669)), x1670), RING(x1671, cons(x1672, x1673), x1674, x1675, x1676, x1677) -> U5^1(empty(map_f(two, x1672)), x1671, cons(x1672, x1673), x1674, x1675, x1676, x1677) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1665, x1666, x1667, tail(x1668), sndsplit(x1670, app(map_f(three, head(x1668)), x1669)), x1670)=RING(x1671, cons(x1672, x1673), x1674, x1675, x1676, x1677)  ==>  RING(x1671, cons(x1672, x1673), x1674, x1675, x1676, x1677)_>=_U5^1(empty(map_f(two, x1672)), x1671, cons(x1672, x1673), x1674, x1675, x1676, x1677))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1665, cons(x1672, x1673), x1667, x1675, x1676, x1670)_>=_U5^1(empty(map_f(two, x1672)), x1665, cons(x1672, x1673), x1667, x1675, x1676, x1670))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U3^1(false, x1684, x1685, x1686, x1687, x1688, x1689) -> RING(x1684, x1685, sndsplit(x1689, x1686), cons(fstsplit(x1689, x1686), x1687), x1688, x1689), RING(x1690, cons(x1691, x1692), x1693, x1694, x1695, x1696) -> U5^1(empty(map_f(two, x1691)), x1690, cons(x1691, x1692), x1693, x1694, x1695, x1696) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1684, x1685, sndsplit(x1689, x1686), cons(fstsplit(x1689, x1686), x1687), x1688, x1689)=RING(x1690, cons(x1691, x1692), x1693, x1694, x1695, x1696)  ==>  RING(x1690, cons(x1691, x1692), x1693, x1694, x1695, x1696)_>=_U5^1(empty(map_f(two, x1691)), x1690, cons(x1691, x1692), x1693, x1694, x1695, x1696))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1684, cons(x1691, x1692), x1693, x1694, x1688, x1689)_>=_U5^1(empty(map_f(two, x1691)), x1684, cons(x1691, x1692), x1693, x1694, x1688, x1689))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	For Pair RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5) the following chains were created:
166.37/63.27	*We consider the chain U1^1(false, x1711, x1712, x1713, x1714, x1715, x1716) -> RING(sndsplit(x1716, x1711), cons(fstsplit(x1716, x1711), x1712), x1713, x1714, x1715, x1716), RING(x1717, x1718, x1719, cons(x1720, x1721), x1722, x1723) -> U9^1(empty(map_f(three, x1720)), x1717, x1718, x1719, cons(x1720, x1721), x1722, x1723) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(sndsplit(x1716, x1711), cons(fstsplit(x1716, x1711), x1712), x1713, x1714, x1715, x1716)=RING(x1717, x1718, x1719, cons(x1720, x1721), x1722, x1723)  ==>  RING(x1717, x1718, x1719, cons(x1720, x1721), x1722, x1723)_>=_U9^1(empty(map_f(three, x1720)), x1717, x1718, x1719, cons(x1720, x1721), x1722, x1723))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.27	
166.37/63.27	(2)    (RING(x1717, x1718, x1713, cons(x1720, x1721), x1715, x1716)_>=_U9^1(empty(map_f(three, x1720)), x1717, x1718, x1713, cons(x1720, x1721), x1715, x1716))
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	
166.37/63.27	*We consider the chain U4^1(false, x1742, x1743, x1744, x1745, x1746, x1747) -> RING(x1742, tail(x1743), sndsplit(x1747, app(map_f(two, head(x1743)), x1744)), cons(fstsplit(x1747, app(map_f(two, head(x1743)), x1744)), x1745), x1746, x1747), RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754) -> U9^1(empty(map_f(three, x1751)), x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754) which results in the following constraint:
166.37/63.27	
166.37/63.27	(1)    (RING(x1742, tail(x1743), sndsplit(x1747, app(map_f(two, head(x1743)), x1744)), cons(fstsplit(x1747, app(map_f(two, head(x1743)), x1744)), x1745), x1746, x1747)=RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754)  ==>  RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754)_>=_U9^1(empty(map_f(three, x1751)), x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747)_>=_U9^1(empty(map_f(three, x1751)), x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U5^1(true, x1755, x1756, x1757, x1758, x1759, x1760) -> RING(x1755, tail(x1756), x1757, x1758, x1759, x1760), RING(x1761, x1762, x1763, cons(x1764, x1765), x1766, x1767) -> U9^1(empty(map_f(three, x1764)), x1761, x1762, x1763, cons(x1764, x1765), x1766, x1767) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1755, tail(x1756), x1757, x1758, x1759, x1760)=RING(x1761, x1762, x1763, cons(x1764, x1765), x1766, x1767)  ==>  RING(x1761, x1762, x1763, cons(x1764, x1765), x1766, x1767)_>=_U9^1(empty(map_f(three, x1764)), x1761, x1762, x1763, cons(x1764, x1765), x1766, x1767))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1755, x1762, x1757, cons(x1764, x1765), x1759, x1760)_>=_U9^1(empty(map_f(three, x1764)), x1755, x1762, x1757, cons(x1764, x1765), x1759, x1760))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U7^1(false, x1780, x1781, x1782, x1783, x1784, x1785) -> RING(x1780, x1781, x1782, x1783, sndsplit(x1785, x1784), x1785), RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792) -> U9^1(empty(map_f(three, x1789)), x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1780, x1781, x1782, x1783, sndsplit(x1785, x1784), x1785)=RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792)  ==>  RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792)_>=_U9^1(empty(map_f(three, x1789)), x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785)_>=_U9^1(empty(map_f(three, x1789)), x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U9^1(true, x1793, x1794, x1795, x1796, x1797, x1798) -> RING(x1793, x1794, x1795, tail(x1796), x1797, x1798), RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805) -> U9^1(empty(map_f(three, x1802)), x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1793, x1794, x1795, tail(x1796), x1797, x1798)=RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805)  ==>  RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805)_>=_U9^1(empty(map_f(three, x1802)), x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (tail(x1796)=cons(x1802, x1803)  ==>  RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1796)=cons(x1802, x1803) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (x2118=cons(x1802, x1803)  ==>  RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rule (III) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U8^1(false, x1812, x1813, x1814, x1815, x1816, x1817) -> RING(x1812, x1813, x1814, tail(x1815), sndsplit(x1817, app(map_f(three, head(x1815)), x1816)), x1817), RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824) -> U9^1(empty(map_f(three, x1821)), x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1812, x1813, x1814, tail(x1815), sndsplit(x1817, app(map_f(three, head(x1815)), x1816)), x1817)=RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824)  ==>  RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824)_>=_U9^1(empty(map_f(three, x1821)), x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (tail(x1815)=cons(x1821, x1822) & app(x2121, x1816)=x2120 & sndsplit(x1817, x2120)=x1823  ==>  RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1815)=cons(x1821, x1822) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (x2124=cons(x1821, x1822) & app(x2121, x1816)=x2120 & sndsplit(x1817, x2120)=x1823  ==>  RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U3^1(false, x1831, x1832, x1833, x1834, x1835, x1836) -> RING(x1831, x1832, sndsplit(x1836, x1833), cons(fstsplit(x1836, x1833), x1834), x1835, x1836), RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843) -> U9^1(empty(map_f(three, x1840)), x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1831, x1832, sndsplit(x1836, x1833), cons(fstsplit(x1836, x1833), x1834), x1835, x1836)=RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843)  ==>  RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843)_>=_U9^1(empty(map_f(three, x1840)), x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836)_>=_U9^1(empty(map_f(three, x1840)), x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	To summarize, we get the following constraints P__>=_ for the following pairs.
166.37/63.28	
166.37/63.28	*U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U1^1(false, cons(x1864, x1863), x7, x8, x9, x10, s(x1865))_>=_RING(sndsplit(s(x1865), cons(x1864, x1863)), cons(fstsplit(s(x1865), cons(x1864, x1863)), x7), x8, x9, x10, s(x1865)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(RING(x110, x111, x106, x107, x108, x109)_>=_U1^1(empty(fstsplit(x109, x110)), x110, x111, x106, x107, x108, x109))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x134, x141, x142, x143, x138, x139)_>=_U1^1(empty(fstsplit(x139, x134)), x134, x141, x142, x143, x138, x139))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x146, x153, x148, x149, x150, x151)_>=_U1^1(empty(fstsplit(x151, x146)), x146, x153, x148, x149, x150, x151))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x170, x171, x172, x173, x180, x175)_>=_U1^1(empty(fstsplit(x175, x170)), x170, x171, x172, x173, x180, x175))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x182, x183, x184, x191, x186, x187)_>=_U1^1(empty(fstsplit(x187, x182)), x182, x183, x184, x191, x186, x187))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x200, x201, x202, x209, x210, x205)_>=_U1^1(empty(fstsplit(x205, x200)), x200, x201, x202, x209, x210, x205))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x218, x219, x226, x227, x222, x223)_>=_U1^1(empty(fstsplit(x223, x218)), x218, x219, x226, x227, x222, x223))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(RING(x250, x251, x246, x247, x248, x249)_>=_U2^1(leq(x249, length(x246)), x250, x251, x246, x247, x248, x249))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x274, x281, x282, x283, x278, x279)_>=_U2^1(leq(x279, length(x282)), x274, x281, x282, x283, x278, x279))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x286, x293, x288, x289, x290, x291)_>=_U2^1(leq(x291, length(x288)), x286, x293, x288, x289, x290, x291))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x310, x311, x312, x313, x320, x315)_>=_U2^1(leq(x315, length(x312)), x310, x311, x312, x313, x320, x315))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x322, x323, x324, x331, x326, x327)_>=_U2^1(leq(x327, length(x324)), x322, x323, x324, x331, x326, x327))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x340, x341, x342, x349, x350, x345)_>=_U2^1(leq(x345, length(x342)), x340, x341, x342, x349, x350, x345))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x358, x359, x366, x367, x362, x363)_>=_U2^1(leq(x363, length(x366)), x358, x359, x366, x367, x362, x363))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892)  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U5^1(true, x676, cons(nil, x678), x679, x680, x681, x682)_>=_RING(x676, tail(cons(nil, x678)), x679, x680, x681, x682))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(RING(x702, x703, x698, x699, x700, x701)_>=_U6^1(leq(x701, length(x700)), x702, x703, x698, x699, x700, x701))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x726, x733, x734, x735, x730, x731)_>=_U6^1(leq(x731, length(x730)), x726, x733, x734, x735, x730, x731))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x738, x745, x740, x741, x742, x743)_>=_U6^1(leq(x743, length(x742)), x738, x745, x740, x741, x742, x743))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x762, x763, x764, x765, x772, x767)_>=_U6^1(leq(x767, length(x772)), x762, x763, x764, x765, x772, x767))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x774, x775, x776, x783, x778, x779)_>=_U6^1(leq(x779, length(x778)), x774, x775, x776, x783, x778, x779))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x792, x793, x794, x801, x802, x797)_>=_U6^1(leq(x797, length(x802)), x792, x793, x794, x801, x802, x797))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x810, x811, x818, x819, x814, x815)_>=_U6^1(leq(x815, length(x814)), x810, x811, x818, x819, x814, x815))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964)  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	
166.37/63.28	*(U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021)  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	*(U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065)  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.28	
166.37/63.28	*(RING(x1570, cons(x1571, x1565), x1566, x1567, x1568, x1569)_>=_U5^1(empty(map_f(two, x1571)), x1570, cons(x1571, x1565), x1566, x1567, x1568, x1569))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600)_>=_U5^1(empty(map_f(two, x1602)), x1595, cons(x1602, x1603), x1604, x1605, x1599, x1600))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613)_>=_U5^1(empty(map_f(two, x1615)), x1608, cons(x1615, x1616), x1610, x1611, x1612, x1613))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1633, cons(x1640, x1641), x1635, x1636, x1644, x1638)_>=_U5^1(empty(map_f(two, x1640)), x1633, cons(x1640, x1641), x1635, x1636, x1644, x1638))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1646, cons(x1653, x1654), x1648, x1656, x1650, x1651)_>=_U5^1(empty(map_f(two, x1653)), x1646, cons(x1653, x1654), x1648, x1656, x1650, x1651))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1665, cons(x1672, x1673), x1667, x1675, x1676, x1670)_>=_U5^1(empty(map_f(two, x1672)), x1665, cons(x1672, x1673), x1667, x1675, x1676, x1670))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1684, cons(x1691, x1692), x1693, x1694, x1688, x1689)_>=_U5^1(empty(map_f(two, x1691)), x1684, cons(x1691, x1692), x1693, x1694, x1688, x1689))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	
166.37/63.28	*(RING(x1717, x1718, x1713, cons(x1720, x1721), x1715, x1716)_>=_U9^1(empty(map_f(three, x1720)), x1717, x1718, x1713, cons(x1720, x1721), x1715, x1716))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747)_>=_U9^1(empty(map_f(three, x1751)), x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1755, x1762, x1757, cons(x1764, x1765), x1759, x1760)_>=_U9^1(empty(map_f(three, x1764)), x1755, x1762, x1757, cons(x1764, x1765), x1759, x1760))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785)_>=_U9^1(empty(map_f(three, x1789)), x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836)_>=_U9^1(empty(map_f(three, x1840)), x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
166.37/63.28	
166.37/63.28	Using the following integer polynomial ordering the  resulting constraints can be solved 
166.37/63.28	
166.37/63.28	Polynomial interpretation [NONINF]:
166.37/63.28	
166.37/63.28	   POL(0) = 0
166.37/63.28	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 - x_1 - x_6
166.37/63.28	   POL(U1^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_2 - x_7
166.37/63.28	   POL(U2^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_1 - x_2 - x_7
166.37/63.28	   POL(U3^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_7
166.37/63.28	   POL(U4^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_7
166.37/63.28	   POL(U5^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_2 - x_3 - x_7
166.37/63.28	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_1 - x_2 - x_7
166.37/63.28	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_7
166.37/63.28	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_7
166.37/63.28	   POL(U9^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_2 - x_5 - x_7
166.37/63.28	   POL(app(x_1, x_2)) = 0
166.37/63.28	   POL(c) = -2
166.37/63.28	   POL(cons(x_1, x_2)) = 0
166.37/63.28	   POL(empty(x_1)) = 0
166.37/63.28	   POL(f(x_1, x_2)) = x_1 + x_2
166.37/63.28	   POL(false) = 0
166.37/63.28	   POL(fstsplit(x_1, x_2)) = 0
166.37/63.28	   POL(head(x_1)) = 0
166.37/63.28	   POL(length(x_1)) = 0
166.37/63.28	   POL(leq(x_1, x_2)) = 0
166.37/63.28	   POL(map_f(x_1, x_2)) = 0
166.37/63.28	   POL(nil) = 1
166.37/63.28	   POL(s(x_1)) = 1
166.37/63.28	   POL(sndsplit(x_1, x_2)) = x_1
166.37/63.28	   POL(tail(x_1)) = 0
166.37/63.28	   POL(three) = 0
166.37/63.28	   POL(true) = 0
166.37/63.28	   POL(two) = 0
166.37/63.28	
166.37/63.28	
166.37/63.28	The following pairs  are in P_>:
166.37/63.28	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.28	The following pairs are in P_bound:
166.37/63.28	   U1^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
166.37/63.28	The following rules are usable:
166.37/63.28	   x -> sndsplit(0, x)
166.37/63.28	   nil -> sndsplit(s(n), nil)
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   true -> leq(0, m)
166.37/63.28	   false -> leq(s(n), 0)
166.37/63.28	   leq(n, m) -> leq(s(n), s(m))
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(63)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U1^1(empty(fstsplit(m, st_1)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(64) DependencyGraphProof (EQUIVALENT)
166.37/63.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(65)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.28	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(66) QDPOrderProof (EQUIVALENT)
166.37/63.28	We use the reduction pair processor [LPAR04,JAR06].
166.37/63.28	
166.37/63.28	
166.37/63.28	The following pairs can be oriented strictly and are deleted.
166.37/63.28	
166.37/63.28	   RING(y0, cons(x0, x1), y2, y3, y4, y5) -> U5^1(empty(map_f(two, x0)), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.28	The remaining pairs can at least be oriented weakly.
166.37/63.28	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
166.37/63.28	
166.37/63.28	   POL(0) = 0
166.37/63.28	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = [1/2]x_2
166.37/63.28	   POL(U2^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(U3^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(U4^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/4]x_3
166.37/63.28	   POL(U5^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/4]x_3
166.37/63.28	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(U9^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = [1/2]x_3
166.37/63.28	   POL(app(x_1, x_2)) = 0
166.37/63.28	   POL(cons(x_1, x_2)) = [1/4] + [4]x_2
166.37/63.28	   POL(empty(x_1)) = 0
166.37/63.28	   POL(f(x_1, x_2)) = 0
166.37/63.28	   POL(false) = 0
166.37/63.28	   POL(fstsplit(x_1, x_2)) = 0
166.37/63.28	   POL(head(x_1)) = 0
166.37/63.28	   POL(length(x_1)) = 0
166.37/63.28	   POL(leq(x_1, x_2)) = 0
166.37/63.28	   POL(map_f(x_1, x_2)) = 0
166.37/63.28	   POL(nil) = 0
166.37/63.28	   POL(s(x_1)) = 0
166.37/63.28	   POL(sndsplit(x_1, x_2)) = 0
166.37/63.28	   POL(tail(x_1)) = [1/2]x_1
166.37/63.28	   POL(three) = 0
166.37/63.28	   POL(true) = 0
166.37/63.28	   POL(two) = 0
166.37/63.28	The value of delta used in the strict ordering is 1/16.
166.37/63.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
166.37/63.28	
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(67)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   U5^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), st_2, in_3, st_3, m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(68) DependencyGraphProof (EQUIVALENT)
166.37/63.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(69)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(70) NonInfProof (EQUIVALENT)
166.37/63.28	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
166.37/63.28	
166.37/63.28	Note that final constraints are written in bold face.
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain U4^1(false, x274, x275, x276, x277, x278, x279) -> RING(x274, tail(x275), sndsplit(x279, app(map_f(two, head(x275)), x276)), cons(fstsplit(x279, app(map_f(two, head(x275)), x276)), x277), x278, x279), RING(x280, x281, x282, x283, x284, x285) -> U2^1(leq(x285, length(x282)), x280, x281, x282, x283, x284, x285) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x274, tail(x275), sndsplit(x279, app(map_f(two, head(x275)), x276)), cons(fstsplit(x279, app(map_f(two, head(x275)), x276)), x277), x278, x279)=RING(x280, x281, x282, x283, x284, x285)  ==>  RING(x280, x281, x282, x283, x284, x285)_>=_U2^1(leq(x285, length(x282)), x280, x281, x282, x283, x284, x285))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x274, x281, x282, x283, x278, x279)_>=_U2^1(leq(x279, length(x282)), x274, x281, x282, x283, x278, x279))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U7^1(false, x310, x311, x312, x313, x314, x315) -> RING(x310, x311, x312, x313, sndsplit(x315, x314), x315), RING(x316, x317, x318, x319, x320, x321) -> U2^1(leq(x321, length(x318)), x316, x317, x318, x319, x320, x321) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x310, x311, x312, x313, sndsplit(x315, x314), x315)=RING(x316, x317, x318, x319, x320, x321)  ==>  RING(x316, x317, x318, x319, x320, x321)_>=_U2^1(leq(x321, length(x318)), x316, x317, x318, x319, x320, x321))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x310, x311, x312, x313, x320, x315)_>=_U2^1(leq(x315, length(x312)), x310, x311, x312, x313, x320, x315))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U9^1(true, x322, x323, x324, x325, x326, x327) -> RING(x322, x323, x324, tail(x325), x326, x327), RING(x328, x329, x330, x331, x332, x333) -> U2^1(leq(x333, length(x330)), x328, x329, x330, x331, x332, x333) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x322, x323, x324, tail(x325), x326, x327)=RING(x328, x329, x330, x331, x332, x333)  ==>  RING(x328, x329, x330, x331, x332, x333)_>=_U2^1(leq(x333, length(x330)), x328, x329, x330, x331, x332, x333))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x322, x323, x324, x331, x326, x327)_>=_U2^1(leq(x327, length(x324)), x322, x323, x324, x331, x326, x327))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U8^1(false, x340, x341, x342, x343, x344, x345) -> RING(x340, x341, x342, tail(x343), sndsplit(x345, app(map_f(three, head(x343)), x344)), x345), RING(x346, x347, x348, x349, x350, x351) -> U2^1(leq(x351, length(x348)), x346, x347, x348, x349, x350, x351) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x340, x341, x342, tail(x343), sndsplit(x345, app(map_f(three, head(x343)), x344)), x345)=RING(x346, x347, x348, x349, x350, x351)  ==>  RING(x346, x347, x348, x349, x350, x351)_>=_U2^1(leq(x351, length(x348)), x346, x347, x348, x349, x350, x351))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x340, x341, x342, x349, x350, x345)_>=_U2^1(leq(x345, length(x342)), x340, x341, x342, x349, x350, x345))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U3^1(false, x358, x359, x360, x361, x362, x363) -> RING(x358, x359, sndsplit(x363, x360), cons(fstsplit(x363, x360), x361), x362, x363), RING(x364, x365, x366, x367, x368, x369) -> U2^1(leq(x369, length(x366)), x364, x365, x366, x367, x368, x369) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x358, x359, sndsplit(x363, x360), cons(fstsplit(x363, x360), x361), x362, x363)=RING(x364, x365, x366, x367, x368, x369)  ==>  RING(x364, x365, x366, x367, x368, x369)_>=_U2^1(leq(x369, length(x366)), x364, x365, x366, x367, x368, x369))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x358, x359, x366, x367, x362, x363)_>=_U2^1(leq(x363, length(x366)), x358, x359, x366, x367, x362, x363))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain RING(x396, x397, x398, x399, x400, x401) -> U2^1(leq(x401, length(x398)), x396, x397, x398, x399, x400, x401), U2^1(false, x402, x403, x404, x405, x406, x407) -> U4^1(empty(fstsplit(x407, app(map_f(two, head(x403)), x404))), x402, x403, x404, x405, x406, x407) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U2^1(leq(x401, length(x398)), x396, x397, x398, x399, x400, x401)=U2^1(false, x402, x403, x404, x405, x406, x407)  ==>  U2^1(false, x402, x403, x404, x405, x406, x407)_>=_U4^1(empty(fstsplit(x407, app(map_f(two, head(x403)), x404))), x402, x403, x404, x405, x406, x407))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (length(x398)=x1888 & leq(x401, x1888)=false  ==>  U2^1(false, x396, x397, x398, x399, x400, x401)_>=_U4^1(empty(fstsplit(x401, app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, x401))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x401, x1888)=false which results in the following new constraints:
166.37/63.28	
166.37/63.28	(3)    (false=false & length(x398)=0  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	(4)    (leq(x1892, x1891)=false & length(x398)=s(x1891) & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892))  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (length(x398)=0  ==>  U2^1(false, x396, x397, x398, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), x398))), x396, x397, x398, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x398)=s(x1891) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (s(length(x1900))=s(x1891) & leq(x1892, x1891)=false & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on length(x398)=0 which results in the following new constraint:
166.37/63.28	
166.37/63.28	(7)    (0=0  ==>  U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(8)    (U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(9)    (length(x1900)=x1891 & leq(x1892, x1891)=false & (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (9) using rule (VI) where we applied the induction hypothesis (\/x1893,x1894,x1895,x1896,x1897:leq(x1892, x1891)=false & length(x1893)=x1891  ==>  U2^1(false, x1894, x1895, x1893, x1896, x1897, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x1895)), x1893))), x1894, x1895, x1893, x1896, x1897, x1892)) with sigma = [x1893 / x1900, x1894 / x396, x1895 / x397, x1896 / x399, x1897 / x400] which results in the following new constraint:
166.37/63.28	
166.37/63.28	(10)    (U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892) & (\/x1902,x1903,x1904,x1905,x1906,x1907,x1908,x1909,x1910,x1911,x1912:length(x1900)=s(x1902) & leq(x1903, x1902)=false & (\/x1904,x1905,x1906,x1907,x1908:leq(x1903, x1902)=false & length(x1904)=x1902  ==>  U2^1(false, x1905, x1906, x1904, x1907, x1908, x1903)_>=_U4^1(empty(fstsplit(x1903, app(map_f(two, head(x1906)), x1904))), x1905, x1906, x1904, x1907, x1908, x1903))  ==>  U2^1(false, x1909, x1910, x1900, x1911, x1912, s(x1903))_>=_U4^1(empty(fstsplit(s(x1903), app(map_f(two, head(x1910)), x1900))), x1909, x1910, x1900, x1911, x1912, s(x1903)))  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (10) using rule (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(11)    (U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892)  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) the following chains were created:
166.37/63.28	*We consider the chain U2^1(false, x506, x507, x508, x509, x510, x511) -> U4^1(empty(fstsplit(x511, app(map_f(two, head(x507)), x508))), x506, x507, x508, x509, x510, x511), U4^1(false, x512, x513, x514, x515, x516, x517) -> RING(x512, tail(x513), sndsplit(x517, app(map_f(two, head(x513)), x514)), cons(fstsplit(x517, app(map_f(two, head(x513)), x514)), x515), x516, x517) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U4^1(empty(fstsplit(x511, app(map_f(two, head(x507)), x508))), x506, x507, x508, x509, x510, x511)=U4^1(false, x512, x513, x514, x515, x516, x517)  ==>  U4^1(false, x512, x513, x514, x515, x516, x517)_>=_RING(x512, tail(x513), sndsplit(x517, app(map_f(two, head(x513)), x514)), cons(fstsplit(x517, app(map_f(two, head(x513)), x514)), x515), x516, x517))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (fstsplit(x511, x1914)=x1913 & empty(x1913)=false  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1913)=false which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (false=false & fstsplit(x511, x1914)=cons(x1919, x1918)  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (fstsplit(x511, x1914)=cons(x1919, x1918)  ==>  U4^1(false, x506, x507, x508, x509, x510, x511)_>=_RING(x506, tail(x507), sndsplit(x511, app(map_f(two, head(x507)), x508)), cons(fstsplit(x511, app(map_f(two, head(x507)), x508)), x509), x510, x511))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x511, x1914)=cons(x1919, x1918) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (cons(x1923, fstsplit(x1924, x1922))=cons(x1919, x1918) & (\/x1925,x1926,x1927,x1928,x1929,x1930,x1931:fstsplit(x1924, x1922)=cons(x1925, x1926)  ==>  U4^1(false, x1927, x1928, x1929, x1930, x1931, x1924)_>=_RING(x1927, tail(x1928), sndsplit(x1924, app(map_f(two, head(x1928)), x1929)), cons(fstsplit(x1924, app(map_f(two, head(x1928)), x1929)), x1930), x1931, x1924))  ==>  U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain U4^1(false, x726, x727, x728, x729, x730, x731) -> RING(x726, tail(x727), sndsplit(x731, app(map_f(two, head(x727)), x728)), cons(fstsplit(x731, app(map_f(two, head(x727)), x728)), x729), x730, x731), RING(x732, x733, x734, x735, x736, x737) -> U6^1(leq(x737, length(x736)), x732, x733, x734, x735, x736, x737) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x726, tail(x727), sndsplit(x731, app(map_f(two, head(x727)), x728)), cons(fstsplit(x731, app(map_f(two, head(x727)), x728)), x729), x730, x731)=RING(x732, x733, x734, x735, x736, x737)  ==>  RING(x732, x733, x734, x735, x736, x737)_>=_U6^1(leq(x737, length(x736)), x732, x733, x734, x735, x736, x737))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x726, x733, x734, x735, x730, x731)_>=_U6^1(leq(x731, length(x730)), x726, x733, x734, x735, x730, x731))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U7^1(false, x762, x763, x764, x765, x766, x767) -> RING(x762, x763, x764, x765, sndsplit(x767, x766), x767), RING(x768, x769, x770, x771, x772, x773) -> U6^1(leq(x773, length(x772)), x768, x769, x770, x771, x772, x773) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x762, x763, x764, x765, sndsplit(x767, x766), x767)=RING(x768, x769, x770, x771, x772, x773)  ==>  RING(x768, x769, x770, x771, x772, x773)_>=_U6^1(leq(x773, length(x772)), x768, x769, x770, x771, x772, x773))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x762, x763, x764, x765, x772, x767)_>=_U6^1(leq(x767, length(x772)), x762, x763, x764, x765, x772, x767))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U9^1(true, x774, x775, x776, x777, x778, x779) -> RING(x774, x775, x776, tail(x777), x778, x779), RING(x780, x781, x782, x783, x784, x785) -> U6^1(leq(x785, length(x784)), x780, x781, x782, x783, x784, x785) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x774, x775, x776, tail(x777), x778, x779)=RING(x780, x781, x782, x783, x784, x785)  ==>  RING(x780, x781, x782, x783, x784, x785)_>=_U6^1(leq(x785, length(x784)), x780, x781, x782, x783, x784, x785))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x774, x775, x776, x783, x778, x779)_>=_U6^1(leq(x779, length(x778)), x774, x775, x776, x783, x778, x779))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U8^1(false, x792, x793, x794, x795, x796, x797) -> RING(x792, x793, x794, tail(x795), sndsplit(x797, app(map_f(three, head(x795)), x796)), x797), RING(x798, x799, x800, x801, x802, x803) -> U6^1(leq(x803, length(x802)), x798, x799, x800, x801, x802, x803) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x792, x793, x794, tail(x795), sndsplit(x797, app(map_f(three, head(x795)), x796)), x797)=RING(x798, x799, x800, x801, x802, x803)  ==>  RING(x798, x799, x800, x801, x802, x803)_>=_U6^1(leq(x803, length(x802)), x798, x799, x800, x801, x802, x803))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x792, x793, x794, x801, x802, x797)_>=_U6^1(leq(x797, length(x802)), x792, x793, x794, x801, x802, x797))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U3^1(false, x810, x811, x812, x813, x814, x815) -> RING(x810, x811, sndsplit(x815, x812), cons(fstsplit(x815, x812), x813), x814, x815), RING(x816, x817, x818, x819, x820, x821) -> U6^1(leq(x821, length(x820)), x816, x817, x818, x819, x820, x821) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x810, x811, sndsplit(x815, x812), cons(fstsplit(x815, x812), x813), x814, x815)=RING(x816, x817, x818, x819, x820, x821)  ==>  RING(x816, x817, x818, x819, x820, x821)_>=_U6^1(leq(x821, length(x820)), x816, x817, x818, x819, x820, x821))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x810, x811, x818, x819, x814, x815)_>=_U6^1(leq(x815, length(x814)), x810, x811, x818, x819, x814, x815))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain RING(x872, x873, x874, x875, x876, x877) -> U6^1(leq(x877, length(x876)), x872, x873, x874, x875, x876, x877), U6^1(true, x878, x879, x880, x881, x882, x883) -> U7^1(empty(fstsplit(x883, x882)), x878, x879, x880, x881, x882, x883) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U6^1(leq(x877, length(x876)), x872, x873, x874, x875, x876, x877)=U6^1(true, x878, x879, x880, x881, x882, x883)  ==>  U6^1(true, x878, x879, x880, x881, x882, x883)_>=_U7^1(empty(fstsplit(x883, x882)), x878, x879, x880, x881, x882, x883))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (length(x876)=x1960 & leq(x877, x1960)=true  ==>  U6^1(true, x872, x873, x874, x875, x876, x877)_>=_U7^1(empty(fstsplit(x877, x876)), x872, x873, x874, x875, x876, x877))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x877, x1960)=true which results in the following new constraints:
166.37/63.28	
166.37/63.28	(3)    (true=true & length(x876)=x1961  ==>  U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.28	
166.37/63.28	(4)    (leq(x1964, x1963)=true & length(x876)=s(x1963) & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964))  ==>  U6^1(true, x872, x873, x874, x875, x876, s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), x876)), x872, x873, x874, x875, x876, s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x876)=s(x1963) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (s(length(x1970))=s(x1963) & leq(x1964, x1963)=true & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(7)    (length(x1970)=x1963 & leq(x1964, x1963)=true & (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (7) using rule (VI) where we applied the induction hypothesis (\/x1965,x1966,x1967,x1968,x1969:leq(x1964, x1963)=true & length(x1965)=x1963  ==>  U6^1(true, x1966, x1967, x1968, x1969, x1965, x1964)_>=_U7^1(empty(fstsplit(x1964, x1965)), x1966, x1967, x1968, x1969, x1965, x1964)) with sigma = [x1965 / x1970, x1966 / x872, x1967 / x873, x1968 / x874, x1969 / x875] which results in the following new constraint:
166.37/63.28	
166.37/63.28	(8)    (U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964) & (\/x1972,x1973,x1974,x1975,x1976,x1977,x1978,x1979,x1980,x1981,x1982:length(x1970)=s(x1972) & leq(x1973, x1972)=true & (\/x1974,x1975,x1976,x1977,x1978:leq(x1973, x1972)=true & length(x1974)=x1972  ==>  U6^1(true, x1975, x1976, x1977, x1978, x1974, x1973)_>=_U7^1(empty(fstsplit(x1973, x1974)), x1975, x1976, x1977, x1978, x1974, x1973))  ==>  U6^1(true, x1979, x1980, x1981, x1982, x1970, s(x1973))_>=_U7^1(empty(fstsplit(s(x1973), x1970)), x1979, x1980, x1981, x1982, x1970, s(x1973)))  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (8) using rule (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(9)    (U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964)  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) the following chains were created:
166.37/63.28	*We consider the chain U6^1(true, x982, x983, x984, x985, x986, x987) -> U7^1(empty(fstsplit(x987, x986)), x982, x983, x984, x985, x986, x987), U7^1(false, x988, x989, x990, x991, x992, x993) -> RING(x988, x989, x990, x991, sndsplit(x993, x992), x993) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U7^1(empty(fstsplit(x987, x986)), x982, x983, x984, x985, x986, x987)=U7^1(false, x988, x989, x990, x991, x992, x993)  ==>  U7^1(false, x988, x989, x990, x991, x992, x993)_>=_RING(x988, x989, x990, x991, sndsplit(x993, x992), x993))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (fstsplit(x987, x986)=x1983 & empty(x1983)=false  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1983)=false which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (false=false & fstsplit(x987, x986)=cons(x1985, x1984)  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (fstsplit(x987, x986)=cons(x1985, x1984)  ==>  U7^1(false, x982, x983, x984, x985, x986, x987)_>=_RING(x982, x983, x984, x985, sndsplit(x987, x986), x987))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x987, x986)=cons(x1985, x1984) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (cons(x1989, fstsplit(x1990, x1988))=cons(x1985, x1984) & (\/x1991,x1992,x1993,x1994,x1995,x1996:fstsplit(x1990, x1988)=cons(x1991, x1992)  ==>  U7^1(false, x1993, x1994, x1995, x1996, x1988, x1990)_>=_RING(x1993, x1994, x1995, x1996, sndsplit(x1990, x1988), x1990))  ==>  U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m) the following chains were created:
166.37/63.28	*We consider the chain RING(x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141) -> U9^1(empty(map_f(three, x1138)), x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141), U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147) -> RING(x1142, x1143, x1144, tail(x1145), x1146, x1147) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U9^1(empty(map_f(three, x1138)), x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)=U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147)  ==>  U9^1(true, x1142, x1143, x1144, x1145, x1146, x1147)_>=_RING(x1142, x1143, x1144, tail(x1145), x1146, x1147))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (three=x1998 & map_f(x1998, x1138)=x1997 & empty(x1997)=true  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x1997)=true which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (true=true & three=x1998 & map_f(x1998, x1138)=nil  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (three=x1998 & map_f(x1998, x1138)=nil  ==>  U9^1(true, x1135, x1136, x1137, cons(x1138, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(x1138, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on map_f(x1998, x1138)=nil which results in the following new constraints:
166.37/63.28	
166.37/63.28	(5)    (nil=nil & three=x2001  ==>  U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	(6)    (app(f(x2004, x2003), map_f(x2004, x2002))=nil & three=x2004 & (\/x2005,x2006,x2007,x2008,x2009,x2010:map_f(x2004, x2002)=nil & three=x2004  ==>  U9^1(true, x2005, x2006, x2007, cons(x2002, x2008), x2009, x2010)_>=_RING(x2005, x2006, x2007, tail(cons(x2002, x2008)), x2009, x2010))  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(7)    (U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (6) using rules (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(8)    (f(x2004, x2003)=x2011 & map_f(x2004, x2002)=x2012 & app(x2011, x2012)=nil & three=x2004  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on app(x2011, x2012)=nil which results in the following new constraint:
166.37/63.28	
166.37/63.28	(9)    (x2013=nil & f(x2004, x2003)=nil & map_f(x2004, x2002)=x2013 & three=x2004  ==>  U9^1(true, x1135, x1136, x1137, cons(cons(x2003, x2002), x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(cons(x2003, x2002), x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We solved constraint (9) using rules (I), (II).
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain RING(x1184, x1185, x1186, x1187, x1188, x1189) -> U6^1(leq(x1189, length(x1188)), x1184, x1185, x1186, x1187, x1188, x1189), U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195) -> U8^1(empty(fstsplit(x1195, app(map_f(three, head(x1193)), x1194))), x1190, x1191, x1192, x1193, x1194, x1195) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U6^1(leq(x1189, length(x1188)), x1184, x1185, x1186, x1187, x1188, x1189)=U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195)  ==>  U6^1(false, x1190, x1191, x1192, x1193, x1194, x1195)_>=_U8^1(empty(fstsplit(x1195, app(map_f(three, head(x1193)), x1194))), x1190, x1191, x1192, x1193, x1194, x1195))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (length(x1188)=x2017 & leq(x1189, x2017)=false  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, x1189)_>=_U8^1(empty(fstsplit(x1189, app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, x1189))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x1189, x2017)=false which results in the following new constraints:
166.37/63.28	
166.37/63.28	(3)    (false=false & length(x1188)=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2019)))
166.37/63.28	
166.37/63.28	(4)    (leq(x2021, x2020)=false & length(x1188)=s(x2020) & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021))  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (length(x1188)=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, x1188, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), x1188))), x1184, x1185, x1186, x1187, x1188, s(x2019)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x1188)=s(x2020) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (s(length(x2029))=s(x2020) & leq(x2021, x2020)=false & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on length(x1188)=0 which results in the following new constraint:
166.37/63.28	
166.37/63.28	(7)    (0=0  ==>  U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(8)    (U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(9)    (length(x2029)=x2020 & leq(x2021, x2020)=false & (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (9) using rule (VI) where we applied the induction hypothesis (\/x2022,x2023,x2024,x2025,x2026:leq(x2021, x2020)=false & length(x2022)=x2020  ==>  U6^1(false, x2023, x2024, x2025, x2026, x2022, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x2026)), x2022))), x2023, x2024, x2025, x2026, x2022, x2021)) with sigma = [x2022 / x2029, x2023 / x1184, x2024 / x1185, x2025 / x1186, x2026 / x1187] which results in the following new constraint:
166.37/63.28	
166.37/63.28	(10)    (U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021) & (\/x2031,x2032,x2033,x2034,x2035,x2036,x2037,x2038,x2039,x2040,x2041:length(x2029)=s(x2031) & leq(x2032, x2031)=false & (\/x2033,x2034,x2035,x2036,x2037:leq(x2032, x2031)=false & length(x2033)=x2031  ==>  U6^1(false, x2034, x2035, x2036, x2037, x2033, x2032)_>=_U8^1(empty(fstsplit(x2032, app(map_f(three, head(x2037)), x2033))), x2034, x2035, x2036, x2037, x2033, x2032))  ==>  U6^1(false, x2038, x2039, x2040, x2041, x2029, s(x2032))_>=_U8^1(empty(fstsplit(s(x2032), app(map_f(three, head(x2041)), x2029))), x2038, x2039, x2040, x2041, x2029, s(x2032)))  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (10) using rule (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(11)    (U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021)  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) the following chains were created:
166.37/63.28	*We consider the chain U6^1(false, x1312, x1313, x1314, x1315, x1316, x1317) -> U8^1(empty(fstsplit(x1317, app(map_f(three, head(x1315)), x1316))), x1312, x1313, x1314, x1315, x1316, x1317), U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323) -> RING(x1318, x1319, x1320, tail(x1321), sndsplit(x1323, app(map_f(three, head(x1321)), x1322)), x1323) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U8^1(empty(fstsplit(x1317, app(map_f(three, head(x1315)), x1316))), x1312, x1313, x1314, x1315, x1316, x1317)=U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323)  ==>  U8^1(false, x1318, x1319, x1320, x1321, x1322, x1323)_>=_RING(x1318, x1319, x1320, tail(x1321), sndsplit(x1323, app(map_f(three, head(x1321)), x1322)), x1323))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (fstsplit(x1317, x2043)=x2042 & empty(x2042)=false  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x2042)=false which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (false=false & fstsplit(x1317, x2043)=cons(x2048, x2047)  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (fstsplit(x1317, x2043)=cons(x2048, x2047)  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, x1317)_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(x1317, app(map_f(three, head(x1315)), x1316)), x1317))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x1317, x2043)=cons(x2048, x2047) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (cons(x2052, fstsplit(x2053, x2051))=cons(x2048, x2047) & (\/x2054,x2055,x2056,x2057,x2058,x2059,x2060:fstsplit(x2053, x2051)=cons(x2054, x2055)  ==>  U8^1(false, x2056, x2057, x2058, x2059, x2060, x2053)_>=_RING(x2056, x2057, x2058, tail(x2059), sndsplit(x2053, app(map_f(three, head(x2059)), x2060)), x2053))  ==>  U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.37/63.28	*We consider the chain RING(x1368, x1369, x1370, x1371, x1372, x1373) -> U2^1(leq(x1373, length(x1370)), x1368, x1369, x1370, x1371, x1372, x1373), U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379) -> U3^1(empty(fstsplit(x1379, x1376)), x1374, x1375, x1376, x1377, x1378, x1379) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U2^1(leq(x1373, length(x1370)), x1368, x1369, x1370, x1371, x1372, x1373)=U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379)  ==>  U2^1(true, x1374, x1375, x1376, x1377, x1378, x1379)_>=_U3^1(empty(fstsplit(x1379, x1376)), x1374, x1375, x1376, x1377, x1378, x1379))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (length(x1370)=x2061 & leq(x1373, x2061)=true  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, x1373)_>=_U3^1(empty(fstsplit(x1373, x1370)), x1368, x1369, x1370, x1371, x1372, x1373))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x1373, x2061)=true which results in the following new constraints:
166.37/63.28	
166.37/63.28	(3)    (true=true & length(x1370)=x2062  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.28	
166.37/63.28	(4)    (leq(x2065, x2064)=true & length(x1370)=s(x2064) & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065))  ==>  U2^1(true, x1368, x1369, x1370, x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), x1370)), x1368, x1369, x1370, x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x1370)=s(x2064) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (s(length(x2071))=s(x2064) & leq(x2065, x2064)=true & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(7)    (length(x2071)=x2064 & leq(x2065, x2064)=true & (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (7) using rule (VI) where we applied the induction hypothesis (\/x2066,x2067,x2068,x2069,x2070:leq(x2065, x2064)=true & length(x2066)=x2064  ==>  U2^1(true, x2067, x2068, x2066, x2069, x2070, x2065)_>=_U3^1(empty(fstsplit(x2065, x2066)), x2067, x2068, x2066, x2069, x2070, x2065)) with sigma = [x2066 / x2071, x2067 / x1368, x2068 / x1369, x2069 / x1371, x2070 / x1372] which results in the following new constraint:
166.37/63.28	
166.37/63.28	(8)    (U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065) & (\/x2073,x2074,x2075,x2076,x2077,x2078,x2079,x2080,x2081,x2082,x2083:length(x2071)=s(x2073) & leq(x2074, x2073)=true & (\/x2075,x2076,x2077,x2078,x2079:leq(x2074, x2073)=true & length(x2075)=x2073  ==>  U2^1(true, x2076, x2077, x2075, x2078, x2079, x2074)_>=_U3^1(empty(fstsplit(x2074, x2075)), x2076, x2077, x2075, x2078, x2079, x2074))  ==>  U2^1(true, x2080, x2081, x2071, x2082, x2083, s(x2074))_>=_U3^1(empty(fstsplit(s(x2074), x2071)), x2080, x2081, x2071, x2082, x2083, s(x2074)))  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (8) using rule (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(9)    (U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065)  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m) the following chains were created:
166.37/63.28	*We consider the chain U2^1(true, x1532, x1533, x1534, x1535, x1536, x1537) -> U3^1(empty(fstsplit(x1537, x1534)), x1532, x1533, x1534, x1535, x1536, x1537), U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543) -> RING(x1538, x1539, sndsplit(x1543, x1540), cons(fstsplit(x1543, x1540), x1541), x1542, x1543) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (U3^1(empty(fstsplit(x1537, x1534)), x1532, x1533, x1534, x1535, x1536, x1537)=U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543)  ==>  U3^1(false, x1538, x1539, x1540, x1541, x1542, x1543)_>=_RING(x1538, x1539, sndsplit(x1543, x1540), cons(fstsplit(x1543, x1540), x1541), x1542, x1543))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (fstsplit(x1537, x1534)=x2084 & empty(x2084)=false  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x2084)=false which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (false=false & fstsplit(x1537, x1534)=cons(x2086, x2085)  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (fstsplit(x1537, x1534)=cons(x2086, x2085)  ==>  U3^1(false, x1532, x1533, x1534, x1535, x1536, x1537)_>=_RING(x1532, x1533, sndsplit(x1537, x1534), cons(fstsplit(x1537, x1534), x1535), x1536, x1537))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x1537, x1534)=cons(x2086, x2085) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(5)    (cons(x2090, fstsplit(x2091, x2089))=cons(x2086, x2085) & (\/x2092,x2093,x2094,x2095,x2096,x2097:fstsplit(x2091, x2089)=cons(x2092, x2093)  ==>  U3^1(false, x2094, x2095, x2089, x2096, x2097, x2091)_>=_RING(x2094, x2095, sndsplit(x2091, x2089), cons(fstsplit(x2091, x2089), x2096), x2097, x2091))  ==>  U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(6)    (U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	For Pair RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5) the following chains were created:
166.37/63.28	*We consider the chain U4^1(false, x1742, x1743, x1744, x1745, x1746, x1747) -> RING(x1742, tail(x1743), sndsplit(x1747, app(map_f(two, head(x1743)), x1744)), cons(fstsplit(x1747, app(map_f(two, head(x1743)), x1744)), x1745), x1746, x1747), RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754) -> U9^1(empty(map_f(three, x1751)), x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1742, tail(x1743), sndsplit(x1747, app(map_f(two, head(x1743)), x1744)), cons(fstsplit(x1747, app(map_f(two, head(x1743)), x1744)), x1745), x1746, x1747)=RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754)  ==>  RING(x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754)_>=_U9^1(empty(map_f(three, x1751)), x1748, x1749, x1750, cons(x1751, x1752), x1753, x1754))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747)_>=_U9^1(empty(map_f(three, x1751)), x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U7^1(false, x1780, x1781, x1782, x1783, x1784, x1785) -> RING(x1780, x1781, x1782, x1783, sndsplit(x1785, x1784), x1785), RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792) -> U9^1(empty(map_f(three, x1789)), x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1780, x1781, x1782, x1783, sndsplit(x1785, x1784), x1785)=RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792)  ==>  RING(x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792)_>=_U9^1(empty(map_f(three, x1789)), x1786, x1787, x1788, cons(x1789, x1790), x1791, x1792))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785)_>=_U9^1(empty(map_f(three, x1789)), x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U9^1(true, x1793, x1794, x1795, x1796, x1797, x1798) -> RING(x1793, x1794, x1795, tail(x1796), x1797, x1798), RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805) -> U9^1(empty(map_f(three, x1802)), x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1793, x1794, x1795, tail(x1796), x1797, x1798)=RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805)  ==>  RING(x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805)_>=_U9^1(empty(map_f(three, x1802)), x1799, x1800, x1801, cons(x1802, x1803), x1804, x1805))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (tail(x1796)=cons(x1802, x1803)  ==>  RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1796)=cons(x1802, x1803) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (x2118=cons(x1802, x1803)  ==>  RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rule (III) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U8^1(false, x1812, x1813, x1814, x1815, x1816, x1817) -> RING(x1812, x1813, x1814, tail(x1815), sndsplit(x1817, app(map_f(three, head(x1815)), x1816)), x1817), RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824) -> U9^1(empty(map_f(three, x1821)), x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1812, x1813, x1814, tail(x1815), sndsplit(x1817, app(map_f(three, head(x1815)), x1816)), x1817)=RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824)  ==>  RING(x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824)_>=_U9^1(empty(map_f(three, x1821)), x1818, x1819, x1820, cons(x1821, x1822), x1823, x1824))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (tail(x1815)=cons(x1821, x1822) & app(x2121, x1816)=x2120 & sndsplit(x1817, x2120)=x1823  ==>  RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on tail(x1815)=cons(x1821, x1822) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(3)    (x2124=cons(x1821, x1822) & app(x2121, x1816)=x2120 & sndsplit(x1817, x2120)=x1823  ==>  RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (3) using rules (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(4)    (RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*We consider the chain U3^1(false, x1831, x1832, x1833, x1834, x1835, x1836) -> RING(x1831, x1832, sndsplit(x1836, x1833), cons(fstsplit(x1836, x1833), x1834), x1835, x1836), RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843) -> U9^1(empty(map_f(three, x1840)), x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843) which results in the following constraint:
166.37/63.28	
166.37/63.28	(1)    (RING(x1831, x1832, sndsplit(x1836, x1833), cons(fstsplit(x1836, x1833), x1834), x1835, x1836)=RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843)  ==>  RING(x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843)_>=_U9^1(empty(map_f(three, x1840)), x1837, x1838, x1839, cons(x1840, x1841), x1842, x1843))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.37/63.28	
166.37/63.28	(2)    (RING(x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836)_>=_U9^1(empty(map_f(three, x1840)), x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	To summarize, we get the following constraints P__>=_ for the following pairs.
166.37/63.28	
166.37/63.28	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(RING(x274, x281, x282, x283, x278, x279)_>=_U2^1(leq(x279, length(x282)), x274, x281, x282, x283, x278, x279))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x310, x311, x312, x313, x320, x315)_>=_U2^1(leq(x315, length(x312)), x310, x311, x312, x313, x320, x315))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x322, x323, x324, x331, x326, x327)_>=_U2^1(leq(x327, length(x324)), x322, x323, x324, x331, x326, x327))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x340, x341, x342, x349, x350, x345)_>=_U2^1(leq(x345, length(x342)), x340, x341, x342, x349, x350, x345))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x358, x359, x366, x367, x362, x363)_>=_U2^1(leq(x363, length(x366)), x358, x359, x366, x367, x362, x363))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U2^1(false, x396, x397, nil, x399, x400, s(x1890))_>=_U4^1(empty(fstsplit(s(x1890), app(map_f(two, head(x397)), nil))), x396, x397, nil, x399, x400, s(x1890)))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U2^1(false, x396, x397, x1900, x399, x400, x1892)_>=_U4^1(empty(fstsplit(x1892, app(map_f(two, head(x397)), x1900))), x396, x397, x1900, x399, x400, x1892)  ==>  U2^1(false, x396, x397, cons(x1901, x1900), x399, x400, s(x1892))_>=_U4^1(empty(fstsplit(s(x1892), app(map_f(two, head(x397)), cons(x1901, x1900)))), x396, x397, cons(x1901, x1900), x399, x400, s(x1892)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U4^1(false, x506, x507, x508, x509, x510, s(x1924))_>=_RING(x506, tail(x507), sndsplit(s(x1924), app(map_f(two, head(x507)), x508)), cons(fstsplit(s(x1924), app(map_f(two, head(x507)), x508)), x509), x510, s(x1924)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(RING(x726, x733, x734, x735, x730, x731)_>=_U6^1(leq(x731, length(x730)), x726, x733, x734, x735, x730, x731))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x762, x763, x764, x765, x772, x767)_>=_U6^1(leq(x767, length(x772)), x762, x763, x764, x765, x772, x767))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x774, x775, x776, x783, x778, x779)_>=_U6^1(leq(x779, length(x778)), x774, x775, x776, x783, x778, x779))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x792, x793, x794, x801, x802, x797)_>=_U6^1(leq(x797, length(x802)), x792, x793, x794, x801, x802, x797))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x810, x811, x818, x819, x814, x815)_>=_U6^1(leq(x815, length(x814)), x810, x811, x818, x819, x814, x815))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U6^1(true, x872, x873, x874, x875, x876, 0)_>=_U7^1(empty(fstsplit(0, x876)), x872, x873, x874, x875, x876, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U6^1(true, x872, x873, x874, x875, x1970, x1964)_>=_U7^1(empty(fstsplit(x1964, x1970)), x872, x873, x874, x875, x1970, x1964)  ==>  U6^1(true, x872, x873, x874, x875, cons(x1971, x1970), s(x1964))_>=_U7^1(empty(fstsplit(s(x1964), cons(x1971, x1970))), x872, x873, x874, x875, cons(x1971, x1970), s(x1964)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	
166.37/63.28	*(U7^1(false, x982, x983, x984, x985, cons(x1989, x1988), s(x1990))_>=_RING(x982, x983, x984, x985, sndsplit(s(x1990), cons(x1989, x1988)), s(x1990)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U9^1(true, x1135, x1136, x1137, cons(nil, x1139), x1140, x1141)_>=_RING(x1135, x1136, x1137, tail(cons(nil, x1139)), x1140, x1141))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U6^1(false, x1184, x1185, x1186, x1187, nil, s(x2019))_>=_U8^1(empty(fstsplit(s(x2019), app(map_f(three, head(x1187)), nil))), x1184, x1185, x1186, x1187, nil, s(x2019)))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U6^1(false, x1184, x1185, x1186, x1187, x2029, x2021)_>=_U8^1(empty(fstsplit(x2021, app(map_f(three, head(x1187)), x2029))), x1184, x1185, x1186, x1187, x2029, x2021)  ==>  U6^1(false, x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021))_>=_U8^1(empty(fstsplit(s(x2021), app(map_f(three, head(x1187)), cons(x2030, x2029)))), x1184, x1185, x1186, x1187, cons(x2030, x2029), s(x2021)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	*(U8^1(false, x1312, x1313, x1314, x1315, x1316, s(x2053))_>=_RING(x1312, x1313, x1314, tail(x1315), sndsplit(s(x2053), app(map_f(three, head(x1315)), x1316)), s(x2053)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	
166.37/63.28	*(U2^1(true, x1368, x1369, x1370, x1371, x1372, 0)_>=_U3^1(empty(fstsplit(0, x1370)), x1368, x1369, x1370, x1371, x1372, 0))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(U2^1(true, x1368, x1369, x2071, x1371, x1372, x2065)_>=_U3^1(empty(fstsplit(x2065, x2071)), x1368, x1369, x2071, x1371, x1372, x2065)  ==>  U2^1(true, x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065))_>=_U3^1(empty(fstsplit(s(x2065), cons(x2072, x2071))), x1368, x1369, cons(x2072, x2071), x1371, x1372, s(x2065)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	
166.37/63.28	*(U3^1(false, x1532, x1533, cons(x2090, x2089), x1535, x1536, s(x2091))_>=_RING(x1532, x1533, sndsplit(s(x2091), cons(x2090, x2089)), cons(fstsplit(s(x2091), cons(x2090, x2089)), x1535), x1536, s(x2091)))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	*RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	
166.37/63.28	*(RING(x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747)_>=_U9^1(empty(map_f(three, x1751)), x1742, x1749, x1750, cons(x1751, x1745), x1746, x1747))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785)_>=_U9^1(empty(map_f(three, x1789)), x1780, x1781, x1782, cons(x1789, x1790), x1791, x1785))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798)_>=_U9^1(empty(map_f(three, x1802)), x1793, x1794, x1795, cons(x1802, x1803), x1797, x1798))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817)_>=_U9^1(empty(map_f(three, x1821)), x1812, x1813, x1814, cons(x1821, x1822), x1823, x1817))
166.37/63.28	
166.37/63.28	
166.37/63.28	*(RING(x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836)_>=_U9^1(empty(map_f(three, x1840)), x1831, x1832, x1839, cons(x1840, x1834), x1835, x1836))
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	
166.37/63.28	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
166.37/63.28	
166.37/63.28	Using the following integer polynomial ordering the  resulting constraints can be solved 
166.37/63.28	
166.37/63.28	Polynomial interpretation [NONINF]:
166.37/63.28	
166.37/63.28	   POL(0) = 0
166.37/63.28	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 - x_3 - x_6
166.37/63.28	   POL(U2^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 - x_4 - x_7
166.37/63.28	   POL(U3^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 + x_2 - x_4 - x_7
166.37/63.28	   POL(U4^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 + x_2 - x_7
166.37/63.28	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 - x_4 - x_7
166.37/63.28	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_1 + x_2 - x_4 - x_7
166.37/63.28	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_1 + x_2 - x_4 - x_7
166.37/63.28	   POL(U9^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_1 + x_2 - x_4 - x_5 - x_7
166.37/63.28	   POL(app(x_1, x_2)) = 0
166.37/63.28	   POL(c) = -2
166.37/63.28	   POL(cons(x_1, x_2)) = 0
166.37/63.28	   POL(empty(x_1)) = 0
166.37/63.28	   POL(f(x_1, x_2)) = x_1 + x_2
166.37/63.28	   POL(false) = 0
166.37/63.28	   POL(fstsplit(x_1, x_2)) = 0
166.37/63.28	   POL(head(x_1)) = 0
166.37/63.28	   POL(length(x_1)) = 0
166.37/63.28	   POL(leq(x_1, x_2)) = 0
166.37/63.28	   POL(map_f(x_1, x_2)) = 0
166.37/63.28	   POL(nil) = 1
166.37/63.28	   POL(s(x_1)) = 1
166.37/63.28	   POL(sndsplit(x_1, x_2)) = x_1
166.37/63.28	   POL(tail(x_1)) = 0
166.37/63.28	   POL(three) = 0
166.37/63.28	   POL(true) = 0
166.37/63.28	   POL(two) = 0
166.37/63.28	
166.37/63.28	
166.37/63.28	The following pairs  are in P_>:
166.37/63.28	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	The following pairs are in P_bound:
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	The following rules are usable:
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   x -> sndsplit(0, x)
166.37/63.28	   nil -> sndsplit(s(n), nil)
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(71)
166.37/63.28	Complex Obligation (AND)
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(72)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U3^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(73) DependencyGraphProof (EQUIVALENT)
166.37/63.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(74)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(75) TransformationProof (EQUIVALENT)
166.37/63.28	By narrowing [LPAR04] the rule RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.28	
166.37/63.28	   (RING(y0, y1, y2, y3, y4, 0) -> U2^1(true, y0, y1, y2, y3, y4, 0),RING(y0, y1, y2, y3, y4, 0) -> U2^1(true, y0, y1, y2, y3, y4, 0))
166.37/63.28	   (RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5),RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5))
166.37/63.28	   (RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5),RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5))
166.37/63.28	
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(76)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   RING(y0, y1, y2, y3, y4, 0) -> U2^1(true, y0, y1, y2, y3, y4, 0)
166.37/63.28	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.28	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(77) DependencyGraphProof (EQUIVALENT)
166.37/63.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(78)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(79) TransformationProof (EQUIVALENT)
166.37/63.28	By narrowing [LPAR04] the rule RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.28	
166.37/63.28	   (RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0),RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0))
166.37/63.28	   (RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5),RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5))
166.37/63.28	   (RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5),RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5))
166.37/63.28	
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(80)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0)
166.37/63.28	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.28	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(81) TransformationProof (EQUIVALENT)
166.37/63.28	By narrowing [LPAR04] the rule U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.28	
166.37/63.28	   (U6^1(true, y0, y1, y2, y3, x0, 0) -> U7^1(empty(nil), y0, y1, y2, y3, x0, 0),U6^1(true, y0, y1, y2, y3, x0, 0) -> U7^1(empty(nil), y0, y1, y2, y3, x0, 0))
166.37/63.28	   (U6^1(true, y0, y1, y2, y3, nil, s(x0)) -> U7^1(empty(nil), y0, y1, y2, y3, nil, s(x0)),U6^1(true, y0, y1, y2, y3, nil, s(x0)) -> U7^1(empty(nil), y0, y1, y2, y3, nil, s(x0)))
166.37/63.28	   (U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(empty(cons(x1, fstsplit(x0, x2))), y0, y1, y2, y3, cons(x1, x2), s(x0)),U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(empty(cons(x1, fstsplit(x0, x2))), y0, y1, y2, y3, cons(x1, x2), s(x0)))
166.37/63.28	
166.37/63.28	
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(82)
166.37/63.28	Obligation:
166.37/63.28	Q DP problem:
166.37/63.28	The TRS P consists of the following rules:
166.37/63.28	
166.37/63.28	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.28	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.28	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.28	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.28	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.28	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.28	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.28	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.28	   RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0)
166.37/63.28	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.28	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.28	   U6^1(true, y0, y1, y2, y3, x0, 0) -> U7^1(empty(nil), y0, y1, y2, y3, x0, 0)
166.37/63.28	   U6^1(true, y0, y1, y2, y3, nil, s(x0)) -> U7^1(empty(nil), y0, y1, y2, y3, nil, s(x0))
166.37/63.28	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(empty(cons(x1, fstsplit(x0, x2))), y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.28	
166.37/63.28	The TRS R consists of the following rules:
166.37/63.28	
166.37/63.28	   sndsplit(0, x) -> x
166.37/63.28	   sndsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(0, x) -> nil
166.37/63.28	   fstsplit(s(n), nil) -> nil
166.37/63.28	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.28	   empty(nil) -> true
166.37/63.28	   empty(cons(h, t)) -> false
166.37/63.28	   tail(cons(h, t)) -> t
166.37/63.28	   head(cons(h, t)) -> h
166.37/63.28	   map_f(pid, nil) -> nil
166.37/63.28	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.28	   app(nil, x) -> x
166.37/63.28	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.28	   length(nil) -> 0
166.37/63.28	   length(cons(h, t)) -> s(length(t))
166.37/63.28	   leq(0, m) -> true
166.37/63.28	   leq(s(n), 0) -> false
166.37/63.28	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.28	
166.37/63.28	The set Q consists of the following terms:
166.37/63.28	
166.37/63.28	   fstsplit(0, x0)
166.37/63.28	   fstsplit(s(x0), nil)
166.37/63.28	   fstsplit(s(x0), cons(x1, x2))
166.37/63.28	   sndsplit(0, x0)
166.37/63.28	   sndsplit(s(x0), nil)
166.37/63.28	   sndsplit(s(x0), cons(x1, x2))
166.37/63.28	   empty(nil)
166.37/63.28	   empty(cons(x0, x1))
166.37/63.28	   leq(0, x0)
166.37/63.28	   leq(s(x0), 0)
166.37/63.28	   leq(s(x0), s(x1))
166.37/63.28	   length(nil)
166.37/63.28	   length(cons(x0, x1))
166.37/63.28	   app(nil, x0)
166.37/63.28	   app(cons(x0, x1), x2)
166.37/63.28	   map_f(x0, nil)
166.37/63.28	   map_f(x0, cons(x1, x2))
166.37/63.28	   head(cons(x0, x1))
166.37/63.28	   tail(cons(x0, x1))
166.37/63.28	
166.37/63.28	We have to consider all minimal (P,Q,R)-chains.
166.37/63.28	----------------------------------------
166.37/63.28	
166.37/63.28	(83) DependencyGraphProof (EQUIVALENT)
166.37/63.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(84)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(empty(cons(x1, fstsplit(x0, x2))), y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(85) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(empty(cons(x1, fstsplit(x0, x2))), y0, y1, y2, y3, cons(x1, x2), s(x0)) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0)),U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0)))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(86)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(87) TransformationProof (EQUIVALENT)
166.37/63.29	By narrowing [LPAR04] the rule U2^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U4^1(empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U2^1(false, y0, y1, y2, y3, y4, 0) -> U4^1(empty(nil), y0, y1, y2, y3, y4, 0),U2^1(false, y0, y1, y2, y3, y4, 0) -> U4^1(empty(nil), y0, y1, y2, y3, y4, 0))
166.37/63.29	   (U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5),U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(88)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U2^1(false, y0, y1, y2, y3, y4, 0) -> U4^1(empty(nil), y0, y1, y2, y3, y4, 0)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(89) DependencyGraphProof (EQUIVALENT)
166.37/63.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(90)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(91) TransformationProof (EQUIVALENT)
166.37/63.29	By narrowing [LPAR04] the rule U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0),U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0))
166.37/63.29	   (U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5),U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(92)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(93) DependencyGraphProof (EQUIVALENT)
166.37/63.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(94)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(95) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6),U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(96)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(97) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6) at position [3] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6),U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(98)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(99) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U2^1(false, y0, cons(x0, x1), y2, y3, y4, y5) -> U4^1(empty(fstsplit(y5, app(map_f(two, x0), y2))), y0, cons(x0, x1), y2, y3, y4, y5) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4),U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4))
166.37/63.29	   (U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6),U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(100)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(101) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U4^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, tail(cons(z1, z2)), sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5),U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, tail(cons(z1, z2)), sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5))
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, tail(cons(z1, z2)), sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7),U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, tail(cons(z1, z2)), sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(102)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, tail(cons(z1, z2)), sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, tail(cons(z1, z2)), sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(103) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, tail(cons(z1, z2)), sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5) at position [1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5),U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(104)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, tail(cons(z1, z2)), sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(105) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, tail(cons(z1, z2)), sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7) at position [1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7),U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(106)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(107) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5) at position [2,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5),U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(108)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(109) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7) at position [2,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7),U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(110)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(111) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, head(cons(z1, z2))), nil)), z3), z4, z5) at position [3,0,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5),U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(112)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(113) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, head(cons(z1, z2))), cons(z3, z4))), z5), z6, z7) at position [3,0,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7),U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(114)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(115) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4),U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4))
166.37/63.29	   (U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6),U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(116)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(117) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5),U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5))
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7),U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(118)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(119) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5) at position [3] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5),U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(120)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(121) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, tail(cons(z3, z4)), sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7) at position [3] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7),U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(122)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   tail(cons(h, t)) -> t
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(123) UsableRulesProof (EQUIVALENT)
166.37/63.29	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(124)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(125) QReductionProof (EQUIVALENT)
166.37/63.29	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.29	
166.37/63.29	   tail(cons(x0, x1))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(126)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(127) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, head(cons(z3, z4))), nil)), z5) at position [4,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5),U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(128)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(129) TransformationProof (EQUIVALENT)
166.37/63.29	By rewriting [LPAR04] the rule U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, head(cons(z3, z4))), cons(z5, z6))), z7) at position [4,1,0,1] we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7),U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(130)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   head(cons(h, t)) -> h
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(131) UsableRulesProof (EQUIVALENT)
166.37/63.29	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(132)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(133) QReductionProof (EQUIVALENT)
166.37/63.29	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.37/63.29	
166.37/63.29	   head(cons(x0, x1))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(134)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(135) TransformationProof (EQUIVALENT)
166.37/63.29	By instantiating [LPAR04] the rule U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U7^1(false, z0, z1, z2, z3, cons(z4, z5), s(z6)) -> RING(z0, z1, z2, z3, sndsplit(s(z6), cons(z4, z5)), s(z6)),U7^1(false, z0, z1, z2, z3, cons(z4, z5), s(z6)) -> RING(z0, z1, z2, z3, sndsplit(s(z6), cons(z4, z5)), s(z6)))
166.37/63.29	
166.37/63.29	
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(136)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(true, y0, y1, y2, y3, cons(x1, x2), s(x0)) -> U7^1(false, y0, y1, y2, y3, cons(x1, x2), s(x0))
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.29	   U7^1(false, z0, z1, z2, z3, cons(z4, z5), s(z6)) -> RING(z0, z1, z2, z3, sndsplit(s(z6), cons(z4, z5)), s(z6))
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(137) DependencyGraphProof (EQUIVALENT)
166.37/63.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(138)
166.37/63.29	Obligation:
166.37/63.29	Q DP problem:
166.37/63.29	The TRS P consists of the following rules:
166.37/63.29	
166.37/63.29	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6)
166.37/63.29	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.29	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.29	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.29	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.29	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.29	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.29	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.29	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.29	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.29	
166.37/63.29	The TRS R consists of the following rules:
166.37/63.29	
166.37/63.29	   map_f(pid, nil) -> nil
166.37/63.29	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.29	   app(nil, x) -> x
166.37/63.29	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.29	   sndsplit(0, x) -> x
166.37/63.29	   sndsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(0, x) -> nil
166.37/63.29	   fstsplit(s(n), nil) -> nil
166.37/63.29	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.29	   empty(nil) -> true
166.37/63.29	   empty(cons(h, t)) -> false
166.37/63.29	   length(nil) -> 0
166.37/63.29	   length(cons(h, t)) -> s(length(t))
166.37/63.29	   leq(0, m) -> true
166.37/63.29	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.29	   leq(s(n), 0) -> false
166.37/63.29	
166.37/63.29	The set Q consists of the following terms:
166.37/63.29	
166.37/63.29	   fstsplit(0, x0)
166.37/63.29	   fstsplit(s(x0), nil)
166.37/63.29	   fstsplit(s(x0), cons(x1, x2))
166.37/63.29	   sndsplit(0, x0)
166.37/63.29	   sndsplit(s(x0), nil)
166.37/63.29	   sndsplit(s(x0), cons(x1, x2))
166.37/63.29	   empty(nil)
166.37/63.29	   empty(cons(x0, x1))
166.37/63.29	   leq(0, x0)
166.37/63.29	   leq(s(x0), 0)
166.37/63.29	   leq(s(x0), s(x1))
166.37/63.29	   length(nil)
166.37/63.29	   length(cons(x0, x1))
166.37/63.29	   app(nil, x0)
166.37/63.29	   app(cons(x0, x1), x2)
166.37/63.29	   map_f(x0, nil)
166.37/63.29	   map_f(x0, cons(x1, x2))
166.37/63.29	
166.37/63.29	We have to consider all minimal (P,Q,R)-chains.
166.37/63.29	----------------------------------------
166.37/63.29	
166.37/63.29	(139) TransformationProof (EQUIVALENT)
166.37/63.29	By forward instantiating [JAR06] the rule U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, z4, z5, z6) we obtained the following new rules [LPAR04]:
166.37/63.29	
166.37/63.29	   (U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6),U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6))
166.37/63.29	   (U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6),U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6))
166.37/63.29	   (U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6),U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6))
166.37/63.30	   (U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6),U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6))
166.37/63.30	   (U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6),U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6))
166.37/63.30	
166.37/63.30	
166.37/63.30	----------------------------------------
166.37/63.30	
166.37/63.30	(140)
166.37/63.30	Obligation:
166.37/63.30	Q DP problem:
166.37/63.30	The TRS P consists of the following rules:
166.37/63.30	
166.37/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.30	   RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.30	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.37/63.30	   U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.37/63.30	   U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.37/63.30	
166.37/63.30	The TRS R consists of the following rules:
166.37/63.30	
166.37/63.30	   map_f(pid, nil) -> nil
166.37/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.30	   app(nil, x) -> x
166.37/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.30	   sndsplit(0, x) -> x
166.37/63.30	   sndsplit(s(n), nil) -> nil
166.37/63.30	   fstsplit(0, x) -> nil
166.37/63.30	   fstsplit(s(n), nil) -> nil
166.37/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.30	   empty(nil) -> true
166.37/63.30	   empty(cons(h, t)) -> false
166.37/63.30	   length(nil) -> 0
166.37/63.30	   length(cons(h, t)) -> s(length(t))
166.37/63.30	   leq(0, m) -> true
166.37/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.30	   leq(s(n), 0) -> false
166.37/63.30	
166.37/63.30	The set Q consists of the following terms:
166.37/63.30	
166.37/63.30	   fstsplit(0, x0)
166.37/63.30	   fstsplit(s(x0), nil)
166.37/63.30	   fstsplit(s(x0), cons(x1, x2))
166.37/63.30	   sndsplit(0, x0)
166.37/63.30	   sndsplit(s(x0), nil)
166.37/63.30	   sndsplit(s(x0), cons(x1, x2))
166.37/63.30	   empty(nil)
166.37/63.30	   empty(cons(x0, x1))
166.37/63.30	   leq(0, x0)
166.37/63.30	   leq(s(x0), 0)
166.37/63.30	   leq(s(x0), s(x1))
166.37/63.30	   length(nil)
166.37/63.30	   length(cons(x0, x1))
166.37/63.30	   app(nil, x0)
166.37/63.30	   app(cons(x0, x1), x2)
166.37/63.30	   map_f(x0, nil)
166.37/63.30	   map_f(x0, cons(x1, x2))
166.37/63.30	
166.37/63.30	We have to consider all minimal (P,Q,R)-chains.
166.37/63.30	----------------------------------------
166.37/63.30	
166.37/63.30	(141) TransformationProof (EQUIVALENT)
166.37/63.30	By forward instantiating [JAR06] the rule RING(y0, y1, nil, y3, y4, y5) -> U2^1(leq(y5, 0), y0, y1, nil, y3, y4, y5) we obtained the following new rules [LPAR04]:
166.37/63.30	
166.37/63.30	   (RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4),RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4))
166.37/63.30	
166.37/63.30	
166.37/63.30	----------------------------------------
166.37/63.30	
166.37/63.30	(142)
166.37/63.30	Obligation:
166.37/63.30	Q DP problem:
166.37/63.30	The TRS P consists of the following rules:
166.37/63.30	
166.37/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.30	   RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.37/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.37/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.37/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.37/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.37/63.30	   U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.37/63.30	   U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.37/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.37/63.30	   RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4)
166.37/63.30	
166.37/63.30	The TRS R consists of the following rules:
166.37/63.30	
166.37/63.30	   map_f(pid, nil) -> nil
166.37/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.37/63.30	   app(nil, x) -> x
166.37/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.37/63.30	   sndsplit(0, x) -> x
166.37/63.30	   sndsplit(s(n), nil) -> nil
166.37/63.30	   fstsplit(0, x) -> nil
166.37/63.30	   fstsplit(s(n), nil) -> nil
166.37/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.37/63.30	   empty(nil) -> true
166.37/63.30	   empty(cons(h, t)) -> false
166.37/63.30	   length(nil) -> 0
166.37/63.30	   length(cons(h, t)) -> s(length(t))
166.37/63.30	   leq(0, m) -> true
166.37/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.37/63.30	   leq(s(n), 0) -> false
166.37/63.30	
166.37/63.30	The set Q consists of the following terms:
166.37/63.30	
166.37/63.30	   fstsplit(0, x0)
166.37/63.30	   fstsplit(s(x0), nil)
166.37/63.30	   fstsplit(s(x0), cons(x1, x2))
166.37/63.30	   sndsplit(0, x0)
166.37/63.30	   sndsplit(s(x0), nil)
166.37/63.30	   sndsplit(s(x0), cons(x1, x2))
166.37/63.30	   empty(nil)
166.37/63.30	   empty(cons(x0, x1))
166.37/63.30	   leq(0, x0)
166.37/63.30	   leq(s(x0), 0)
166.37/63.30	   leq(s(x0), s(x1))
166.37/63.30	   length(nil)
166.37/63.30	   length(cons(x0, x1))
166.37/63.30	   app(nil, x0)
166.37/63.30	   app(cons(x0, x1), x2)
166.37/63.30	   map_f(x0, nil)
166.37/63.30	   map_f(x0, cons(x1, x2))
166.37/63.30	
166.37/63.30	We have to consider all minimal (P,Q,R)-chains.
166.37/63.30	----------------------------------------
166.37/63.30	
166.37/63.30	(143) TransformationProof (EQUIVALENT)
166.37/63.30	By forward instantiating [JAR06] the rule RING(y0, y1, cons(x0, x1), y3, y4, y5) -> U2^1(leq(y5, s(length(x1))), y0, y1, cons(x0, x1), y3, y4, y5) we obtained the following new rules [LPAR04]:
166.37/63.30	
166.37/63.30	   (RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6),RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6))
166.37/63.30	
166.37/63.30	
166.37/63.30	----------------------------------------
166.37/63.30	
166.37/63.30	(144)
166.37/63.30	Obligation:
166.37/63.30	Q DP problem:
166.37/63.30	The TRS P consists of the following rules:
166.37/63.30	
166.37/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.37/63.30	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.37/63.30	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.37/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.37/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.56/63.30	   U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.56/63.30	   RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4)
166.56/63.30	   RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(145) TransformationProof (EQUIVALENT)
166.56/63.30	By forward instantiating [JAR06] the rule RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5) we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (RING(x0, x1, x2, cons(y_4, y_5), nil, x4) -> U6^1(leq(x4, 0), x0, x1, x2, cons(y_4, y_5), nil, x4),RING(x0, x1, x2, cons(y_4, y_5), nil, x4) -> U6^1(leq(x4, 0), x0, x1, x2, cons(y_4, y_5), nil, x4))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(146)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.56/63.30	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.56/63.30	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.56/63.30	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.56/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.56/63.30	   U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.56/63.30	   RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4)
166.56/63.30	   RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6)
166.56/63.30	   RING(x0, x1, x2, cons(y_4, y_5), nil, x4) -> U6^1(leq(x4, 0), x0, x1, x2, cons(y_4, y_5), nil, x4)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(147) TransformationProof (EQUIVALENT)
166.56/63.30	By forward instantiating [JAR06] the rule RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5) we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (RING(x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6) -> U6^1(leq(x6, s(length(x5))), x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6),RING(x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6) -> U6^1(leq(x6, s(length(x5))), x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(148)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.56/63.30	   U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.56/63.30	   U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.56/63.30	   U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.56/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.56/63.30	   U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.56/63.30	   U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.56/63.30	   U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.56/63.30	   U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.56/63.30	   RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4)
166.56/63.30	   RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6)
166.56/63.30	   RING(x0, x1, x2, cons(y_4, y_5), nil, x4) -> U6^1(leq(x4, 0), x0, x1, x2, cons(y_4, y_5), nil, x4)
166.56/63.30	   RING(x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6) -> U6^1(leq(x6, s(length(x5))), x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(149) QDPSizeChangeProof (EQUIVALENT)
166.56/63.30	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
166.56/63.30	
166.56/63.30	From the DPs we obtained the following set of size-change graphs:
166.56/63.30	*U4^1(false, z0, cons(z1, z2), nil, z3, z4, z5) -> RING(z0, z2, sndsplit(z5, app(map_f(two, z1), nil)), cons(fstsplit(z5, app(map_f(two, z1), nil)), z3), z4, z5)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 > 2, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(x0, cons(y_2, y_3), nil, x2, x3, x4) -> U2^1(leq(x4, 0), x0, cons(y_2, y_3), nil, x2, x3, x4)
166.56/63.30	The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5, 5 >= 6, 6 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U2^1(false, z0, cons(x1, x2), nil, z2, z3, z4) -> U4^1(empty(fstsplit(z4, app(map_f(two, x1), nil))), z0, cons(x1, x2), nil, z2, z3, z4)
166.56/63.30	The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U4^1(false, z0, cons(z1, z2), cons(z3, z4), z5, z6, z7) -> RING(z0, z2, sndsplit(z7, app(map_f(two, z1), cons(z3, z4))), cons(fstsplit(z7, app(map_f(two, z1), cons(z3, z4))), z5), z6, z7)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 > 2, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6) -> U2^1(leq(x6, s(length(x3))), x0, cons(y_2, y_3), cons(x2, x3), x4, x5, x6)
166.56/63.30	The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5, 5 >= 6, 6 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U2^1(false, z0, cons(x1, x2), cons(z2, z3), z4, z5, z6) -> U4^1(empty(fstsplit(z6, app(map_f(two, x1), cons(z2, z3)))), z0, cons(x1, x2), cons(z2, z3), z4, z5, z6)
166.56/63.30	The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U8^1(false, z0, z1, z2, cons(z3, z4), nil, z5) -> RING(z0, z1, z2, z4, sndsplit(z5, app(map_f(three, z3), nil)), z5)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*U8^1(false, z0, z1, z2, cons(z3, z4), cons(z5, z6), z7) -> RING(z0, z1, z2, z4, sndsplit(z7, app(map_f(three, z3), cons(z5, z6))), z7)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(x0, x1, x2, cons(y_4, y_5), nil, x4) -> U6^1(leq(x4, 0), x0, x1, x2, cons(y_4, y_5), nil, x4)
166.56/63.30	The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5, 5 >= 6, 6 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U6^1(false, z0, z1, z2, cons(x3, x4), nil, z4) -> U8^1(empty(fstsplit(z4, app(map_f(three, x3), nil))), z0, z1, z2, cons(x3, x4), nil, z4)
166.56/63.30	The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U6^1(false, z0, z1, z2, cons(x3, x4), cons(z4, z5), z6) -> U8^1(empty(fstsplit(z6, app(map_f(three, x3), cons(z4, z5)))), z0, z1, z2, cons(x3, x4), cons(z4, z5), z6)
166.56/63.30	The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6) -> U6^1(leq(x6, s(length(x5))), x0, x1, x2, cons(y_4, y_5), cons(x4, x5), x6)
166.56/63.30	The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5, 5 >= 6, 6 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5, 5 >= 6, 6 >= 7
166.56/63.30	
166.56/63.30	
166.56/63.30	*U9^1(true, x0, x1, cons(y_2, y_3), cons(x3, x4), x5, x6) -> RING(x0, x1, cons(y_2, y_3), x4, x5, x6)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*U9^1(true, x0, x1, nil, cons(x3, x4), x5, x6) -> RING(x0, x1, nil, x4, x5, x6)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*U9^1(true, x0, x1, x2, cons(x3, x4), cons(y_4, y_5), x6) -> RING(x0, x1, x2, x4, cons(y_4, y_5), x6)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*U9^1(true, x0, x1, x2, cons(x3, cons(y_3, y_4)), x5, x6) -> RING(x0, x1, x2, cons(y_3, y_4), x5, x6)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	*U9^1(true, x0, x1, x2, cons(x3, x4), nil, x6) -> RING(x0, x1, x2, x4, nil, x6)
166.56/63.30	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 > 4, 6 >= 5, 7 >= 6
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(150)
166.56/63.30	YES
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(151)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U2^1(leq(m, length(st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U2^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U3^1(empty(fstsplit(m, st_2)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   tail(cons(h, t)) -> t
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(152) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(153)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   tail(cons(h, t)) -> t
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(154) QDPQMonotonicMRRProof (EQUIVALENT)
166.56/63.30	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
166.56/63.30	
166.56/63.30	
166.56/63.30	Strictly oriented rules of the TRS R:
166.56/63.30	
166.56/63.30	   tail(cons(h, t)) -> t
166.56/63.30	
166.56/63.30	Used ordering: Polynomial interpretation [POLO]:
166.56/63.30	
166.56/63.30	   POL(0) = 0
166.56/63.30	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + 2*x_2 + x_3 + 2*x_4
166.56/63.30	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_3 + x_4 + 2*x_5
166.56/63.30	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_3 + x_4 + 2*x_5
166.56/63.30	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_3 + x_4 + 2*x_5
166.56/63.30	   POL(U9^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_2 + 2*x_3 + x_4 + 2*x_5
166.56/63.30	   POL(app(x_1, x_2)) = x_1 + 2*x_2
166.56/63.30	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
166.56/63.30	   POL(empty(x_1)) = 0
166.56/63.30	   POL(f(x_1, x_2)) = 0
166.56/63.30	   POL(false) = 0
166.56/63.30	   POL(fstsplit(x_1, x_2)) = 2*x_2
166.56/63.30	   POL(head(x_1)) = 1 + x_1
166.56/63.30	   POL(length(x_1)) = 0
166.56/63.30	   POL(leq(x_1, x_2)) = 0
166.56/63.30	   POL(map_f(x_1, x_2)) = 0
166.56/63.30	   POL(nil) = 0
166.56/63.30	   POL(s(x_1)) = 0
166.56/63.30	   POL(sndsplit(x_1, x_2)) = 2 + x_2
166.56/63.30	   POL(tail(x_1)) = x_1
166.56/63.30	   POL(three) = 0
166.56/63.30	   POL(true) = 0
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(155)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(156) TransformationProof (EQUIVALENT)
166.56/63.30	By instantiating [LPAR04] the rule U9^1(true, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), st_3, m) we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6),U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(157)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(y0, y1, y2, cons(x0, x1), y4, y5) -> U9^1(empty(map_f(three, x0)), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	   U9^1(true, z0, z1, z2, cons(z3, z4), z5, z6) -> RING(z0, z1, z2, tail(cons(z3, z4)), z5, z6)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(158) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(159)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(160) NonInfProof (EQUIVALENT)
166.56/63.30	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
166.56/63.30	
166.56/63.30	Note that final constraints are written in bold face.
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	For Pair U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.56/63.30	*We consider the chain RING(x12, x13, x14, x15, x16, x17) -> U6^1(leq(x17, length(x16)), x12, x13, x14, x15, x16, x17), U6^1(true, x18, x19, x20, x21, x22, x23) -> U7^1(empty(fstsplit(x23, x22)), x18, x19, x20, x21, x22, x23) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (U6^1(leq(x17, length(x16)), x12, x13, x14, x15, x16, x17)=U6^1(true, x18, x19, x20, x21, x22, x23)  ==>  U6^1(true, x18, x19, x20, x21, x22, x23)_>=_U7^1(empty(fstsplit(x23, x22)), x18, x19, x20, x21, x22, x23))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (length(x16)=x186 & leq(x17, x186)=true  ==>  U6^1(true, x12, x13, x14, x15, x16, x17)_>=_U7^1(empty(fstsplit(x17, x16)), x12, x13, x14, x15, x16, x17))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x17, x186)=true which results in the following new constraints:
166.56/63.30	
166.56/63.30	(3)    (true=true & length(x16)=x187  ==>  U6^1(true, x12, x13, x14, x15, x16, 0)_>=_U7^1(empty(fstsplit(0, x16)), x12, x13, x14, x15, x16, 0))
166.56/63.30	
166.56/63.30	(4)    (leq(x190, x189)=true & length(x16)=s(x189) & (\/x191,x192,x193,x194,x195:leq(x190, x189)=true & length(x191)=x189  ==>  U6^1(true, x192, x193, x194, x195, x191, x190)_>=_U7^1(empty(fstsplit(x190, x191)), x192, x193, x194, x195, x191, x190))  ==>  U6^1(true, x12, x13, x14, x15, x16, s(x190))_>=_U7^1(empty(fstsplit(s(x190), x16)), x12, x13, x14, x15, x16, s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(5)    (U6^1(true, x12, x13, x14, x15, x16, 0)_>=_U7^1(empty(fstsplit(0, x16)), x12, x13, x14, x15, x16, 0))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x16)=s(x189) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(6)    (s(length(x196))=s(x189) & leq(x190, x189)=true & (\/x191,x192,x193,x194,x195:leq(x190, x189)=true & length(x191)=x189  ==>  U6^1(true, x192, x193, x194, x195, x191, x190)_>=_U7^1(empty(fstsplit(x190, x191)), x192, x193, x194, x195, x191, x190)) & (\/x198,x199,x200,x201,x202,x203,x204,x205,x206,x207,x208:length(x196)=s(x198) & leq(x199, x198)=true & (\/x200,x201,x202,x203,x204:leq(x199, x198)=true & length(x200)=x198  ==>  U6^1(true, x201, x202, x203, x204, x200, x199)_>=_U7^1(empty(fstsplit(x199, x200)), x201, x202, x203, x204, x200, x199))  ==>  U6^1(true, x205, x206, x207, x208, x196, s(x199))_>=_U7^1(empty(fstsplit(s(x199), x196)), x205, x206, x207, x208, x196, s(x199)))  ==>  U6^1(true, x12, x13, x14, x15, cons(x197, x196), s(x190))_>=_U7^1(empty(fstsplit(s(x190), cons(x197, x196))), x12, x13, x14, x15, cons(x197, x196), s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(7)    (length(x196)=x189 & leq(x190, x189)=true & (\/x191,x192,x193,x194,x195:leq(x190, x189)=true & length(x191)=x189  ==>  U6^1(true, x192, x193, x194, x195, x191, x190)_>=_U7^1(empty(fstsplit(x190, x191)), x192, x193, x194, x195, x191, x190)) & (\/x198,x199,x200,x201,x202,x203,x204,x205,x206,x207,x208:length(x196)=s(x198) & leq(x199, x198)=true & (\/x200,x201,x202,x203,x204:leq(x199, x198)=true & length(x200)=x198  ==>  U6^1(true, x201, x202, x203, x204, x200, x199)_>=_U7^1(empty(fstsplit(x199, x200)), x201, x202, x203, x204, x200, x199))  ==>  U6^1(true, x205, x206, x207, x208, x196, s(x199))_>=_U7^1(empty(fstsplit(s(x199), x196)), x205, x206, x207, x208, x196, s(x199)))  ==>  U6^1(true, x12, x13, x14, x15, cons(x197, x196), s(x190))_>=_U7^1(empty(fstsplit(s(x190), cons(x197, x196))), x12, x13, x14, x15, cons(x197, x196), s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (7) using rule (VI) where we applied the induction hypothesis (\/x191,x192,x193,x194,x195:leq(x190, x189)=true & length(x191)=x189  ==>  U6^1(true, x192, x193, x194, x195, x191, x190)_>=_U7^1(empty(fstsplit(x190, x191)), x192, x193, x194, x195, x191, x190)) with sigma = [x191 / x196, x192 / x12, x193 / x13, x194 / x14, x195 / x15] which results in the following new constraint:
166.56/63.30	
166.56/63.30	(8)    (U6^1(true, x12, x13, x14, x15, x196, x190)_>=_U7^1(empty(fstsplit(x190, x196)), x12, x13, x14, x15, x196, x190) & (\/x198,x199,x200,x201,x202,x203,x204,x205,x206,x207,x208:length(x196)=s(x198) & leq(x199, x198)=true & (\/x200,x201,x202,x203,x204:leq(x199, x198)=true & length(x200)=x198  ==>  U6^1(true, x201, x202, x203, x204, x200, x199)_>=_U7^1(empty(fstsplit(x199, x200)), x201, x202, x203, x204, x200, x199))  ==>  U6^1(true, x205, x206, x207, x208, x196, s(x199))_>=_U7^1(empty(fstsplit(s(x199), x196)), x205, x206, x207, x208, x196, s(x199)))  ==>  U6^1(true, x12, x13, x14, x15, cons(x197, x196), s(x190))_>=_U7^1(empty(fstsplit(s(x190), cons(x197, x196))), x12, x13, x14, x15, cons(x197, x196), s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (8) using rule (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(9)    (U6^1(true, x12, x13, x14, x15, x196, x190)_>=_U7^1(empty(fstsplit(x190, x196)), x12, x13, x14, x15, x196, x190)  ==>  U6^1(true, x12, x13, x14, x15, cons(x197, x196), s(x190))_>=_U7^1(empty(fstsplit(s(x190), cons(x197, x196))), x12, x13, x14, x15, cons(x197, x196), s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	For Pair U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m) the following chains were created:
166.56/63.30	*We consider the chain U6^1(true, x36, x37, x38, x39, x40, x41) -> U7^1(empty(fstsplit(x41, x40)), x36, x37, x38, x39, x40, x41), U7^1(false, x42, x43, x44, x45, x46, x47) -> RING(x42, x43, x44, x45, sndsplit(x47, x46), x47) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (U7^1(empty(fstsplit(x41, x40)), x36, x37, x38, x39, x40, x41)=U7^1(false, x42, x43, x44, x45, x46, x47)  ==>  U7^1(false, x42, x43, x44, x45, x46, x47)_>=_RING(x42, x43, x44, x45, sndsplit(x47, x46), x47))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (fstsplit(x41, x40)=x209 & empty(x209)=false  ==>  U7^1(false, x36, x37, x38, x39, x40, x41)_>=_RING(x36, x37, x38, x39, sndsplit(x41, x40), x41))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x209)=false which results in the following new constraint:
166.56/63.30	
166.56/63.30	(3)    (false=false & fstsplit(x41, x40)=cons(x211, x210)  ==>  U7^1(false, x36, x37, x38, x39, x40, x41)_>=_RING(x36, x37, x38, x39, sndsplit(x41, x40), x41))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(4)    (fstsplit(x41, x40)=cons(x211, x210)  ==>  U7^1(false, x36, x37, x38, x39, x40, x41)_>=_RING(x36, x37, x38, x39, sndsplit(x41, x40), x41))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x41, x40)=cons(x211, x210) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(5)    (cons(x215, fstsplit(x216, x214))=cons(x211, x210) & (\/x217,x218,x219,x220,x221,x222:fstsplit(x216, x214)=cons(x217, x218)  ==>  U7^1(false, x219, x220, x221, x222, x214, x216)_>=_RING(x219, x220, x221, x222, sndsplit(x216, x214), x216))  ==>  U7^1(false, x36, x37, x38, x39, cons(x215, x214), s(x216))_>=_RING(x36, x37, x38, x39, sndsplit(s(x216), cons(x215, x214)), s(x216)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(6)    (U7^1(false, x36, x37, x38, x39, cons(x215, x214), s(x216))_>=_RING(x36, x37, x38, x39, sndsplit(s(x216), cons(x215, x214)), s(x216)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	For Pair RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.56/63.30	*We consider the chain U7^1(false, x78, x79, x80, x81, x82, x83) -> RING(x78, x79, x80, x81, sndsplit(x83, x82), x83), RING(x84, x85, x86, x87, x88, x89) -> U6^1(leq(x89, length(x88)), x84, x85, x86, x87, x88, x89) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (RING(x78, x79, x80, x81, sndsplit(x83, x82), x83)=RING(x84, x85, x86, x87, x88, x89)  ==>  RING(x84, x85, x86, x87, x88, x89)_>=_U6^1(leq(x89, length(x88)), x84, x85, x86, x87, x88, x89))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (RING(x78, x79, x80, x81, x88, x83)_>=_U6^1(leq(x83, length(x88)), x78, x79, x80, x81, x88, x83))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	*We consider the chain U8^1(false, x102, x103, x104, x105, x106, x107) -> RING(x102, x103, x104, tail(x105), sndsplit(x107, app(map_f(three, head(x105)), x106)), x107), RING(x108, x109, x110, x111, x112, x113) -> U6^1(leq(x113, length(x112)), x108, x109, x110, x111, x112, x113) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (RING(x102, x103, x104, tail(x105), sndsplit(x107, app(map_f(three, head(x105)), x106)), x107)=RING(x108, x109, x110, x111, x112, x113)  ==>  RING(x108, x109, x110, x111, x112, x113)_>=_U6^1(leq(x113, length(x112)), x108, x109, x110, x111, x112, x113))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (RING(x102, x103, x104, tail(x105), x112, x107)_>=_U6^1(leq(x107, length(x112)), x102, x103, x104, tail(x105), x112, x107))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	For Pair U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m) the following chains were created:
166.56/63.30	*We consider the chain RING(x126, x127, x128, x129, x130, x131) -> U6^1(leq(x131, length(x130)), x126, x127, x128, x129, x130, x131), U6^1(false, x132, x133, x134, x135, x136, x137) -> U8^1(empty(fstsplit(x137, app(map_f(three, head(x135)), x136))), x132, x133, x134, x135, x136, x137) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (U6^1(leq(x131, length(x130)), x126, x127, x128, x129, x130, x131)=U6^1(false, x132, x133, x134, x135, x136, x137)  ==>  U6^1(false, x132, x133, x134, x135, x136, x137)_>=_U8^1(empty(fstsplit(x137, app(map_f(three, head(x135)), x136))), x132, x133, x134, x135, x136, x137))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (length(x130)=x227 & leq(x131, x227)=false  ==>  U6^1(false, x126, x127, x128, x129, x130, x131)_>=_U8^1(empty(fstsplit(x131, app(map_f(three, head(x129)), x130))), x126, x127, x128, x129, x130, x131))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on leq(x131, x227)=false which results in the following new constraints:
166.56/63.30	
166.56/63.30	(3)    (false=false & length(x130)=0  ==>  U6^1(false, x126, x127, x128, x129, x130, s(x229))_>=_U8^1(empty(fstsplit(s(x229), app(map_f(three, head(x129)), x130))), x126, x127, x128, x129, x130, s(x229)))
166.56/63.30	
166.56/63.30	(4)    (leq(x231, x230)=false & length(x130)=s(x230) & (\/x232,x233,x234,x235,x236:leq(x231, x230)=false & length(x232)=x230  ==>  U6^1(false, x233, x234, x235, x236, x232, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x236)), x232))), x233, x234, x235, x236, x232, x231))  ==>  U6^1(false, x126, x127, x128, x129, x130, s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), x130))), x126, x127, x128, x129, x130, s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(5)    (length(x130)=0  ==>  U6^1(false, x126, x127, x128, x129, x130, s(x229))_>=_U8^1(empty(fstsplit(s(x229), app(map_f(three, head(x129)), x130))), x126, x127, x128, x129, x130, s(x229)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on length(x130)=s(x230) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(6)    (s(length(x239))=s(x230) & leq(x231, x230)=false & (\/x232,x233,x234,x235,x236:leq(x231, x230)=false & length(x232)=x230  ==>  U6^1(false, x233, x234, x235, x236, x232, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x236)), x232))), x233, x234, x235, x236, x232, x231)) & (\/x241,x242,x243,x244,x245,x246,x247,x248,x249,x250,x251:length(x239)=s(x241) & leq(x242, x241)=false & (\/x243,x244,x245,x246,x247:leq(x242, x241)=false & length(x243)=x241  ==>  U6^1(false, x244, x245, x246, x247, x243, x242)_>=_U8^1(empty(fstsplit(x242, app(map_f(three, head(x247)), x243))), x244, x245, x246, x247, x243, x242))  ==>  U6^1(false, x248, x249, x250, x251, x239, s(x242))_>=_U8^1(empty(fstsplit(s(x242), app(map_f(three, head(x251)), x239))), x248, x249, x250, x251, x239, s(x242)))  ==>  U6^1(false, x126, x127, x128, x129, cons(x240, x239), s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), cons(x240, x239)))), x126, x127, x128, x129, cons(x240, x239), s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on length(x130)=0 which results in the following new constraint:
166.56/63.30	
166.56/63.30	(7)    (0=0  ==>  U6^1(false, x126, x127, x128, x129, nil, s(x229))_>=_U8^1(empty(fstsplit(s(x229), app(map_f(three, head(x129)), nil))), x126, x127, x128, x129, nil, s(x229)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(8)    (U6^1(false, x126, x127, x128, x129, nil, s(x229))_>=_U8^1(empty(fstsplit(s(x229), app(map_f(three, head(x129)), nil))), x126, x127, x128, x129, nil, s(x229)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (6) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(9)    (length(x239)=x230 & leq(x231, x230)=false & (\/x232,x233,x234,x235,x236:leq(x231, x230)=false & length(x232)=x230  ==>  U6^1(false, x233, x234, x235, x236, x232, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x236)), x232))), x233, x234, x235, x236, x232, x231)) & (\/x241,x242,x243,x244,x245,x246,x247,x248,x249,x250,x251:length(x239)=s(x241) & leq(x242, x241)=false & (\/x243,x244,x245,x246,x247:leq(x242, x241)=false & length(x243)=x241  ==>  U6^1(false, x244, x245, x246, x247, x243, x242)_>=_U8^1(empty(fstsplit(x242, app(map_f(three, head(x247)), x243))), x244, x245, x246, x247, x243, x242))  ==>  U6^1(false, x248, x249, x250, x251, x239, s(x242))_>=_U8^1(empty(fstsplit(s(x242), app(map_f(three, head(x251)), x239))), x248, x249, x250, x251, x239, s(x242)))  ==>  U6^1(false, x126, x127, x128, x129, cons(x240, x239), s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), cons(x240, x239)))), x126, x127, x128, x129, cons(x240, x239), s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (9) using rule (VI) where we applied the induction hypothesis (\/x232,x233,x234,x235,x236:leq(x231, x230)=false & length(x232)=x230  ==>  U6^1(false, x233, x234, x235, x236, x232, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x236)), x232))), x233, x234, x235, x236, x232, x231)) with sigma = [x232 / x239, x233 / x126, x234 / x127, x235 / x128, x236 / x129] which results in the following new constraint:
166.56/63.30	
166.56/63.30	(10)    (U6^1(false, x126, x127, x128, x129, x239, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x129)), x239))), x126, x127, x128, x129, x239, x231) & (\/x241,x242,x243,x244,x245,x246,x247,x248,x249,x250,x251:length(x239)=s(x241) & leq(x242, x241)=false & (\/x243,x244,x245,x246,x247:leq(x242, x241)=false & length(x243)=x241  ==>  U6^1(false, x244, x245, x246, x247, x243, x242)_>=_U8^1(empty(fstsplit(x242, app(map_f(three, head(x247)), x243))), x244, x245, x246, x247, x243, x242))  ==>  U6^1(false, x248, x249, x250, x251, x239, s(x242))_>=_U8^1(empty(fstsplit(s(x242), app(map_f(three, head(x251)), x239))), x248, x249, x250, x251, x239, s(x242)))  ==>  U6^1(false, x126, x127, x128, x129, cons(x240, x239), s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), cons(x240, x239)))), x126, x127, x128, x129, cons(x240, x239), s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (10) using rule (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(11)    (U6^1(false, x126, x127, x128, x129, x239, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x129)), x239))), x126, x127, x128, x129, x239, x231)  ==>  U6^1(false, x126, x127, x128, x129, cons(x240, x239), s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), cons(x240, x239)))), x126, x127, x128, x129, cons(x240, x239), s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	For Pair U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) the following chains were created:
166.56/63.30	*We consider the chain U6^1(false, x168, x169, x170, x171, x172, x173) -> U8^1(empty(fstsplit(x173, app(map_f(three, head(x171)), x172))), x168, x169, x170, x171, x172, x173), U8^1(false, x174, x175, x176, x177, x178, x179) -> RING(x174, x175, x176, tail(x177), sndsplit(x179, app(map_f(three, head(x177)), x178)), x179) which results in the following constraint:
166.56/63.30	
166.56/63.30	(1)    (U8^1(empty(fstsplit(x173, app(map_f(three, head(x171)), x172))), x168, x169, x170, x171, x172, x173)=U8^1(false, x174, x175, x176, x177, x178, x179)  ==>  U8^1(false, x174, x175, x176, x177, x178, x179)_>=_RING(x174, x175, x176, tail(x177), sndsplit(x179, app(map_f(three, head(x177)), x178)), x179))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(2)    (fstsplit(x173, x253)=x252 & empty(x252)=false  ==>  U8^1(false, x168, x169, x170, x171, x172, x173)_>=_RING(x168, x169, x170, tail(x171), sndsplit(x173, app(map_f(three, head(x171)), x172)), x173))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on empty(x252)=false which results in the following new constraint:
166.56/63.30	
166.56/63.30	(3)    (false=false & fstsplit(x173, x253)=cons(x258, x257)  ==>  U8^1(false, x168, x169, x170, x171, x172, x173)_>=_RING(x168, x169, x170, tail(x171), sndsplit(x173, app(map_f(three, head(x171)), x172)), x173))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(4)    (fstsplit(x173, x253)=cons(x258, x257)  ==>  U8^1(false, x168, x169, x170, x171, x172, x173)_>=_RING(x168, x169, x170, tail(x171), sndsplit(x173, app(map_f(three, head(x171)), x172)), x173))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on fstsplit(x173, x253)=cons(x258, x257) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(5)    (cons(x262, fstsplit(x263, x261))=cons(x258, x257) & (\/x264,x265,x266,x267,x268,x269,x270:fstsplit(x263, x261)=cons(x264, x265)  ==>  U8^1(false, x266, x267, x268, x269, x270, x263)_>=_RING(x266, x267, x268, tail(x269), sndsplit(x263, app(map_f(three, head(x269)), x270)), x263))  ==>  U8^1(false, x168, x169, x170, x171, x172, s(x263))_>=_RING(x168, x169, x170, tail(x171), sndsplit(s(x263), app(map_f(three, head(x171)), x172)), s(x263)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
166.56/63.30	
166.56/63.30	(6)    (U8^1(false, x168, x169, x170, x171, x172, s(x263))_>=_RING(x168, x169, x170, tail(x171), sndsplit(s(x263), app(map_f(three, head(x171)), x172)), s(x263)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	To summarize, we get the following constraints P__>=_ for the following pairs.
166.56/63.30	
166.56/63.30	*U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	
166.56/63.30	*(U6^1(true, x12, x13, x14, x15, x16, 0)_>=_U7^1(empty(fstsplit(0, x16)), x12, x13, x14, x15, x16, 0))
166.56/63.30	
166.56/63.30	
166.56/63.30	*(U6^1(true, x12, x13, x14, x15, x196, x190)_>=_U7^1(empty(fstsplit(x190, x196)), x12, x13, x14, x15, x196, x190)  ==>  U6^1(true, x12, x13, x14, x15, cons(x197, x196), s(x190))_>=_U7^1(empty(fstsplit(s(x190), cons(x197, x196))), x12, x13, x14, x15, cons(x197, x196), s(x190)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	*U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	
166.56/63.30	*(U7^1(false, x36, x37, x38, x39, cons(x215, x214), s(x216))_>=_RING(x36, x37, x38, x39, sndsplit(s(x216), cons(x215, x214)), s(x216)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	*RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	
166.56/63.30	*(RING(x78, x79, x80, x81, x88, x83)_>=_U6^1(leq(x83, length(x88)), x78, x79, x80, x81, x88, x83))
166.56/63.30	
166.56/63.30	
166.56/63.30	*(RING(x102, x103, x104, tail(x105), x112, x107)_>=_U6^1(leq(x107, length(x112)), x102, x103, x104, tail(x105), x112, x107))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	*U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	
166.56/63.30	*(U6^1(false, x126, x127, x128, x129, nil, s(x229))_>=_U8^1(empty(fstsplit(s(x229), app(map_f(three, head(x129)), nil))), x126, x127, x128, x129, nil, s(x229)))
166.56/63.30	
166.56/63.30	
166.56/63.30	*(U6^1(false, x126, x127, x128, x129, x239, x231)_>=_U8^1(empty(fstsplit(x231, app(map_f(three, head(x129)), x239))), x126, x127, x128, x129, x239, x231)  ==>  U6^1(false, x126, x127, x128, x129, cons(x240, x239), s(x231))_>=_U8^1(empty(fstsplit(s(x231), app(map_f(three, head(x129)), cons(x240, x239)))), x126, x127, x128, x129, cons(x240, x239), s(x231)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	*U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	*(U8^1(false, x168, x169, x170, x171, x172, s(x263))_>=_RING(x168, x169, x170, tail(x171), sndsplit(s(x263), app(map_f(three, head(x171)), x172)), s(x263)))
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	
166.56/63.30	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
166.56/63.30	
166.56/63.30	Using the following integer polynomial ordering the  resulting constraints can be solved 
166.56/63.30	
166.56/63.30	Polynomial interpretation [NONINF]:
166.56/63.30	
166.56/63.30	   POL(0) = 0
166.56/63.30	   POL(RING(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + x_4 - x_5 - x_6
166.56/63.30	   POL(U6^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 1 - x_1 + x_5 - x_6 - x_7
166.56/63.30	   POL(U7^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -x_1 + x_5 - x_6 - x_7
166.56/63.30	   POL(U8^1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -x_1 + x_5 - x_7
166.56/63.30	   POL(app(x_1, x_2)) = 0
166.56/63.30	   POL(c) = -1
166.56/63.30	   POL(cons(x_1, x_2)) = 0
166.56/63.30	   POL(empty(x_1)) = 0
166.56/63.30	   POL(f(x_1, x_2)) = x_1 + x_2
166.56/63.30	   POL(false) = 0
166.56/63.30	   POL(fstsplit(x_1, x_2)) = 0
166.56/63.30	   POL(head(x_1)) = 0
166.56/63.30	   POL(length(x_1)) = 0
166.56/63.30	   POL(leq(x_1, x_2)) = 0
166.56/63.30	   POL(map_f(x_1, x_2)) = 0
166.56/63.30	   POL(nil) = 1
166.56/63.30	   POL(s(x_1)) = 1
166.56/63.30	   POL(sndsplit(x_1, x_2)) = x_1
166.56/63.30	   POL(tail(x_1)) = 0
166.56/63.30	   POL(three) = 0
166.56/63.30	   POL(true) = 0
166.56/63.30	
166.56/63.30	
166.56/63.30	The following pairs  are in P_>:
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	The following pairs are in P_bound:
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	The following rules are usable:
166.56/63.30	   true -> empty(nil)
166.56/63.30	   false -> empty(cons(h, t))
166.56/63.30	   x -> sndsplit(0, x)
166.56/63.30	   nil -> sndsplit(s(n), nil)
166.56/63.30	   true -> leq(0, m)
166.56/63.30	   false -> leq(s(n), 0)
166.56/63.30	   leq(n, m) -> leq(s(n), s(m))
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(161)
166.56/63.30	Complex Obligation (AND)
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(162)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U7^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(163) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(164)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(165) TransformationProof (EQUIVALENT)
166.56/63.30	By narrowing [LPAR04] the rule U6^1(false, st_1, in_2, st_2, in_3, st_3, m) -> U8^1(empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0),U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0))
166.56/63.30	   (U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5),U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(166)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(false, y0, y1, y2, y3, y4, 0) -> U8^1(empty(nil), y0, y1, y2, y3, y4, 0)
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(167) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(168)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(169) TransformationProof (EQUIVALENT)
166.56/63.30	By narrowing [LPAR04] the rule RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m) at position [0] we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0),RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0))
166.56/63.30	   (RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5),RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5))
166.56/63.30	   (RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5),RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(170)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	   RING(y0, y1, y2, y3, y4, 0) -> U6^1(true, y0, y1, y2, y3, y4, 0)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(171) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(172)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(173) TransformationProof (EQUIVALENT)
166.56/63.30	By narrowing [LPAR04] the rule U8^1(false, st_1, in_2, st_2, in_3, st_3, m) -> RING(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m) at position [4] we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0),U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0))
166.56/63.30	   (U8^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> RING(y0, y1, y2, tail(cons(x0, x1)), sndsplit(y5, app(map_f(three, x0), y4)), y5),U8^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> RING(y0, y1, y2, tail(cons(x0, x1)), sndsplit(y5, app(map_f(three, x0), y4)), y5))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(174)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	   U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0)
166.56/63.30	   U8^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> RING(y0, y1, y2, tail(cons(x0, x1)), sndsplit(y5, app(map_f(three, x0), y4)), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(175) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(176)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(177) UsableRulesProof (EQUIVALENT)
166.56/63.30	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(178)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(179) QReductionProof (EQUIVALENT)
166.56/63.30	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.56/63.30	
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(180)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(181) TransformationProof (EQUIVALENT)
166.56/63.30	By narrowing [LPAR04] the rule U8^1(false, y0, y1, y2, y3, y4, 0) -> RING(y0, y1, y2, tail(y3), app(map_f(three, head(y3)), y4), 0) at position [4] we obtained the following new rules [LPAR04]:
166.56/63.30	
166.56/63.30	   (U8^1(false, y0, y1, y2, cons(x0, x1), y4, 0) -> RING(y0, y1, y2, tail(cons(x0, x1)), app(map_f(three, x0), y4), 0),U8^1(false, y0, y1, y2, cons(x0, x1), y4, 0) -> RING(y0, y1, y2, tail(cons(x0, x1)), app(map_f(three, x0), y4), 0))
166.56/63.30	
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(182)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(false, y0, y1, y2, cons(x0, x1), y4, y5) -> U8^1(empty(fstsplit(y5, app(map_f(three, x0), y4))), y0, y1, y2, cons(x0, x1), y4, y5)
166.56/63.30	   RING(y0, y1, y2, y3, nil, y5) -> U6^1(leq(y5, 0), y0, y1, y2, y3, nil, y5)
166.56/63.30	   RING(y0, y1, y2, y3, cons(x0, x1), y5) -> U6^1(leq(y5, s(length(x1))), y0, y1, y2, y3, cons(x0, x1), y5)
166.56/63.30	   U8^1(false, y0, y1, y2, cons(x0, x1), y4, 0) -> RING(y0, y1, y2, tail(cons(x0, x1)), app(map_f(three, x0), y4), 0)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(183) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(184)
166.56/63.30	TRUE
166.56/63.30	
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(185)
166.56/63.30	Obligation:
166.56/63.30	Q DP problem:
166.56/63.30	The TRS P consists of the following rules:
166.56/63.30	
166.56/63.30	   U6^1(true, st_1, in_2, st_2, in_3, st_3, m) -> U7^1(empty(fstsplit(m, st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	   RING(st_1, in_2, st_2, in_3, st_3, m) -> U6^1(leq(m, length(st_3)), st_1, in_2, st_2, in_3, st_3, m)
166.56/63.30	
166.56/63.30	The TRS R consists of the following rules:
166.56/63.30	
166.56/63.30	   sndsplit(0, x) -> x
166.56/63.30	   sndsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(0, x) -> nil
166.56/63.30	   fstsplit(s(n), nil) -> nil
166.56/63.30	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
166.56/63.30	   empty(nil) -> true
166.56/63.30	   empty(cons(h, t)) -> false
166.56/63.30	   head(cons(h, t)) -> h
166.56/63.30	   map_f(pid, nil) -> nil
166.56/63.30	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
166.56/63.30	   app(nil, x) -> x
166.56/63.30	   app(cons(h, t), x) -> cons(h, app(t, x))
166.56/63.30	   length(nil) -> 0
166.56/63.30	   length(cons(h, t)) -> s(length(t))
166.56/63.30	   leq(0, m) -> true
166.56/63.30	   leq(s(n), 0) -> false
166.56/63.30	   leq(s(n), s(m)) -> leq(n, m)
166.56/63.30	
166.56/63.30	The set Q consists of the following terms:
166.56/63.30	
166.56/63.30	   fstsplit(0, x0)
166.56/63.30	   fstsplit(s(x0), nil)
166.56/63.30	   fstsplit(s(x0), cons(x1, x2))
166.56/63.30	   sndsplit(0, x0)
166.56/63.30	   sndsplit(s(x0), nil)
166.56/63.30	   sndsplit(s(x0), cons(x1, x2))
166.56/63.30	   empty(nil)
166.56/63.30	   empty(cons(x0, x1))
166.56/63.30	   leq(0, x0)
166.56/63.30	   leq(s(x0), 0)
166.56/63.30	   leq(s(x0), s(x1))
166.56/63.30	   length(nil)
166.56/63.30	   length(cons(x0, x1))
166.56/63.30	   app(nil, x0)
166.56/63.30	   app(cons(x0, x1), x2)
166.56/63.30	   map_f(x0, nil)
166.56/63.30	   map_f(x0, cons(x1, x2))
166.56/63.30	   head(cons(x0, x1))
166.56/63.30	   tail(cons(x0, x1))
166.56/63.30	
166.56/63.30	We have to consider all minimal (P,Q,R)-chains.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(186) DependencyGraphProof (EQUIVALENT)
166.56/63.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
166.56/63.30	----------------------------------------
166.56/63.30	
166.56/63.30	(187)
166.56/63.30	TRUE
166.56/63.35	EOF