3.59/1.84 YES 3.59/1.84 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 3.59/1.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.59/1.84 3.59/1.84 3.59/1.84 Left Termination of the query pattern 3.59/1.84 3.59/1.84 reverse(g,a) 3.59/1.84 3.59/1.84 w.r.t. the given Prolog program could successfully be proven: 3.59/1.84 3.59/1.84 (0) Prolog 3.59/1.84 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.59/1.84 (2) PiTRS 3.59/1.84 (3) DependencyPairsProof [EQUIVALENT, 5 ms] 3.59/1.84 (4) PiDP 3.59/1.84 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.59/1.84 (6) PiDP 3.59/1.84 (7) UsableRulesProof [EQUIVALENT, 0 ms] 3.59/1.84 (8) PiDP 3.59/1.84 (9) PiDPToQDPProof [SOUND, 0 ms] 3.59/1.84 (10) QDP 3.59/1.84 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.59/1.84 (12) YES 3.59/1.84 3.59/1.84 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (0) 3.59/1.84 Obligation: 3.59/1.84 Clauses: 3.59/1.84 3.59/1.84 reverse(L, LR) :- revacc(L, LR, []). 3.59/1.84 revacc([], L, L). 3.59/1.84 revacc(.(EL, T), R, A) :- revacc(T, R, .(EL, A)). 3.59/1.84 3.59/1.84 3.59/1.84 Query: reverse(g,a) 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (1) PrologToPiTRSProof (SOUND) 3.59/1.84 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.59/1.84 3.59/1.84 reverse_in_2: (b,f) 3.59/1.84 3.59/1.84 revacc_in_3: (b,f,b) 3.59/1.84 3.59/1.84 Transforming Prolog into the following Term Rewriting System: 3.59/1.84 3.59/1.84 Pi-finite rewrite system: 3.59/1.84 The TRS R consists of the following rules: 3.59/1.84 3.59/1.84 reverse_in_ga(L, LR) -> U1_ga(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 revacc_in_gag([], L, L) -> revacc_out_gag([], L, L) 3.59/1.84 revacc_in_gag(.(EL, T), R, A) -> U2_gag(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 U2_gag(EL, T, R, A, revacc_out_gag(T, R, .(EL, A))) -> revacc_out_gag(.(EL, T), R, A) 3.59/1.84 U1_ga(L, LR, revacc_out_gag(L, LR, [])) -> reverse_out_ga(L, LR) 3.59/1.84 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 reverse_in_ga(x1, x2) = reverse_in_ga(x1) 3.59/1.84 3.59/1.84 U1_ga(x1, x2, x3) = U1_ga(x3) 3.59/1.84 3.59/1.84 revacc_in_gag(x1, x2, x3) = revacc_in_gag(x1, x3) 3.59/1.84 3.59/1.84 [] = [] 3.59/1.84 3.59/1.84 revacc_out_gag(x1, x2, x3) = revacc_out_gag(x2) 3.59/1.84 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) 3.59/1.84 3.59/1.84 reverse_out_ga(x1, x2) = reverse_out_ga(x2) 3.59/1.84 3.59/1.84 3.59/1.84 3.59/1.84 3.59/1.84 3.59/1.84 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.59/1.84 3.59/1.84 3.59/1.84 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (2) 3.59/1.84 Obligation: 3.59/1.84 Pi-finite rewrite system: 3.59/1.84 The TRS R consists of the following rules: 3.59/1.84 3.59/1.84 reverse_in_ga(L, LR) -> U1_ga(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 revacc_in_gag([], L, L) -> revacc_out_gag([], L, L) 3.59/1.84 revacc_in_gag(.(EL, T), R, A) -> U2_gag(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 U2_gag(EL, T, R, A, revacc_out_gag(T, R, .(EL, A))) -> revacc_out_gag(.(EL, T), R, A) 3.59/1.84 U1_ga(L, LR, revacc_out_gag(L, LR, [])) -> reverse_out_ga(L, LR) 3.59/1.84 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 reverse_in_ga(x1, x2) = reverse_in_ga(x1) 3.59/1.84 3.59/1.84 U1_ga(x1, x2, x3) = U1_ga(x3) 3.59/1.84 3.59/1.84 revacc_in_gag(x1, x2, x3) = revacc_in_gag(x1, x3) 3.59/1.84 3.59/1.84 [] = [] 3.59/1.84 3.59/1.84 revacc_out_gag(x1, x2, x3) = revacc_out_gag(x2) 3.59/1.84 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) 3.59/1.84 3.59/1.84 reverse_out_ga(x1, x2) = reverse_out_ga(x2) 3.59/1.84 3.59/1.84 3.59/1.84 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (3) DependencyPairsProof (EQUIVALENT) 3.59/1.84 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.59/1.84 Pi DP problem: 3.59/1.84 The TRS P consists of the following rules: 3.59/1.84 3.59/1.84 REVERSE_IN_GA(L, LR) -> U1_GA(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 REVERSE_IN_GA(L, LR) -> REVACC_IN_GAG(L, LR, []) 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> U2_GAG(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> REVACC_IN_GAG(T, R, .(EL, A)) 3.59/1.84 3.59/1.84 The TRS R consists of the following rules: 3.59/1.84 3.59/1.84 reverse_in_ga(L, LR) -> U1_ga(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 revacc_in_gag([], L, L) -> revacc_out_gag([], L, L) 3.59/1.84 revacc_in_gag(.(EL, T), R, A) -> U2_gag(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 U2_gag(EL, T, R, A, revacc_out_gag(T, R, .(EL, A))) -> revacc_out_gag(.(EL, T), R, A) 3.59/1.84 U1_ga(L, LR, revacc_out_gag(L, LR, [])) -> reverse_out_ga(L, LR) 3.59/1.84 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 reverse_in_ga(x1, x2) = reverse_in_ga(x1) 3.59/1.84 3.59/1.84 U1_ga(x1, x2, x3) = U1_ga(x3) 3.59/1.84 3.59/1.84 revacc_in_gag(x1, x2, x3) = revacc_in_gag(x1, x3) 3.59/1.84 3.59/1.84 [] = [] 3.59/1.84 3.59/1.84 revacc_out_gag(x1, x2, x3) = revacc_out_gag(x2) 3.59/1.84 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) 3.59/1.84 3.59/1.84 reverse_out_ga(x1, x2) = reverse_out_ga(x2) 3.59/1.84 3.59/1.84 REVERSE_IN_GA(x1, x2) = REVERSE_IN_GA(x1) 3.59/1.84 3.59/1.84 U1_GA(x1, x2, x3) = U1_GA(x3) 3.59/1.84 3.59/1.84 REVACC_IN_GAG(x1, x2, x3) = REVACC_IN_GAG(x1, x3) 3.59/1.84 3.59/1.84 U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5) 3.59/1.84 3.59/1.84 3.59/1.84 We have to consider all (P,R,Pi)-chains 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (4) 3.59/1.84 Obligation: 3.59/1.84 Pi DP problem: 3.59/1.84 The TRS P consists of the following rules: 3.59/1.84 3.59/1.84 REVERSE_IN_GA(L, LR) -> U1_GA(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 REVERSE_IN_GA(L, LR) -> REVACC_IN_GAG(L, LR, []) 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> U2_GAG(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> REVACC_IN_GAG(T, R, .(EL, A)) 3.59/1.84 3.59/1.84 The TRS R consists of the following rules: 3.59/1.84 3.59/1.84 reverse_in_ga(L, LR) -> U1_ga(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 revacc_in_gag([], L, L) -> revacc_out_gag([], L, L) 3.59/1.84 revacc_in_gag(.(EL, T), R, A) -> U2_gag(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 U2_gag(EL, T, R, A, revacc_out_gag(T, R, .(EL, A))) -> revacc_out_gag(.(EL, T), R, A) 3.59/1.84 U1_ga(L, LR, revacc_out_gag(L, LR, [])) -> reverse_out_ga(L, LR) 3.59/1.84 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 reverse_in_ga(x1, x2) = reverse_in_ga(x1) 3.59/1.84 3.59/1.84 U1_ga(x1, x2, x3) = U1_ga(x3) 3.59/1.84 3.59/1.84 revacc_in_gag(x1, x2, x3) = revacc_in_gag(x1, x3) 3.59/1.84 3.59/1.84 [] = [] 3.59/1.84 3.59/1.84 revacc_out_gag(x1, x2, x3) = revacc_out_gag(x2) 3.59/1.84 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) 3.59/1.84 3.59/1.84 reverse_out_ga(x1, x2) = reverse_out_ga(x2) 3.59/1.84 3.59/1.84 REVERSE_IN_GA(x1, x2) = REVERSE_IN_GA(x1) 3.59/1.84 3.59/1.84 U1_GA(x1, x2, x3) = U1_GA(x3) 3.59/1.84 3.59/1.84 REVACC_IN_GAG(x1, x2, x3) = REVACC_IN_GAG(x1, x3) 3.59/1.84 3.59/1.84 U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5) 3.59/1.84 3.59/1.84 3.59/1.84 We have to consider all (P,R,Pi)-chains 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (5) DependencyGraphProof (EQUIVALENT) 3.59/1.84 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (6) 3.59/1.84 Obligation: 3.59/1.84 Pi DP problem: 3.59/1.84 The TRS P consists of the following rules: 3.59/1.84 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> REVACC_IN_GAG(T, R, .(EL, A)) 3.59/1.84 3.59/1.84 The TRS R consists of the following rules: 3.59/1.84 3.59/1.84 reverse_in_ga(L, LR) -> U1_ga(L, LR, revacc_in_gag(L, LR, [])) 3.59/1.84 revacc_in_gag([], L, L) -> revacc_out_gag([], L, L) 3.59/1.84 revacc_in_gag(.(EL, T), R, A) -> U2_gag(EL, T, R, A, revacc_in_gag(T, R, .(EL, A))) 3.59/1.84 U2_gag(EL, T, R, A, revacc_out_gag(T, R, .(EL, A))) -> revacc_out_gag(.(EL, T), R, A) 3.59/1.84 U1_ga(L, LR, revacc_out_gag(L, LR, [])) -> reverse_out_ga(L, LR) 3.59/1.84 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 reverse_in_ga(x1, x2) = reverse_in_ga(x1) 3.59/1.84 3.59/1.84 U1_ga(x1, x2, x3) = U1_ga(x3) 3.59/1.84 3.59/1.84 revacc_in_gag(x1, x2, x3) = revacc_in_gag(x1, x3) 3.59/1.84 3.59/1.84 [] = [] 3.59/1.84 3.59/1.84 revacc_out_gag(x1, x2, x3) = revacc_out_gag(x2) 3.59/1.84 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) 3.59/1.84 3.59/1.84 reverse_out_ga(x1, x2) = reverse_out_ga(x2) 3.59/1.84 3.59/1.84 REVACC_IN_GAG(x1, x2, x3) = REVACC_IN_GAG(x1, x3) 3.59/1.84 3.59/1.84 3.59/1.84 We have to consider all (P,R,Pi)-chains 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (7) UsableRulesProof (EQUIVALENT) 3.59/1.84 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (8) 3.59/1.84 Obligation: 3.59/1.84 Pi DP problem: 3.59/1.84 The TRS P consists of the following rules: 3.59/1.84 3.59/1.84 REVACC_IN_GAG(.(EL, T), R, A) -> REVACC_IN_GAG(T, R, .(EL, A)) 3.59/1.84 3.59/1.84 R is empty. 3.59/1.84 The argument filtering Pi contains the following mapping: 3.59/1.84 .(x1, x2) = .(x1, x2) 3.59/1.84 3.59/1.84 REVACC_IN_GAG(x1, x2, x3) = REVACC_IN_GAG(x1, x3) 3.59/1.84 3.59/1.84 3.59/1.84 We have to consider all (P,R,Pi)-chains 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (9) PiDPToQDPProof (SOUND) 3.59/1.84 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (10) 3.59/1.84 Obligation: 3.59/1.84 Q DP problem: 3.59/1.84 The TRS P consists of the following rules: 3.59/1.84 3.59/1.84 REVACC_IN_GAG(.(EL, T), A) -> REVACC_IN_GAG(T, .(EL, A)) 3.59/1.84 3.59/1.84 R is empty. 3.59/1.84 Q is empty. 3.59/1.84 We have to consider all (P,Q,R)-chains. 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (11) QDPSizeChangeProof (EQUIVALENT) 3.59/1.84 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. 3.59/1.84 3.59/1.84 From the DPs we obtained the following set of size-change graphs: 3.59/1.84 *REVACC_IN_GAG(.(EL, T), A) -> REVACC_IN_GAG(T, .(EL, A)) 3.59/1.84 The graph contains the following edges 1 > 1 3.59/1.84 3.59/1.84 3.59/1.84 ---------------------------------------- 3.59/1.84 3.59/1.84 (12) 3.59/1.84 YES 3.94/1.87 EOF