3.41/1.59 YES 3.41/1.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 3.41/1.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.41/1.60 3.41/1.60 3.41/1.60 Left Termination of the query pattern 3.41/1.60 3.41/1.60 app(g,g,a) 3.41/1.60 3.41/1.60 w.r.t. the given Prolog program could successfully be proven: 3.41/1.60 3.41/1.60 (0) Prolog 3.41/1.60 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.41/1.60 (2) PiTRS 3.41/1.60 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 3.41/1.60 (4) PiDP 3.41/1.60 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.41/1.60 (6) PiDP 3.41/1.60 (7) UsableRulesProof [EQUIVALENT, 0 ms] 3.41/1.60 (8) PiDP 3.41/1.60 (9) PiDPToQDPProof [SOUND, 13 ms] 3.41/1.60 (10) QDP 3.41/1.60 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.41/1.60 (12) YES 3.41/1.60 3.41/1.60 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (0) 3.41/1.60 Obligation: 3.41/1.60 Clauses: 3.41/1.60 3.41/1.60 app([], X, X). 3.41/1.60 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 3.41/1.60 3.41/1.60 3.41/1.60 Query: app(g,g,a) 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (1) PrologToPiTRSProof (SOUND) 3.41/1.60 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.41/1.60 3.41/1.60 app_in_3: (b,b,f) 3.41/1.60 3.41/1.60 Transforming Prolog into the following Term Rewriting System: 3.41/1.60 3.41/1.60 Pi-finite rewrite system: 3.41/1.60 The TRS R consists of the following rules: 3.41/1.60 3.41/1.60 app_in_gga([], X, X) -> app_out_gga([], X, X) 3.41/1.60 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 3.41/1.60 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 3.41/1.60 3.41/1.60 [] = [] 3.41/1.60 3.41/1.60 app_out_gga(x1, x2, x3) = app_out_gga(x3) 3.41/1.60 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) 3.41/1.60 3.41/1.60 3.41/1.60 3.41/1.60 3.41/1.60 3.41/1.60 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.41/1.60 3.41/1.60 3.41/1.60 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (2) 3.41/1.60 Obligation: 3.41/1.60 Pi-finite rewrite system: 3.41/1.60 The TRS R consists of the following rules: 3.41/1.60 3.41/1.60 app_in_gga([], X, X) -> app_out_gga([], X, X) 3.41/1.60 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 3.41/1.60 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 3.41/1.60 3.41/1.60 [] = [] 3.41/1.60 3.41/1.60 app_out_gga(x1, x2, x3) = app_out_gga(x3) 3.41/1.60 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) 3.41/1.60 3.41/1.60 3.41/1.60 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (3) DependencyPairsProof (EQUIVALENT) 3.41/1.60 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.41/1.60 Pi DP problem: 3.41/1.60 The TRS P consists of the following rules: 3.41/1.60 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 3.41/1.60 3.41/1.60 The TRS R consists of the following rules: 3.41/1.60 3.41/1.60 app_in_gga([], X, X) -> app_out_gga([], X, X) 3.41/1.60 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 3.41/1.60 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 3.41/1.60 3.41/1.60 [] = [] 3.41/1.60 3.41/1.60 app_out_gga(x1, x2, x3) = app_out_gga(x3) 3.41/1.60 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) 3.41/1.60 3.41/1.60 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 3.41/1.60 3.41/1.60 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5) 3.41/1.60 3.41/1.60 3.41/1.60 We have to consider all (P,R,Pi)-chains 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (4) 3.41/1.60 Obligation: 3.41/1.60 Pi DP problem: 3.41/1.60 The TRS P consists of the following rules: 3.41/1.60 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 3.41/1.60 3.41/1.60 The TRS R consists of the following rules: 3.41/1.60 3.41/1.60 app_in_gga([], X, X) -> app_out_gga([], X, X) 3.41/1.60 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 3.41/1.60 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 3.41/1.60 3.41/1.60 [] = [] 3.41/1.60 3.41/1.60 app_out_gga(x1, x2, x3) = app_out_gga(x3) 3.41/1.60 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) 3.41/1.60 3.41/1.60 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 3.41/1.60 3.41/1.60 U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5) 3.41/1.60 3.41/1.60 3.41/1.60 We have to consider all (P,R,Pi)-chains 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (5) DependencyGraphProof (EQUIVALENT) 3.41/1.60 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (6) 3.41/1.60 Obligation: 3.41/1.60 Pi DP problem: 3.41/1.60 The TRS P consists of the following rules: 3.41/1.60 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 3.41/1.60 3.41/1.60 The TRS R consists of the following rules: 3.41/1.60 3.41/1.60 app_in_gga([], X, X) -> app_out_gga([], X, X) 3.41/1.60 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 3.41/1.60 U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 3.41/1.60 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 3.41/1.60 3.41/1.60 [] = [] 3.41/1.60 3.41/1.60 app_out_gga(x1, x2, x3) = app_out_gga(x3) 3.41/1.60 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) 3.41/1.60 3.41/1.60 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 3.41/1.60 3.41/1.60 3.41/1.60 We have to consider all (P,R,Pi)-chains 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (7) UsableRulesProof (EQUIVALENT) 3.41/1.60 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (8) 3.41/1.60 Obligation: 3.41/1.60 Pi DP problem: 3.41/1.60 The TRS P consists of the following rules: 3.41/1.60 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 3.41/1.60 3.41/1.60 R is empty. 3.41/1.60 The argument filtering Pi contains the following mapping: 3.41/1.60 .(x1, x2) = .(x1, x2) 3.41/1.60 3.41/1.60 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 3.41/1.60 3.41/1.60 3.41/1.60 We have to consider all (P,R,Pi)-chains 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (9) PiDPToQDPProof (SOUND) 3.41/1.60 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (10) 3.41/1.60 Obligation: 3.41/1.60 Q DP problem: 3.41/1.60 The TRS P consists of the following rules: 3.41/1.60 3.41/1.60 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 3.41/1.60 3.41/1.60 R is empty. 3.41/1.60 Q is empty. 3.41/1.60 We have to consider all (P,Q,R)-chains. 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (11) QDPSizeChangeProof (EQUIVALENT) 3.41/1.60 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. 3.41/1.60 3.41/1.60 From the DPs we obtained the following set of size-change graphs: 3.41/1.60 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 3.41/1.60 The graph contains the following edges 1 > 1, 2 >= 2 3.41/1.60 3.41/1.60 3.41/1.60 ---------------------------------------- 3.41/1.60 3.41/1.60 (12) 3.41/1.60 YES 3.41/1.64 EOF