8.02/2.93 MAYBE 8.02/2.96 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 8.02/2.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.02/2.96 8.02/2.96 8.02/2.96 Left Termination of the query pattern 8.02/2.96 8.02/2.96 a() 8.02/2.96 8.02/2.96 w.r.t. the given Prolog program could not be shown: 8.02/2.96 8.02/2.96 (0) Prolog 8.02/2.96 (1) PrologToPiTRSProof [SOUND, 0 ms] 8.02/2.96 (2) PiTRS 8.02/2.96 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 8.02/2.96 (4) PiDP 8.02/2.96 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (6) AND 8.02/2.96 (7) PiDP 8.02/2.96 (8) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (9) PiDP 8.02/2.96 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 8.02/2.96 (11) QDP 8.02/2.96 (12) NonTerminationLoopProof [COMPLETE, 0 ms] 8.02/2.96 (13) NO 8.02/2.96 (14) PiDP 8.02/2.96 (15) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (16) PiDP 8.02/2.96 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 8.02/2.96 (18) QDP 8.02/2.96 (19) PrologToPiTRSProof [SOUND, 0 ms] 8.02/2.96 (20) PiTRS 8.02/2.96 (21) DependencyPairsProof [EQUIVALENT, 0 ms] 8.02/2.96 (22) PiDP 8.02/2.96 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (24) AND 8.02/2.96 (25) PiDP 8.02/2.96 (26) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (27) PiDP 8.02/2.96 (28) PiDPToQDPProof [EQUIVALENT, 1 ms] 8.02/2.96 (29) QDP 8.02/2.96 (30) NonTerminationLoopProof [COMPLETE, 0 ms] 8.02/2.96 (31) NO 8.02/2.96 (32) PiDP 8.02/2.96 (33) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (34) PiDP 8.02/2.96 (35) PiDPToQDPProof [EQUIVALENT, 0 ms] 8.02/2.96 (36) QDP 8.02/2.96 (37) PrologToTRSTransformerProof [SOUND, 0 ms] 8.02/2.96 (38) QTRS 8.02/2.96 (39) QTRSRRRProof [EQUIVALENT, 44 ms] 8.02/2.96 (40) QTRS 8.02/2.96 (41) QTRSRRRProof [EQUIVALENT, 5 ms] 8.02/2.96 (42) QTRS 8.02/2.96 (43) Overlay + Local Confluence [EQUIVALENT, 2 ms] 8.02/2.96 (44) QTRS 8.02/2.96 (45) DependencyPairsProof [EQUIVALENT, 0 ms] 8.02/2.96 (46) QDP 8.02/2.96 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (48) AND 8.02/2.96 (49) QDP 8.02/2.96 (50) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (51) QDP 8.02/2.96 (52) QReductionProof [EQUIVALENT, 1 ms] 8.02/2.96 (53) QDP 8.02/2.96 (54) QDP 8.02/2.96 (55) UsableRulesProof [EQUIVALENT, 0 ms] 8.02/2.96 (56) QDP 8.02/2.96 (57) QReductionProof [EQUIVALENT, 0 ms] 8.02/2.96 (58) QDP 8.02/2.96 (59) NonTerminationLoopProof [COMPLETE, 0 ms] 8.02/2.96 (60) NO 8.02/2.96 (61) PrologToDTProblemTransformerProof [SOUND, 0 ms] 8.02/2.96 (62) TRIPLES 8.02/2.96 (63) TriplesToPiDPProof [SOUND, 0 ms] 8.02/2.96 (64) PiDP 8.02/2.96 (65) DependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (66) AND 8.02/2.96 (67) PiDP 8.02/2.96 (68) PiDPToQDPProof [EQUIVALENT, 0 ms] 8.02/2.96 (69) QDP 8.02/2.96 (70) NonTerminationLoopProof [COMPLETE, 0 ms] 8.02/2.96 (71) NO 8.02/2.96 (72) PiDP 8.02/2.96 (73) PiDPToQDPProof [EQUIVALENT, 0 ms] 8.02/2.96 (74) QDP 8.02/2.96 (75) PrologToIRSwTTransformerProof [SOUND, 0 ms] 8.02/2.96 (76) AND 8.02/2.96 (77) IRSwT 8.02/2.96 (78) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (79) IRSwT 8.02/2.96 (80) IntTRSCompressionProof [EQUIVALENT, 21 ms] 8.02/2.96 (81) IRSwT 8.02/2.96 (82) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 8.02/2.96 (83) IRSwT 8.02/2.96 (84) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] 8.02/2.96 (85) IRSwT 8.02/2.96 (86) FilterProof [EQUIVALENT, 0 ms] 8.02/2.96 (87) IntTRS 8.02/2.96 (88) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] 8.02/2.96 (89) NO 8.02/2.96 (90) IRSwT 8.02/2.96 (91) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (92) IRSwT 8.02/2.96 (93) IntTRSCompressionProof [EQUIVALENT, 3 ms] 8.02/2.96 (94) IRSwT 8.02/2.96 (95) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 8.02/2.96 (96) IRSwT 8.02/2.96 (97) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] 8.02/2.96 (98) IRSwT 8.02/2.96 (99) FilterProof [EQUIVALENT, 0 ms] 8.02/2.96 (100) IntTRS 8.02/2.96 (101) IntTRSPeriodicNontermProof [COMPLETE, 4 ms] 8.02/2.96 (102) NO 8.02/2.96 8.02/2.96 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (0) 8.02/2.96 Obligation: 8.02/2.96 Clauses: 8.02/2.96 8.02/2.96 a :- b. 8.02/2.96 a :- e. 8.02/2.96 b :- c. 8.02/2.96 c :- d. 8.02/2.96 d :- b. 8.02/2.96 e :- f. 8.02/2.96 f :- g. 8.02/2.96 g :- e. 8.02/2.96 8.02/2.96 8.02/2.96 Query: a() 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (1) PrologToPiTRSProof (SOUND) 8.02/2.96 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 8.02/2.96 8.02/2.96 Transforming Prolog into the following Term Rewriting System: 8.02/2.96 8.02/2.96 Pi-finite rewrite system: 8.02/2.96 The TRS R consists of the following rules: 8.02/2.96 8.02/2.96 a_in_ -> U1_(b_in_) 8.02/2.96 b_in_ -> U3_(c_in_) 8.02/2.96 c_in_ -> U4_(d_in_) 8.02/2.96 d_in_ -> U5_(b_in_) 8.02/2.96 U5_(b_out_) -> d_out_ 8.02/2.96 U4_(d_out_) -> c_out_ 8.02/2.96 U3_(c_out_) -> b_out_ 8.02/2.96 U1_(b_out_) -> a_out_ 8.02/2.96 a_in_ -> U2_(e_in_) 8.02/2.96 e_in_ -> U6_(f_in_) 8.02/2.96 f_in_ -> U7_(g_in_) 8.02/2.96 g_in_ -> U8_(e_in_) 8.02/2.96 U8_(e_out_) -> g_out_ 8.02/2.96 U7_(g_out_) -> f_out_ 8.02/2.96 U6_(f_out_) -> e_out_ 8.02/2.96 U2_(e_out_) -> a_out_ 8.02/2.96 8.02/2.96 Pi is empty. 8.02/2.96 8.02/2.96 8.02/2.96 8.02/2.96 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 8.02/2.96 8.02/2.96 8.02/2.96 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (2) 8.02/2.96 Obligation: 8.02/2.96 Pi-finite rewrite system: 8.02/2.96 The TRS R consists of the following rules: 8.02/2.96 8.02/2.96 a_in_ -> U1_(b_in_) 8.02/2.96 b_in_ -> U3_(c_in_) 8.02/2.96 c_in_ -> U4_(d_in_) 8.02/2.96 d_in_ -> U5_(b_in_) 8.02/2.96 U5_(b_out_) -> d_out_ 8.02/2.96 U4_(d_out_) -> c_out_ 8.02/2.96 U3_(c_out_) -> b_out_ 8.02/2.96 U1_(b_out_) -> a_out_ 8.02/2.96 a_in_ -> U2_(e_in_) 8.02/2.96 e_in_ -> U6_(f_in_) 8.02/2.96 f_in_ -> U7_(g_in_) 8.02/2.96 g_in_ -> U8_(e_in_) 8.02/2.96 U8_(e_out_) -> g_out_ 8.02/2.96 U7_(g_out_) -> f_out_ 8.02/2.96 U6_(f_out_) -> e_out_ 8.02/2.96 U2_(e_out_) -> a_out_ 8.02/2.96 8.02/2.96 Pi is empty. 8.02/2.96 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (3) DependencyPairsProof (EQUIVALENT) 8.02/2.96 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 8.02/2.96 Pi DP problem: 8.02/2.96 The TRS P consists of the following rules: 8.02/2.96 8.02/2.96 A_IN_ -> U1_^1(b_in_) 8.02/2.96 A_IN_ -> B_IN_ 8.02/2.96 B_IN_ -> U3_^1(c_in_) 8.02/2.96 B_IN_ -> C_IN_ 8.02/2.96 C_IN_ -> U4_^1(d_in_) 8.02/2.96 C_IN_ -> D_IN_ 8.02/2.96 D_IN_ -> U5_^1(b_in_) 8.02/2.96 D_IN_ -> B_IN_ 8.02/2.96 A_IN_ -> U2_^1(e_in_) 8.02/2.96 A_IN_ -> E_IN_ 8.02/2.96 E_IN_ -> U6_^1(f_in_) 8.02/2.96 E_IN_ -> F_IN_ 8.02/2.96 F_IN_ -> U7_^1(g_in_) 8.02/2.96 F_IN_ -> G_IN_ 8.02/2.96 G_IN_ -> U8_^1(e_in_) 8.02/2.96 G_IN_ -> E_IN_ 8.02/2.96 8.02/2.96 The TRS R consists of the following rules: 8.02/2.96 8.02/2.96 a_in_ -> U1_(b_in_) 8.02/2.96 b_in_ -> U3_(c_in_) 8.02/2.96 c_in_ -> U4_(d_in_) 8.02/2.96 d_in_ -> U5_(b_in_) 8.02/2.96 U5_(b_out_) -> d_out_ 8.02/2.96 U4_(d_out_) -> c_out_ 8.02/2.96 U3_(c_out_) -> b_out_ 8.02/2.96 U1_(b_out_) -> a_out_ 8.02/2.96 a_in_ -> U2_(e_in_) 8.02/2.96 e_in_ -> U6_(f_in_) 8.02/2.96 f_in_ -> U7_(g_in_) 8.02/2.96 g_in_ -> U8_(e_in_) 8.02/2.96 U8_(e_out_) -> g_out_ 8.02/2.96 U7_(g_out_) -> f_out_ 8.02/2.96 U6_(f_out_) -> e_out_ 8.02/2.96 U2_(e_out_) -> a_out_ 8.02/2.96 8.02/2.96 Pi is empty. 8.02/2.96 We have to consider all (P,R,Pi)-chains 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (4) 8.02/2.96 Obligation: 8.02/2.96 Pi DP problem: 8.02/2.96 The TRS P consists of the following rules: 8.02/2.96 8.02/2.96 A_IN_ -> U1_^1(b_in_) 8.02/2.96 A_IN_ -> B_IN_ 8.02/2.96 B_IN_ -> U3_^1(c_in_) 8.02/2.96 B_IN_ -> C_IN_ 8.02/2.96 C_IN_ -> U4_^1(d_in_) 8.02/2.96 C_IN_ -> D_IN_ 8.02/2.96 D_IN_ -> U5_^1(b_in_) 8.02/2.96 D_IN_ -> B_IN_ 8.02/2.96 A_IN_ -> U2_^1(e_in_) 8.02/2.96 A_IN_ -> E_IN_ 8.02/2.96 E_IN_ -> U6_^1(f_in_) 8.02/2.96 E_IN_ -> F_IN_ 8.02/2.96 F_IN_ -> U7_^1(g_in_) 8.02/2.96 F_IN_ -> G_IN_ 8.02/2.96 G_IN_ -> U8_^1(e_in_) 8.02/2.96 G_IN_ -> E_IN_ 8.02/2.96 8.02/2.96 The TRS R consists of the following rules: 8.02/2.96 8.02/2.96 a_in_ -> U1_(b_in_) 8.02/2.96 b_in_ -> U3_(c_in_) 8.02/2.96 c_in_ -> U4_(d_in_) 8.02/2.96 d_in_ -> U5_(b_in_) 8.02/2.96 U5_(b_out_) -> d_out_ 8.02/2.96 U4_(d_out_) -> c_out_ 8.02/2.96 U3_(c_out_) -> b_out_ 8.02/2.96 U1_(b_out_) -> a_out_ 8.02/2.96 a_in_ -> U2_(e_in_) 8.02/2.96 e_in_ -> U6_(f_in_) 8.02/2.96 f_in_ -> U7_(g_in_) 8.02/2.96 g_in_ -> U8_(e_in_) 8.02/2.96 U8_(e_out_) -> g_out_ 8.02/2.96 U7_(g_out_) -> f_out_ 8.02/2.96 U6_(f_out_) -> e_out_ 8.02/2.96 U2_(e_out_) -> a_out_ 8.02/2.96 8.02/2.96 Pi is empty. 8.02/2.96 We have to consider all (P,R,Pi)-chains 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (5) DependencyGraphProof (EQUIVALENT) 8.02/2.96 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes. 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (6) 8.02/2.96 Complex Obligation (AND) 8.02/2.96 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (7) 8.02/2.96 Obligation: 8.02/2.96 Pi DP problem: 8.02/2.96 The TRS P consists of the following rules: 8.02/2.96 8.02/2.96 E_IN_ -> F_IN_ 8.02/2.96 F_IN_ -> G_IN_ 8.02/2.96 G_IN_ -> E_IN_ 8.02/2.96 8.02/2.96 The TRS R consists of the following rules: 8.02/2.96 8.02/2.96 a_in_ -> U1_(b_in_) 8.02/2.96 b_in_ -> U3_(c_in_) 8.02/2.96 c_in_ -> U4_(d_in_) 8.02/2.96 d_in_ -> U5_(b_in_) 8.02/2.96 U5_(b_out_) -> d_out_ 8.02/2.96 U4_(d_out_) -> c_out_ 8.02/2.96 U3_(c_out_) -> b_out_ 8.02/2.96 U1_(b_out_) -> a_out_ 8.02/2.96 a_in_ -> U2_(e_in_) 8.02/2.96 e_in_ -> U6_(f_in_) 8.02/2.96 f_in_ -> U7_(g_in_) 8.02/2.96 g_in_ -> U8_(e_in_) 8.02/2.96 U8_(e_out_) -> g_out_ 8.02/2.96 U7_(g_out_) -> f_out_ 8.02/2.96 U6_(f_out_) -> e_out_ 8.02/2.96 U2_(e_out_) -> a_out_ 8.02/2.96 8.02/2.96 Pi is empty. 8.02/2.96 We have to consider all (P,R,Pi)-chains 8.02/2.96 ---------------------------------------- 8.02/2.96 8.02/2.96 (8) UsableRulesProof (EQUIVALENT) 8.02/2.96 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 8.02/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (9) 8.13/2.96 Obligation: 8.13/2.96 Pi DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 E_IN_ -> F_IN_ 8.13/2.96 F_IN_ -> G_IN_ 8.13/2.96 G_IN_ -> E_IN_ 8.13/2.96 8.13/2.96 R is empty. 8.13/2.96 Pi is empty. 8.13/2.96 We have to consider all (P,R,Pi)-chains 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (10) PiDPToQDPProof (EQUIVALENT) 8.13/2.96 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (11) 8.13/2.96 Obligation: 8.13/2.96 Q DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 E_IN_ -> F_IN_ 8.13/2.96 F_IN_ -> G_IN_ 8.13/2.96 G_IN_ -> E_IN_ 8.13/2.96 8.13/2.96 R is empty. 8.13/2.96 Q is empty. 8.13/2.96 We have to consider all (P,Q,R)-chains. 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (12) NonTerminationLoopProof (COMPLETE) 8.13/2.96 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 8.13/2.96 Found a loop by narrowing to the left: 8.13/2.96 8.13/2.96 s = F_IN_ evaluates to t =F_IN_ 8.13/2.96 8.13/2.96 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 8.13/2.96 * Matcher: [ ] 8.13/2.96 * Semiunifier: [ ] 8.13/2.96 8.13/2.96 -------------------------------------------------------------------------------- 8.13/2.96 Rewriting sequence 8.13/2.96 8.13/2.96 F_IN_ -> G_IN_ 8.13/2.96 with rule F_IN_ -> G_IN_ at position [] and matcher [ ] 8.13/2.96 8.13/2.96 G_IN_ -> E_IN_ 8.13/2.96 with rule G_IN_ -> E_IN_ at position [] and matcher [ ] 8.13/2.96 8.13/2.96 E_IN_ -> F_IN_ 8.13/2.96 with rule E_IN_ -> F_IN_ 8.13/2.96 8.13/2.96 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 8.13/2.96 8.13/2.96 8.13/2.96 All these steps are and every following step will be a correct step w.r.t to Q. 8.13/2.96 8.13/2.96 8.13/2.96 8.13/2.96 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (13) 8.13/2.96 NO 8.13/2.96 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (14) 8.13/2.96 Obligation: 8.13/2.96 Pi DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 B_IN_ -> C_IN_ 8.13/2.96 C_IN_ -> D_IN_ 8.13/2.96 D_IN_ -> B_IN_ 8.13/2.96 8.13/2.96 The TRS R consists of the following rules: 8.13/2.96 8.13/2.96 a_in_ -> U1_(b_in_) 8.13/2.96 b_in_ -> U3_(c_in_) 8.13/2.96 c_in_ -> U4_(d_in_) 8.13/2.96 d_in_ -> U5_(b_in_) 8.13/2.96 U5_(b_out_) -> d_out_ 8.13/2.96 U4_(d_out_) -> c_out_ 8.13/2.96 U3_(c_out_) -> b_out_ 8.13/2.96 U1_(b_out_) -> a_out_ 8.13/2.96 a_in_ -> U2_(e_in_) 8.13/2.96 e_in_ -> U6_(f_in_) 8.13/2.96 f_in_ -> U7_(g_in_) 8.13/2.96 g_in_ -> U8_(e_in_) 8.13/2.96 U8_(e_out_) -> g_out_ 8.13/2.96 U7_(g_out_) -> f_out_ 8.13/2.96 U6_(f_out_) -> e_out_ 8.13/2.96 U2_(e_out_) -> a_out_ 8.13/2.96 8.13/2.96 Pi is empty. 8.13/2.96 We have to consider all (P,R,Pi)-chains 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (15) UsableRulesProof (EQUIVALENT) 8.13/2.96 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (16) 8.13/2.96 Obligation: 8.13/2.96 Pi DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 B_IN_ -> C_IN_ 8.13/2.96 C_IN_ -> D_IN_ 8.13/2.96 D_IN_ -> B_IN_ 8.13/2.96 8.13/2.96 R is empty. 8.13/2.96 Pi is empty. 8.13/2.96 We have to consider all (P,R,Pi)-chains 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (17) PiDPToQDPProof (EQUIVALENT) 8.13/2.96 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (18) 8.13/2.96 Obligation: 8.13/2.96 Q DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 B_IN_ -> C_IN_ 8.13/2.96 C_IN_ -> D_IN_ 8.13/2.96 D_IN_ -> B_IN_ 8.13/2.96 8.13/2.96 R is empty. 8.13/2.96 Q is empty. 8.13/2.96 We have to consider all (P,Q,R)-chains. 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (19) PrologToPiTRSProof (SOUND) 8.13/2.96 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 8.13/2.96 8.13/2.96 Transforming Prolog into the following Term Rewriting System: 8.13/2.96 8.13/2.96 Pi-finite rewrite system: 8.13/2.96 The TRS R consists of the following rules: 8.13/2.96 8.13/2.96 a_in_ -> U1_(b_in_) 8.13/2.96 b_in_ -> U3_(c_in_) 8.13/2.96 c_in_ -> U4_(d_in_) 8.13/2.96 d_in_ -> U5_(b_in_) 8.13/2.96 U5_(b_out_) -> d_out_ 8.13/2.96 U4_(d_out_) -> c_out_ 8.13/2.96 U3_(c_out_) -> b_out_ 8.13/2.96 U1_(b_out_) -> a_out_ 8.13/2.96 a_in_ -> U2_(e_in_) 8.13/2.96 e_in_ -> U6_(f_in_) 8.13/2.96 f_in_ -> U7_(g_in_) 8.13/2.96 g_in_ -> U8_(e_in_) 8.13/2.96 U8_(e_out_) -> g_out_ 8.13/2.96 U7_(g_out_) -> f_out_ 8.13/2.96 U6_(f_out_) -> e_out_ 8.13/2.96 U2_(e_out_) -> a_out_ 8.13/2.96 8.13/2.96 Pi is empty. 8.13/2.96 8.13/2.96 8.13/2.96 8.13/2.96 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 8.13/2.96 8.13/2.96 8.13/2.96 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (20) 8.13/2.96 Obligation: 8.13/2.96 Pi-finite rewrite system: 8.13/2.96 The TRS R consists of the following rules: 8.13/2.96 8.13/2.96 a_in_ -> U1_(b_in_) 8.13/2.96 b_in_ -> U3_(c_in_) 8.13/2.96 c_in_ -> U4_(d_in_) 8.13/2.96 d_in_ -> U5_(b_in_) 8.13/2.96 U5_(b_out_) -> d_out_ 8.13/2.96 U4_(d_out_) -> c_out_ 8.13/2.96 U3_(c_out_) -> b_out_ 8.13/2.96 U1_(b_out_) -> a_out_ 8.13/2.96 a_in_ -> U2_(e_in_) 8.13/2.96 e_in_ -> U6_(f_in_) 8.13/2.96 f_in_ -> U7_(g_in_) 8.13/2.96 g_in_ -> U8_(e_in_) 8.13/2.96 U8_(e_out_) -> g_out_ 8.13/2.96 U7_(g_out_) -> f_out_ 8.13/2.96 U6_(f_out_) -> e_out_ 8.13/2.96 U2_(e_out_) -> a_out_ 8.13/2.96 8.13/2.96 Pi is empty. 8.13/2.96 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (21) DependencyPairsProof (EQUIVALENT) 8.13/2.96 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 8.13/2.96 Pi DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 A_IN_ -> U1_^1(b_in_) 8.13/2.96 A_IN_ -> B_IN_ 8.13/2.96 B_IN_ -> U3_^1(c_in_) 8.13/2.96 B_IN_ -> C_IN_ 8.13/2.96 C_IN_ -> U4_^1(d_in_) 8.13/2.96 C_IN_ -> D_IN_ 8.13/2.96 D_IN_ -> U5_^1(b_in_) 8.13/2.96 D_IN_ -> B_IN_ 8.13/2.96 A_IN_ -> U2_^1(e_in_) 8.13/2.96 A_IN_ -> E_IN_ 8.13/2.96 E_IN_ -> U6_^1(f_in_) 8.13/2.96 E_IN_ -> F_IN_ 8.13/2.96 F_IN_ -> U7_^1(g_in_) 8.13/2.96 F_IN_ -> G_IN_ 8.13/2.96 G_IN_ -> U8_^1(e_in_) 8.13/2.96 G_IN_ -> E_IN_ 8.13/2.96 8.13/2.96 The TRS R consists of the following rules: 8.13/2.96 8.13/2.96 a_in_ -> U1_(b_in_) 8.13/2.96 b_in_ -> U3_(c_in_) 8.13/2.96 c_in_ -> U4_(d_in_) 8.13/2.96 d_in_ -> U5_(b_in_) 8.13/2.96 U5_(b_out_) -> d_out_ 8.13/2.96 U4_(d_out_) -> c_out_ 8.13/2.96 U3_(c_out_) -> b_out_ 8.13/2.96 U1_(b_out_) -> a_out_ 8.13/2.96 a_in_ -> U2_(e_in_) 8.13/2.96 e_in_ -> U6_(f_in_) 8.13/2.96 f_in_ -> U7_(g_in_) 8.13/2.96 g_in_ -> U8_(e_in_) 8.13/2.96 U8_(e_out_) -> g_out_ 8.13/2.96 U7_(g_out_) -> f_out_ 8.13/2.96 U6_(f_out_) -> e_out_ 8.13/2.96 U2_(e_out_) -> a_out_ 8.13/2.96 8.13/2.96 Pi is empty. 8.13/2.96 We have to consider all (P,R,Pi)-chains 8.13/2.96 ---------------------------------------- 8.13/2.96 8.13/2.96 (22) 8.13/2.96 Obligation: 8.13/2.96 Pi DP problem: 8.13/2.96 The TRS P consists of the following rules: 8.13/2.96 8.13/2.96 A_IN_ -> U1_^1(b_in_) 8.13/2.96 A_IN_ -> B_IN_ 8.13/2.96 B_IN_ -> U3_^1(c_in_) 8.13/2.96 B_IN_ -> C_IN_ 8.13/2.96 C_IN_ -> U4_^1(d_in_) 8.13/2.96 C_IN_ -> D_IN_ 8.13/2.96 D_IN_ -> U5_^1(b_in_) 8.13/2.96 D_IN_ -> B_IN_ 8.13/2.96 A_IN_ -> U2_^1(e_in_) 8.13/2.96 A_IN_ -> E_IN_ 8.13/2.96 E_IN_ -> U6_^1(f_in_) 8.13/2.96 E_IN_ -> F_IN_ 8.13/2.96 F_IN_ -> U7_^1(g_in_) 8.13/2.96 F_IN_ -> G_IN_ 8.13/2.96 G_IN_ -> U8_^1(e_in_) 8.13/2.96 G_IN_ -> E_IN_ 8.13/2.96 8.13/2.96 The TRS R consists of the following rules: 8.13/2.96 8.13/2.96 a_in_ -> U1_(b_in_) 8.13/2.96 b_in_ -> U3_(c_in_) 8.13/2.96 c_in_ -> U4_(d_in_) 8.13/2.96 d_in_ -> U5_(b_in_) 8.13/2.96 U5_(b_out_) -> d_out_ 8.13/2.96 U4_(d_out_) -> c_out_ 8.13/2.96 U3_(c_out_) -> b_out_ 8.13/2.97 U1_(b_out_) -> a_out_ 8.13/2.97 a_in_ -> U2_(e_in_) 8.13/2.97 e_in_ -> U6_(f_in_) 8.13/2.97 f_in_ -> U7_(g_in_) 8.13/2.97 g_in_ -> U8_(e_in_) 8.13/2.97 U8_(e_out_) -> g_out_ 8.13/2.97 U7_(g_out_) -> f_out_ 8.13/2.97 U6_(f_out_) -> e_out_ 8.13/2.97 U2_(e_out_) -> a_out_ 8.13/2.97 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (23) DependencyGraphProof (EQUIVALENT) 8.13/2.97 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (24) 8.13/2.97 Complex Obligation (AND) 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (25) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 E_IN_ -> F_IN_ 8.13/2.97 F_IN_ -> G_IN_ 8.13/2.97 G_IN_ -> E_IN_ 8.13/2.97 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 a_in_ -> U1_(b_in_) 8.13/2.97 b_in_ -> U3_(c_in_) 8.13/2.97 c_in_ -> U4_(d_in_) 8.13/2.97 d_in_ -> U5_(b_in_) 8.13/2.97 U5_(b_out_) -> d_out_ 8.13/2.97 U4_(d_out_) -> c_out_ 8.13/2.97 U3_(c_out_) -> b_out_ 8.13/2.97 U1_(b_out_) -> a_out_ 8.13/2.97 a_in_ -> U2_(e_in_) 8.13/2.97 e_in_ -> U6_(f_in_) 8.13/2.97 f_in_ -> U7_(g_in_) 8.13/2.97 g_in_ -> U8_(e_in_) 8.13/2.97 U8_(e_out_) -> g_out_ 8.13/2.97 U7_(g_out_) -> f_out_ 8.13/2.97 U6_(f_out_) -> e_out_ 8.13/2.97 U2_(e_out_) -> a_out_ 8.13/2.97 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (26) UsableRulesProof (EQUIVALENT) 8.13/2.97 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (27) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 E_IN_ -> F_IN_ 8.13/2.97 F_IN_ -> G_IN_ 8.13/2.97 G_IN_ -> E_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (28) PiDPToQDPProof (EQUIVALENT) 8.13/2.97 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (29) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 E_IN_ -> F_IN_ 8.13/2.97 F_IN_ -> G_IN_ 8.13/2.97 G_IN_ -> E_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (30) NonTerminationLoopProof (COMPLETE) 8.13/2.97 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 8.13/2.97 Found a loop by narrowing to the left: 8.13/2.97 8.13/2.97 s = F_IN_ evaluates to t =F_IN_ 8.13/2.97 8.13/2.97 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 8.13/2.97 * Matcher: [ ] 8.13/2.97 * Semiunifier: [ ] 8.13/2.97 8.13/2.97 -------------------------------------------------------------------------------- 8.13/2.97 Rewriting sequence 8.13/2.97 8.13/2.97 F_IN_ -> G_IN_ 8.13/2.97 with rule F_IN_ -> G_IN_ at position [] and matcher [ ] 8.13/2.97 8.13/2.97 G_IN_ -> E_IN_ 8.13/2.97 with rule G_IN_ -> E_IN_ at position [] and matcher [ ] 8.13/2.97 8.13/2.97 E_IN_ -> F_IN_ 8.13/2.97 with rule E_IN_ -> F_IN_ 8.13/2.97 8.13/2.97 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 8.13/2.97 8.13/2.97 8.13/2.97 All these steps are and every following step will be a correct step w.r.t to Q. 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (31) 8.13/2.97 NO 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (32) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 B_IN_ -> C_IN_ 8.13/2.97 C_IN_ -> D_IN_ 8.13/2.97 D_IN_ -> B_IN_ 8.13/2.97 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 a_in_ -> U1_(b_in_) 8.13/2.97 b_in_ -> U3_(c_in_) 8.13/2.97 c_in_ -> U4_(d_in_) 8.13/2.97 d_in_ -> U5_(b_in_) 8.13/2.97 U5_(b_out_) -> d_out_ 8.13/2.97 U4_(d_out_) -> c_out_ 8.13/2.97 U3_(c_out_) -> b_out_ 8.13/2.97 U1_(b_out_) -> a_out_ 8.13/2.97 a_in_ -> U2_(e_in_) 8.13/2.97 e_in_ -> U6_(f_in_) 8.13/2.97 f_in_ -> U7_(g_in_) 8.13/2.97 g_in_ -> U8_(e_in_) 8.13/2.97 U8_(e_out_) -> g_out_ 8.13/2.97 U7_(g_out_) -> f_out_ 8.13/2.97 U6_(f_out_) -> e_out_ 8.13/2.97 U2_(e_out_) -> a_out_ 8.13/2.97 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (33) UsableRulesProof (EQUIVALENT) 8.13/2.97 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (34) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 B_IN_ -> C_IN_ 8.13/2.97 C_IN_ -> D_IN_ 8.13/2.97 D_IN_ -> B_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (35) PiDPToQDPProof (EQUIVALENT) 8.13/2.97 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (36) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 B_IN_ -> C_IN_ 8.13/2.97 C_IN_ -> D_IN_ 8.13/2.97 D_IN_ -> B_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (37) PrologToTRSTransformerProof (SOUND) 8.13/2.97 Transformed Prolog program to TRS. 8.13/2.97 8.13/2.97 { 8.13/2.97 "root": 22, 8.13/2.97 "program": { 8.13/2.97 "directives": [], 8.13/2.97 "clauses": [ 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(e)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(b)", 8.13/2.97 "(c)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(c)", 8.13/2.97 "(d)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(d)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(e)", 8.13/2.97 "(f)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(f)", 8.13/2.97 "(g)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(g)", 8.13/2.97 "(e)" 8.13/2.97 ] 8.13/2.97 ] 8.13/2.97 }, 8.13/2.97 "graph": { 8.13/2.97 "nodes": { 8.13/2.97 "22": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "44": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "23": { 8.13/2.97 "goal": [ 8.13/2.97 { 8.13/2.97 "clause": 0, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "type": "Nodes", 8.13/2.97 "120": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 2, 8.13/2.97 "scope": 2, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "131": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "121": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "133": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 5, 8.13/2.97 "scope": 5, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "123": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 3, 8.13/2.97 "scope": 3, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "124": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(d)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "135": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(f)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "136": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 6, 8.13/2.97 "scope": 6, 8.13/2.97 "term": "(f)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "126": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 4, 8.13/2.97 "scope": 4, 8.13/2.97 "term": "(d)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "137": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(g)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "138": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 7, 8.13/2.97 "scope": 7, 8.13/2.97 "term": "(g)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "128": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "139": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "42": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 0, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "65": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "edges": [ 8.13/2.97 { 8.13/2.97 "from": 22, 8.13/2.97 "to": 23, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 23, 8.13/2.97 "to": 42, 8.13/2.97 "label": "PARALLEL" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 23, 8.13/2.97 "to": 44, 8.13/2.97 "label": "PARALLEL" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 42, 8.13/2.97 "to": 65, 8.13/2.97 "label": "ONLY EVAL with clause\na :- b.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 44, 8.13/2.97 "to": 131, 8.13/2.97 "label": "ONLY EVAL with clause\na :- e.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 65, 8.13/2.97 "to": 120, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 120, 8.13/2.97 "to": 121, 8.13/2.97 "label": "ONLY EVAL with clause\nb :- c.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 121, 8.13/2.97 "to": 123, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 123, 8.13/2.97 "to": 124, 8.13/2.97 "label": "ONLY EVAL with clause\nc :- d.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 124, 8.13/2.97 "to": 126, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 126, 8.13/2.97 "to": 128, 8.13/2.97 "label": "ONLY EVAL with clause\nd :- b.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 128, 8.13/2.97 "to": 65, 8.13/2.97 "label": "INSTANCE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 131, 8.13/2.97 "to": 133, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 133, 8.13/2.97 "to": 135, 8.13/2.97 "label": "ONLY EVAL with clause\ne :- f.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 135, 8.13/2.97 "to": 136, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 136, 8.13/2.97 "to": 137, 8.13/2.97 "label": "ONLY EVAL with clause\nf :- g.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 137, 8.13/2.97 "to": 138, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 138, 8.13/2.97 "to": 139, 8.13/2.97 "label": "ONLY EVAL with clause\ng :- e.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 139, 8.13/2.97 "to": 131, 8.13/2.97 "label": "INSTANCE" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "type": "Graph" 8.13/2.97 } 8.13/2.97 } 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (38) 8.13/2.97 Obligation: 8.13/2.97 Q restricted rewrite system: 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f22_in -> U1(f65_in) 8.13/2.97 U1(f65_out1) -> f22_out1 8.13/2.97 f22_in -> U2(f131_in) 8.13/2.97 U2(f131_out1) -> f22_out1 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 U3(f65_out1) -> f65_out1 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 U4(f131_out1) -> f131_out1 8.13/2.97 8.13/2.97 Q is empty. 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (39) QTRSRRRProof (EQUIVALENT) 8.13/2.97 Used ordering: 8.13/2.97 Polynomial interpretation [POLO]: 8.13/2.97 8.13/2.97 POL(U1(x_1)) = 2 + 2*x_1 8.13/2.97 POL(U2(x_1)) = 2*x_1 8.13/2.97 POL(U3(x_1)) = 2*x_1 8.13/2.97 POL(U4(x_1)) = 2*x_1 8.13/2.97 POL(f131_in) = 0 8.13/2.97 POL(f131_out1) = 1 8.13/2.97 POL(f22_in) = 2 8.13/2.97 POL(f22_out1) = 1 8.13/2.97 POL(f65_in) = 0 8.13/2.97 POL(f65_out1) = 1 8.13/2.97 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 8.13/2.97 8.13/2.97 U1(f65_out1) -> f22_out1 8.13/2.97 f22_in -> U2(f131_in) 8.13/2.97 U2(f131_out1) -> f22_out1 8.13/2.97 U3(f65_out1) -> f65_out1 8.13/2.97 U4(f131_out1) -> f131_out1 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (40) 8.13/2.97 Obligation: 8.13/2.97 Q restricted rewrite system: 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f22_in -> U1(f65_in) 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 Q is empty. 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (41) QTRSRRRProof (EQUIVALENT) 8.13/2.97 Used ordering: 8.13/2.97 f22_in/0) 8.13/2.97 U1/1(YES) 8.13/2.97 f65_in/0) 8.13/2.97 U3/1)YES( 8.13/2.97 f131_in/0) 8.13/2.97 U4/1)YES( 8.13/2.97 8.13/2.97 Quasi precedence: 8.13/2.97 f22_in > U1_1 8.13/2.97 f22_in > f65_in 8.13/2.97 8.13/2.97 8.13/2.97 Status: 8.13/2.97 f22_in: multiset status 8.13/2.97 U1_1: [1] 8.13/2.97 f65_in: multiset status 8.13/2.97 f131_in: multiset status 8.13/2.97 8.13/2.97 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 8.13/2.97 8.13/2.97 f22_in -> U1(f65_in) 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (42) 8.13/2.97 Obligation: 8.13/2.97 Q restricted rewrite system: 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 Q is empty. 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (43) Overlay + Local Confluence (EQUIVALENT) 8.13/2.97 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (44) 8.13/2.97 Obligation: 8.13/2.97 Q restricted rewrite system: 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (45) DependencyPairsProof (EQUIVALENT) 8.13/2.97 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (46) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F65_IN -> F65_IN 8.13/2.97 F131_IN -> F131_IN 8.13/2.97 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (47) DependencyGraphProof (EQUIVALENT) 8.13/2.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (48) 8.13/2.97 Complex Obligation (AND) 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (49) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F131_IN -> F131_IN 8.13/2.97 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (50) UsableRulesProof (EQUIVALENT) 8.13/2.97 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. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (51) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F131_IN -> F131_IN 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (52) QReductionProof (EQUIVALENT) 8.13/2.97 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (53) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F131_IN -> F131_IN 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (54) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F65_IN -> F65_IN 8.13/2.97 8.13/2.97 The TRS R consists of the following rules: 8.13/2.97 8.13/2.97 f65_in -> U3(f65_in) 8.13/2.97 f131_in -> U4(f131_in) 8.13/2.97 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (55) UsableRulesProof (EQUIVALENT) 8.13/2.97 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. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (56) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F65_IN -> F65_IN 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 The set Q consists of the following terms: 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (57) QReductionProof (EQUIVALENT) 8.13/2.97 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 8.13/2.97 8.13/2.97 f65_in 8.13/2.97 f131_in 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (58) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 F65_IN -> F65_IN 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all minimal (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (59) NonTerminationLoopProof (COMPLETE) 8.13/2.97 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 8.13/2.97 Found a loop by semiunifying a rule from P directly. 8.13/2.97 8.13/2.97 s = F65_IN evaluates to t =F65_IN 8.13/2.97 8.13/2.97 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 8.13/2.97 * Matcher: [ ] 8.13/2.97 * Semiunifier: [ ] 8.13/2.97 8.13/2.97 -------------------------------------------------------------------------------- 8.13/2.97 Rewriting sequence 8.13/2.97 8.13/2.97 The DP semiunifies directly so there is only one rewrite step from F65_IN to F65_IN. 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (60) 8.13/2.97 NO 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (61) PrologToDTProblemTransformerProof (SOUND) 8.13/2.97 Built DT problem from termination graph DT10. 8.13/2.97 8.13/2.97 { 8.13/2.97 "root": 1, 8.13/2.97 "program": { 8.13/2.97 "directives": [], 8.13/2.97 "clauses": [ 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(e)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(b)", 8.13/2.97 "(c)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(c)", 8.13/2.97 "(d)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(d)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(e)", 8.13/2.97 "(f)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(f)", 8.13/2.97 "(g)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(g)", 8.13/2.97 "(e)" 8.13/2.97 ] 8.13/2.97 ] 8.13/2.97 }, 8.13/2.97 "graph": { 8.13/2.97 "nodes": { 8.13/2.97 "46": { 8.13/2.97 "goal": [ 8.13/2.97 { 8.13/2.97 "clause": 2, 8.13/2.97 "scope": 2, 8.13/2.97 "term": "(b)" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": -1, 8.13/2.97 "scope": 2, 8.13/2.97 "term": null 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "58": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 2, 8.13/2.97 "scope": 2, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "160": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 6, 8.13/2.97 "scope": 7, 8.13/2.97 "term": "(f)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "type": "Nodes", 8.13/2.97 "161": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(g)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "140": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "162": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 7, 8.13/2.97 "scope": 8, 8.13/2.97 "term": "(g)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "141": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "163": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "110": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "154": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(f)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "1": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "103": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 4, 8.13/2.97 "scope": 4, 8.13/2.97 "term": "(d)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "147": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 5, 8.13/2.97 "scope": 6, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "5": { 8.13/2.97 "goal": [ 8.13/2.97 { 8.13/2.97 "clause": 0, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "117": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 2, 8.13/2.97 "scope": 5, 8.13/2.97 "term": "(b)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "119": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "71": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "61": { 8.13/2.97 "goal": [ 8.13/2.97 { 8.13/2.97 "clause": -1, 8.13/2.97 "scope": 2, 8.13/2.97 "term": null 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "94": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 3, 8.13/2.97 "scope": 3, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "41": { 8.13/2.97 "goal": [ 8.13/2.97 { 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(b)" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "96": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(d)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "edges": [ 8.13/2.97 { 8.13/2.97 "from": 1, 8.13/2.97 "to": 5, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 5, 8.13/2.97 "to": 41, 8.13/2.97 "label": "ONLY EVAL with clause\na :- b.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 41, 8.13/2.97 "to": 46, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 46, 8.13/2.97 "to": 58, 8.13/2.97 "label": "PARALLEL" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 46, 8.13/2.97 "to": 61, 8.13/2.97 "label": "PARALLEL" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 58, 8.13/2.97 "to": 71, 8.13/2.97 "label": "ONLY EVAL with clause\nb :- c.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 61, 8.13/2.97 "to": 140, 8.13/2.97 "label": "FAILURE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 71, 8.13/2.97 "to": 94, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 94, 8.13/2.97 "to": 96, 8.13/2.97 "label": "ONLY EVAL with clause\nc :- d.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 96, 8.13/2.97 "to": 103, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 103, 8.13/2.97 "to": 110, 8.13/2.97 "label": "ONLY EVAL with clause\nd :- b.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 110, 8.13/2.97 "to": 117, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 117, 8.13/2.97 "to": 119, 8.13/2.97 "label": "ONLY EVAL with clause\nb :- c.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 119, 8.13/2.97 "to": 71, 8.13/2.97 "label": "INSTANCE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 140, 8.13/2.97 "to": 141, 8.13/2.97 "label": "ONLY EVAL with clause\na :- e.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 141, 8.13/2.97 "to": 147, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 147, 8.13/2.97 "to": 154, 8.13/2.97 "label": "ONLY EVAL with clause\ne :- f.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 154, 8.13/2.97 "to": 160, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 160, 8.13/2.97 "to": 161, 8.13/2.97 "label": "ONLY EVAL with clause\nf :- g.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 161, 8.13/2.97 "to": 162, 8.13/2.97 "label": "CASE" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 162, 8.13/2.97 "to": 163, 8.13/2.97 "label": "ONLY EVAL with clause\ng :- e.\nand substitution" 8.13/2.97 }, 8.13/2.97 { 8.13/2.97 "from": 163, 8.13/2.97 "to": 141, 8.13/2.97 "label": "INSTANCE" 8.13/2.97 } 8.13/2.97 ], 8.13/2.97 "type": "Graph" 8.13/2.97 } 8.13/2.97 } 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (62) 8.13/2.97 Obligation: 8.13/2.97 Triples: 8.13/2.97 8.13/2.97 cA :- cA. 8.13/2.97 eB :- eB. 8.13/2.97 aC :- cA. 8.13/2.97 aC :- eB. 8.13/2.97 8.13/2.97 Clauses: 8.13/2.97 8.13/2.97 ccA :- ccA. 8.13/2.97 ecB :- ecB. 8.13/2.97 8.13/2.97 Afs: 8.13/2.97 8.13/2.97 aC = aC 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (63) TriplesToPiDPProof (SOUND) 8.13/2.97 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 8.13/2.97 8.13/2.97 Transforming TRIPLES into the following Term Rewriting System: 8.13/2.97 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 AC_IN_ -> U3_^1(cA_in_) 8.13/2.97 AC_IN_ -> CA_IN_ 8.13/2.97 CA_IN_ -> U1_^1(cA_in_) 8.13/2.97 CA_IN_ -> CA_IN_ 8.13/2.97 AC_IN_ -> U4_^1(eB_in_) 8.13/2.97 AC_IN_ -> EB_IN_ 8.13/2.97 EB_IN_ -> U2_^1(eB_in_) 8.13/2.97 EB_IN_ -> EB_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 8.13/2.97 8.13/2.97 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (64) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 AC_IN_ -> U3_^1(cA_in_) 8.13/2.97 AC_IN_ -> CA_IN_ 8.13/2.97 CA_IN_ -> U1_^1(cA_in_) 8.13/2.97 CA_IN_ -> CA_IN_ 8.13/2.97 AC_IN_ -> U4_^1(eB_in_) 8.13/2.97 AC_IN_ -> EB_IN_ 8.13/2.97 EB_IN_ -> U2_^1(eB_in_) 8.13/2.97 EB_IN_ -> EB_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (65) DependencyGraphProof (EQUIVALENT) 8.13/2.97 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (66) 8.13/2.97 Complex Obligation (AND) 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (67) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 EB_IN_ -> EB_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (68) PiDPToQDPProof (EQUIVALENT) 8.13/2.97 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (69) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 EB_IN_ -> EB_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (70) NonTerminationLoopProof (COMPLETE) 8.13/2.97 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 8.13/2.97 Found a loop by semiunifying a rule from P directly. 8.13/2.97 8.13/2.97 s = EB_IN_ evaluates to t =EB_IN_ 8.13/2.97 8.13/2.97 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 8.13/2.97 * Matcher: [ ] 8.13/2.97 * Semiunifier: [ ] 8.13/2.97 8.13/2.97 -------------------------------------------------------------------------------- 8.13/2.97 Rewriting sequence 8.13/2.97 8.13/2.97 The DP semiunifies directly so there is only one rewrite step from EB_IN_ to EB_IN_. 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (71) 8.13/2.97 NO 8.13/2.97 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (72) 8.13/2.97 Obligation: 8.13/2.97 Pi DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 CA_IN_ -> CA_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Pi is empty. 8.13/2.97 We have to consider all (P,R,Pi)-chains 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (73) PiDPToQDPProof (EQUIVALENT) 8.13/2.97 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (74) 8.13/2.97 Obligation: 8.13/2.97 Q DP problem: 8.13/2.97 The TRS P consists of the following rules: 8.13/2.97 8.13/2.97 CA_IN_ -> CA_IN_ 8.13/2.97 8.13/2.97 R is empty. 8.13/2.97 Q is empty. 8.13/2.97 We have to consider all (P,Q,R)-chains. 8.13/2.97 ---------------------------------------- 8.13/2.97 8.13/2.97 (75) PrologToIRSwTTransformerProof (SOUND) 8.13/2.97 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 8.13/2.97 8.13/2.97 { 8.13/2.97 "root": 16, 8.13/2.97 "program": { 8.13/2.97 "directives": [], 8.13/2.97 "clauses": [ 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(a)", 8.13/2.97 "(e)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(b)", 8.13/2.97 "(c)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(c)", 8.13/2.97 "(d)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(d)", 8.13/2.97 "(b)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(e)", 8.13/2.97 "(f)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(f)", 8.13/2.97 "(g)" 8.13/2.97 ], 8.13/2.97 [ 8.13/2.97 "(g)", 8.13/2.97 "(e)" 8.13/2.97 ] 8.13/2.97 ] 8.13/2.97 }, 8.13/2.97 "graph": { 8.13/2.97 "nodes": { 8.13/2.97 "45": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 1, 8.13/2.97 "scope": 1, 8.13/2.97 "term": "(a)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 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"(g)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "122": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "134": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": -1, 8.13/2.97 "scope": -1, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "102": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 3, 8.13/2.97 "scope": 3, 8.13/2.97 "term": "(c)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "125": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 5, 8.13/2.97 "scope": 5, 8.13/2.97 "term": "(e)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.97 "relations": [] 8.13/2.97 }, 8.13/2.97 "ground": [], 8.13/2.97 "free": [], 8.13/2.97 "exprvars": [] 8.13/2.97 } 8.13/2.97 }, 8.13/2.97 "115": { 8.13/2.97 "goal": [{ 8.13/2.97 "clause": 4, 8.13/2.97 "scope": 4, 8.13/2.97 "term": "(d)" 8.13/2.97 }], 8.13/2.97 "kb": { 8.13/2.97 "nonunifying": [], 8.13/2.97 "intvars": {}, 8.13/2.97 "arithmetic": { 8.13/2.97 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "127": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": -1, 8.13/2.98 "scope": -1, 8.13/2.98 "term": "(f)" 8.13/2.98 }], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "118": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": -1, 8.13/2.98 "scope": -1, 8.13/2.98 "term": "(b)" 8.13/2.98 }], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "129": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": 6, 8.13/2.98 "scope": 6, 8.13/2.98 "term": "(f)" 8.13/2.98 }], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "93": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": 2, 8.13/2.98 "scope": 2, 8.13/2.98 "term": "(b)" 8.13/2.98 }], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "109": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": 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"PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "21": { 8.13/2.98 "goal": [ 8.13/2.98 { 8.13/2.98 "clause": 0, 8.13/2.98 "scope": 1, 8.13/2.98 "term": "(a)" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "clause": 1, 8.13/2.98 "scope": 1, 8.13/2.98 "term": "(a)" 8.13/2.98 } 8.13/2.98 ], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "43": { 8.13/2.98 "goal": [{ 8.13/2.98 "clause": 0, 8.13/2.98 "scope": 1, 8.13/2.98 "term": "(a)" 8.13/2.98 }], 8.13/2.98 "kb": { 8.13/2.98 "nonunifying": [], 8.13/2.98 "intvars": {}, 8.13/2.98 "arithmetic": { 8.13/2.98 "type": "PlainIntegerRelationState", 8.13/2.98 "relations": [] 8.13/2.98 }, 8.13/2.98 "ground": [], 8.13/2.98 "free": [], 8.13/2.98 "exprvars": [] 8.13/2.98 } 8.13/2.98 } 8.13/2.98 }, 8.13/2.98 "edges": [ 8.13/2.98 { 8.13/2.98 "from": 16, 8.13/2.98 "to": 21, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 21, 8.13/2.98 "to": 43, 8.13/2.98 "label": "PARALLEL" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 21, 8.13/2.98 "to": 45, 8.13/2.98 "label": "PARALLEL" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 43, 8.13/2.98 "to": 62, 8.13/2.98 "label": "ONLY EVAL with clause\na :- b.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 45, 8.13/2.98 "to": 122, 8.13/2.98 "label": "ONLY EVAL with clause\na :- e.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 62, 8.13/2.98 "to": 93, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 93, 8.13/2.98 "to": 97, 8.13/2.98 "label": "ONLY EVAL with clause\nb :- c.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 97, 8.13/2.98 "to": 102, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 102, 8.13/2.98 "to": 109, 8.13/2.98 "label": "ONLY EVAL with clause\nc :- d.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 109, 8.13/2.98 "to": 115, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 115, 8.13/2.98 "to": 118, 8.13/2.98 "label": "ONLY EVAL with clause\nd :- b.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 118, 8.13/2.98 "to": 62, 8.13/2.98 "label": "INSTANCE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 122, 8.13/2.98 "to": 125, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 125, 8.13/2.98 "to": 127, 8.13/2.98 "label": "ONLY EVAL with clause\ne :- f.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 127, 8.13/2.98 "to": 129, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 129, 8.13/2.98 "to": 130, 8.13/2.98 "label": "ONLY EVAL with clause\nf :- g.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 130, 8.13/2.98 "to": 132, 8.13/2.98 "label": "CASE" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 132, 8.13/2.98 "to": 134, 8.13/2.98 "label": "ONLY EVAL with clause\ng :- e.\nand substitution" 8.13/2.98 }, 8.13/2.98 { 8.13/2.98 "from": 134, 8.13/2.98 "to": 122, 8.13/2.98 "label": "INSTANCE" 8.13/2.98 } 8.13/2.98 ], 8.13/2.98 "type": "Graph" 8.13/2.98 } 8.13/2.98 } 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (76) 8.13/2.98 Complex Obligation (AND) 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (77) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f132_in -> f134_in :|: TRUE 8.13/2.98 f134_out -> f132_out :|: TRUE 8.13/2.98 f129_out -> f127_out :|: TRUE 8.13/2.98 f127_in -> f129_in :|: TRUE 8.13/2.98 f134_in -> f122_in :|: TRUE 8.13/2.98 f122_out -> f134_out :|: TRUE 8.13/2.98 f132_out -> f130_out :|: TRUE 8.13/2.98 f130_in -> f132_in :|: TRUE 8.13/2.98 f127_out -> f125_out :|: TRUE 8.13/2.98 f125_in -> f127_in :|: TRUE 8.13/2.98 f125_out -> f122_out :|: TRUE 8.13/2.98 f122_in -> f125_in :|: TRUE 8.13/2.98 f130_out -> f129_out :|: TRUE 8.13/2.98 f129_in -> f130_in :|: TRUE 8.13/2.98 f21_out -> f16_out :|: TRUE 8.13/2.98 f16_in -> f21_in :|: TRUE 8.13/2.98 f21_in -> f45_in :|: TRUE 8.13/2.98 f43_out -> f21_out :|: TRUE 8.13/2.98 f45_out -> f21_out :|: TRUE 8.13/2.98 f21_in -> f43_in :|: TRUE 8.13/2.98 f122_out -> f45_out :|: TRUE 8.13/2.98 f45_in -> f122_in :|: TRUE 8.13/2.98 Start term: f16_in 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (78) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 8.13/2.98 Constructed simple dependency graph. 8.13/2.98 8.13/2.98 Simplified to the following IRSwTs: 8.13/2.98 8.13/2.98 intTRSProblem: 8.13/2.98 f132_in -> f134_in :|: TRUE 8.13/2.98 f127_in -> f129_in :|: TRUE 8.13/2.98 f134_in -> f122_in :|: TRUE 8.13/2.98 f130_in -> f132_in :|: TRUE 8.13/2.98 f125_in -> f127_in :|: TRUE 8.13/2.98 f122_in -> f125_in :|: TRUE 8.13/2.98 f129_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (79) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f132_in -> f134_in :|: TRUE 8.13/2.98 f127_in -> f129_in :|: TRUE 8.13/2.98 f134_in -> f122_in :|: TRUE 8.13/2.98 f130_in -> f132_in :|: TRUE 8.13/2.98 f125_in -> f127_in :|: TRUE 8.13/2.98 f122_in -> f125_in :|: TRUE 8.13/2.98 f129_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (80) IntTRSCompressionProof (EQUIVALENT) 8.13/2.98 Compressed rules. 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (81) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f130_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (82) IRSFormatTransformerProof (EQUIVALENT) 8.13/2.98 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (83) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f130_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (84) IRSwTTerminationDigraphProof (EQUIVALENT) 8.13/2.98 Constructed termination digraph! 8.13/2.98 Nodes: 8.13/2.98 (1) f130_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 Arcs: 8.13/2.98 (1) -> (1) 8.13/2.98 8.13/2.98 This digraph is fully evaluated! 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (85) 8.13/2.98 Obligation: 8.13/2.98 8.13/2.98 Termination digraph: 8.13/2.98 Nodes: 8.13/2.98 (1) f130_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 Arcs: 8.13/2.98 (1) -> (1) 8.13/2.98 8.13/2.98 This digraph is fully evaluated! 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (86) FilterProof (EQUIVALENT) 8.13/2.98 Used the following sort dictionary for filtering: 8.13/2.98 f130_in() 8.13/2.98 Replaced non-predefined constructor symbols by 0. 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (87) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f130_in -> f130_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (88) IntTRSNonPeriodicNontermProof (COMPLETE) 8.13/2.98 Normalized system to the following form: 8.13/2.98 f(pc) -> f(1) :|: pc = 1 && TRUE 8.13/2.98 Proved unsatisfiability of the following formula, indicating that the system is never left after entering: 8.13/2.98 ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) 8.13/2.98 Proved satisfiability of the following formula, indicating that the system is entered at least once: 8.13/2.98 (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (89) 8.13/2.98 NO 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (90) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f109_in -> f115_in :|: TRUE 8.13/2.98 f115_out -> f109_out :|: TRUE 8.13/2.98 f118_out -> f115_out :|: TRUE 8.13/2.98 f115_in -> f118_in :|: TRUE 8.13/2.98 f97_out -> f93_out :|: TRUE 8.13/2.98 f93_in -> f97_in :|: TRUE 8.13/2.98 f62_out -> f118_out :|: TRUE 8.13/2.98 f118_in -> f62_in :|: TRUE 8.13/2.98 f102_out -> f97_out :|: TRUE 8.13/2.98 f97_in -> f102_in :|: TRUE 8.13/2.98 f62_in -> f93_in :|: TRUE 8.13/2.98 f93_out -> f62_out :|: TRUE 8.13/2.98 f102_in -> f109_in :|: TRUE 8.13/2.98 f109_out -> f102_out :|: TRUE 8.13/2.98 f21_out -> f16_out :|: TRUE 8.13/2.98 f16_in -> f21_in :|: TRUE 8.13/2.98 f21_in -> f45_in :|: TRUE 8.13/2.98 f43_out -> f21_out :|: TRUE 8.13/2.98 f45_out -> f21_out :|: TRUE 8.13/2.98 f21_in -> f43_in :|: TRUE 8.13/2.98 f62_out -> f43_out :|: TRUE 8.13/2.98 f43_in -> f62_in :|: TRUE 8.13/2.98 Start term: f16_in 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (91) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 8.13/2.98 Constructed simple dependency graph. 8.13/2.98 8.13/2.98 Simplified to the following IRSwTs: 8.13/2.98 8.13/2.98 intTRSProblem: 8.13/2.98 f109_in -> f115_in :|: TRUE 8.13/2.98 f115_in -> f118_in :|: TRUE 8.13/2.98 f93_in -> f97_in :|: TRUE 8.13/2.98 f118_in -> f62_in :|: TRUE 8.13/2.98 f97_in -> f102_in :|: TRUE 8.13/2.98 f62_in -> f93_in :|: TRUE 8.13/2.98 f102_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (92) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f109_in -> f115_in :|: TRUE 8.13/2.98 f115_in -> f118_in :|: TRUE 8.13/2.98 f93_in -> f97_in :|: TRUE 8.13/2.98 f118_in -> f62_in :|: TRUE 8.13/2.98 f97_in -> f102_in :|: TRUE 8.13/2.98 f62_in -> f93_in :|: TRUE 8.13/2.98 f102_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (93) IntTRSCompressionProof (EQUIVALENT) 8.13/2.98 Compressed rules. 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (94) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f109_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (95) IRSFormatTransformerProof (EQUIVALENT) 8.13/2.98 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (96) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f109_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (97) IRSwTTerminationDigraphProof (EQUIVALENT) 8.13/2.98 Constructed termination digraph! 8.13/2.98 Nodes: 8.13/2.98 (1) f109_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 Arcs: 8.13/2.98 (1) -> (1) 8.13/2.98 8.13/2.98 This digraph is fully evaluated! 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (98) 8.13/2.98 Obligation: 8.13/2.98 8.13/2.98 Termination digraph: 8.13/2.98 Nodes: 8.13/2.98 (1) f109_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 Arcs: 8.13/2.98 (1) -> (1) 8.13/2.98 8.13/2.98 This digraph is fully evaluated! 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (99) FilterProof (EQUIVALENT) 8.13/2.98 Used the following sort dictionary for filtering: 8.13/2.98 f109_in() 8.13/2.98 Replaced non-predefined constructor symbols by 0. 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (100) 8.13/2.98 Obligation: 8.13/2.98 Rules: 8.13/2.98 f109_in -> f109_in :|: TRUE 8.13/2.98 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (101) IntTRSPeriodicNontermProof (COMPLETE) 8.13/2.98 Normalized system to the following form: 8.13/2.98 f(pc) -> f(1) :|: pc = 1 && TRUE 8.13/2.98 Witness term starting non-terminating reduction: f(1) 8.13/2.98 ---------------------------------------- 8.13/2.98 8.13/2.98 (102) 8.13/2.98 NO 8.13/3.01 EOF