3.51/1.75 YES 3.78/1.76 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 3.78/1.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.78/1.76 3.78/1.76 3.78/1.76 Left Termination of the query pattern 3.78/1.76 3.78/1.76 list(g) 3.78/1.76 3.78/1.76 w.r.t. the given Prolog program could successfully be proven: 3.78/1.76 3.78/1.76 (0) Prolog 3.78/1.76 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.78/1.76 (2) PiTRS 3.78/1.76 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 3.78/1.76 (4) PiDP 3.78/1.76 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.78/1.76 (6) PiDP 3.78/1.76 (7) UsableRulesProof [EQUIVALENT, 0 ms] 3.78/1.76 (8) PiDP 3.78/1.76 (9) PiDPToQDPProof [EQUIVALENT, 0 ms] 3.78/1.76 (10) QDP 3.78/1.76 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.78/1.76 (12) YES 3.78/1.76 3.78/1.76 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (0) 3.78/1.76 Obligation: 3.78/1.76 Clauses: 3.78/1.76 3.78/1.76 list(.(H, Ts)) :- list(Ts). 3.78/1.76 list([]). 3.78/1.76 3.78/1.76 3.78/1.76 Query: list(g) 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (1) PrologToPiTRSProof (SOUND) 3.78/1.76 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.78/1.76 3.78/1.76 list_in_1: (b) 3.78/1.76 3.78/1.76 Transforming Prolog into the following Term Rewriting System: 3.78/1.76 3.78/1.76 Pi-finite rewrite system: 3.78/1.76 The TRS R consists of the following rules: 3.78/1.76 3.78/1.76 list_in_g(.(H, Ts)) -> U1_g(H, Ts, list_in_g(Ts)) 3.78/1.76 list_in_g([]) -> list_out_g([]) 3.78/1.76 U1_g(H, Ts, list_out_g(Ts)) -> list_out_g(.(H, Ts)) 3.78/1.76 3.78/1.76 Pi is empty. 3.78/1.76 3.78/1.76 3.78/1.76 3.78/1.76 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.78/1.76 3.78/1.76 3.78/1.76 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (2) 3.78/1.76 Obligation: 3.78/1.76 Pi-finite rewrite system: 3.78/1.76 The TRS R consists of the following rules: 3.78/1.76 3.78/1.76 list_in_g(.(H, Ts)) -> U1_g(H, Ts, list_in_g(Ts)) 3.78/1.76 list_in_g([]) -> list_out_g([]) 3.78/1.76 U1_g(H, Ts, list_out_g(Ts)) -> list_out_g(.(H, Ts)) 3.78/1.76 3.78/1.76 Pi is empty. 3.78/1.76 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (3) DependencyPairsProof (EQUIVALENT) 3.78/1.76 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.78/1.76 Pi DP problem: 3.78/1.76 The TRS P consists of the following rules: 3.78/1.76 3.78/1.76 LIST_IN_G(.(H, Ts)) -> U1_G(H, Ts, list_in_g(Ts)) 3.78/1.76 LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 3.78/1.76 The TRS R consists of the following rules: 3.78/1.76 3.78/1.76 list_in_g(.(H, Ts)) -> U1_g(H, Ts, list_in_g(Ts)) 3.78/1.76 list_in_g([]) -> list_out_g([]) 3.78/1.76 U1_g(H, Ts, list_out_g(Ts)) -> list_out_g(.(H, Ts)) 3.78/1.76 3.78/1.76 Pi is empty. 3.78/1.76 We have to consider all (P,R,Pi)-chains 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (4) 3.78/1.76 Obligation: 3.78/1.76 Pi DP problem: 3.78/1.76 The TRS P consists of the following rules: 3.78/1.76 3.78/1.76 LIST_IN_G(.(H, Ts)) -> U1_G(H, Ts, list_in_g(Ts)) 3.78/1.76 LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 3.78/1.76 The TRS R consists of the following rules: 3.78/1.76 3.78/1.76 list_in_g(.(H, Ts)) -> U1_g(H, Ts, list_in_g(Ts)) 3.78/1.76 list_in_g([]) -> list_out_g([]) 3.78/1.76 U1_g(H, Ts, list_out_g(Ts)) -> list_out_g(.(H, Ts)) 3.78/1.76 3.78/1.76 Pi is empty. 3.78/1.76 We have to consider all (P,R,Pi)-chains 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (5) DependencyGraphProof (EQUIVALENT) 3.78/1.76 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (6) 3.78/1.76 Obligation: 3.78/1.76 Pi DP problem: 3.78/1.76 The TRS P consists of the following rules: 3.78/1.76 3.78/1.76 LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 3.78/1.76 The TRS R consists of the following rules: 3.78/1.76 3.78/1.76 list_in_g(.(H, Ts)) -> U1_g(H, Ts, list_in_g(Ts)) 3.78/1.76 list_in_g([]) -> list_out_g([]) 3.78/1.76 U1_g(H, Ts, list_out_g(Ts)) -> list_out_g(.(H, Ts)) 3.78/1.76 3.78/1.76 Pi is empty. 3.78/1.76 We have to consider all (P,R,Pi)-chains 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (7) UsableRulesProof (EQUIVALENT) 3.78/1.76 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (8) 3.78/1.76 Obligation: 3.78/1.76 Pi DP problem: 3.78/1.76 The TRS P consists of the following rules: 3.78/1.76 3.78/1.76 LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 3.78/1.76 R is empty. 3.78/1.76 Pi is empty. 3.78/1.76 We have to consider all (P,R,Pi)-chains 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (9) PiDPToQDPProof (EQUIVALENT) 3.78/1.76 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (10) 3.78/1.76 Obligation: 3.78/1.76 Q DP problem: 3.78/1.76 The TRS P consists of the following rules: 3.78/1.76 3.78/1.76 LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 3.78/1.76 R is empty. 3.78/1.76 Q is empty. 3.78/1.76 We have to consider all (P,Q,R)-chains. 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (11) QDPSizeChangeProof (EQUIVALENT) 3.78/1.76 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. 3.78/1.76 3.78/1.76 From the DPs we obtained the following set of size-change graphs: 3.78/1.76 *LIST_IN_G(.(H, Ts)) -> LIST_IN_G(Ts) 3.78/1.76 The graph contains the following edges 1 > 1 3.78/1.76 3.78/1.76 3.78/1.76 ---------------------------------------- 3.78/1.76 3.78/1.76 (12) 3.78/1.76 YES 3.82/1.79 EOF