4.83/2.12 YES 4.83/2.13 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.83/2.13 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.83/2.13 4.83/2.13 4.83/2.13 Left Termination of the query pattern 4.83/2.13 4.83/2.13 plus(g,a,a) 4.83/2.13 4.83/2.13 w.r.t. the given Prolog program could successfully be proven: 4.83/2.13 4.83/2.13 (0) Prolog 4.83/2.13 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.83/2.13 (2) PiTRS 4.83/2.13 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.83/2.13 (4) PiDP 4.83/2.13 (5) DependencyGraphProof [EQUIVALENT, 2 ms] 4.83/2.13 (6) AND 4.83/2.13 (7) PiDP 4.83/2.13 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.83/2.13 (9) PiDP 4.83/2.13 (10) PiDPToQDPProof [SOUND, 0 ms] 4.83/2.13 (11) QDP 4.83/2.13 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.83/2.13 (13) YES 4.83/2.13 (14) PiDP 4.83/2.13 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.83/2.13 (16) PiDP 4.83/2.13 (17) PiDPToQDPProof [SOUND, 0 ms] 4.83/2.13 (18) QDP 4.83/2.13 (19) MRRProof [EQUIVALENT, 3 ms] 4.83/2.13 (20) QDP 4.83/2.13 (21) PisEmptyProof [EQUIVALENT, 0 ms] 4.83/2.13 (22) YES 4.83/2.13 4.83/2.13 4.83/2.13 ---------------------------------------- 4.83/2.13 4.83/2.13 (0) 4.83/2.13 Obligation: 4.83/2.13 Clauses: 4.83/2.13 4.83/2.13 p(s(0), 0). 4.83/2.13 p(s(s(X)), s(s(Y))) :- p(s(X), s(Y)). 4.83/2.13 plus(0, Y, Y). 4.83/2.13 plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z)). 4.83/2.13 4.83/2.13 4.83/2.13 Query: plus(g,a,a) 4.83/2.13 ---------------------------------------- 4.83/2.13 4.83/2.13 (1) PrologToPiTRSProof (SOUND) 4.83/2.13 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.83/2.13 4.83/2.13 plus_in_3: (b,f,f) 4.83/2.13 4.83/2.13 p_in_2: (b,f) 4.83/2.13 4.83/2.13 Transforming Prolog into the following Term Rewriting System: 4.83/2.13 4.83/2.13 Pi-finite rewrite system: 4.83/2.13 The TRS R consists of the following rules: 4.83/2.13 4.83/2.13 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.13 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.13 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.13 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.13 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.13 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.13 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.13 4.83/2.13 The argument filtering Pi contains the following mapping: 4.83/2.13 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.13 4.83/2.13 0 = 0 4.83/2.13 4.83/2.13 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.13 4.83/2.13 s(x1) = s(x1) 4.83/2.13 4.83/2.13 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.13 4.83/2.13 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.13 4.83/2.13 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.13 4.83/2.13 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.13 4.83/2.13 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.13 4.83/2.13 4.83/2.13 4.83/2.13 4.83/2.13 4.83/2.13 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.83/2.13 4.83/2.13 4.83/2.13 4.83/2.13 ---------------------------------------- 4.83/2.13 4.83/2.13 (2) 4.83/2.13 Obligation: 4.83/2.13 Pi-finite rewrite system: 4.83/2.13 The TRS R consists of the following rules: 4.83/2.13 4.83/2.13 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.13 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.13 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.13 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.13 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.13 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.13 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.13 4.83/2.13 The argument filtering Pi contains the following mapping: 4.83/2.13 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.13 4.83/2.13 0 = 0 4.83/2.13 4.83/2.13 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.13 4.83/2.13 s(x1) = s(x1) 4.83/2.13 4.83/2.13 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.13 4.83/2.13 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.13 4.83/2.13 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.13 4.83/2.13 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.13 4.83/2.13 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.13 4.83/2.13 4.83/2.13 4.83/2.13 ---------------------------------------- 4.83/2.13 4.83/2.13 (3) DependencyPairsProof (EQUIVALENT) 4.83/2.13 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.83/2.13 Pi DP problem: 4.83/2.13 The TRS P consists of the following rules: 4.83/2.13 4.83/2.13 PLUS_IN_GAA(s(X), Y, s(Z)) -> U2_GAA(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.13 PLUS_IN_GAA(s(X), Y, s(Z)) -> P_IN_GA(s(X), U) 4.83/2.13 P_IN_GA(s(s(X)), s(s(Y))) -> U1_GA(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.13 P_IN_GA(s(s(X)), s(s(Y))) -> P_IN_GA(s(X), s(Y)) 4.83/2.13 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> U3_GAA(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.13 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> PLUS_IN_GAA(U, Y, Z) 4.83/2.13 4.83/2.13 The TRS R consists of the following rules: 4.83/2.13 4.83/2.13 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.13 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.13 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.13 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.13 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.13 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.13 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.13 4.83/2.13 The argument filtering Pi contains the following mapping: 4.83/2.13 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.13 4.83/2.13 0 = 0 4.83/2.13 4.83/2.13 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.13 4.83/2.13 s(x1) = s(x1) 4.83/2.13 4.83/2.13 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.13 4.83/2.13 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.13 4.83/2.13 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.13 4.83/2.13 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.13 4.83/2.13 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.13 4.83/2.13 PLUS_IN_GAA(x1, x2, x3) = PLUS_IN_GAA(x1) 4.83/2.13 4.83/2.13 U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) 4.83/2.13 4.83/2.13 P_IN_GA(x1, x2) = P_IN_GA(x1) 4.83/2.13 4.83/2.13 U1_GA(x1, x2, x3) = U1_GA(x3) 4.83/2.13 4.83/2.13 U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) 4.83/2.13 4.83/2.13 4.83/2.13 We have to consider all (P,R,Pi)-chains 4.83/2.13 ---------------------------------------- 4.83/2.13 4.83/2.13 (4) 4.83/2.13 Obligation: 4.83/2.13 Pi DP problem: 4.83/2.13 The TRS P consists of the following rules: 4.83/2.13 4.83/2.13 PLUS_IN_GAA(s(X), Y, s(Z)) -> U2_GAA(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 PLUS_IN_GAA(s(X), Y, s(Z)) -> P_IN_GA(s(X), U) 4.83/2.14 P_IN_GA(s(s(X)), s(s(Y))) -> U1_GA(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.14 P_IN_GA(s(s(X)), s(s(Y))) -> P_IN_GA(s(X), s(Y)) 4.83/2.14 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> U3_GAA(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.14 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> PLUS_IN_GAA(U, Y, Z) 4.83/2.14 4.83/2.14 The TRS R consists of the following rules: 4.83/2.14 4.83/2.14 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.14 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.14 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.14 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.14 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.14 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.14 4.83/2.14 The argument filtering Pi contains the following mapping: 4.83/2.14 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.14 4.83/2.14 0 = 0 4.83/2.14 4.83/2.14 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.14 4.83/2.14 s(x1) = s(x1) 4.83/2.14 4.83/2.14 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.14 4.83/2.14 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.14 4.83/2.14 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.14 4.83/2.14 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.14 4.83/2.14 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.14 4.83/2.14 PLUS_IN_GAA(x1, x2, x3) = PLUS_IN_GAA(x1) 4.83/2.14 4.83/2.14 U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) 4.83/2.14 4.83/2.14 P_IN_GA(x1, x2) = P_IN_GA(x1) 4.83/2.14 4.83/2.14 U1_GA(x1, x2, x3) = U1_GA(x3) 4.83/2.14 4.83/2.14 U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) 4.83/2.14 4.83/2.14 4.83/2.14 We have to consider all (P,R,Pi)-chains 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (5) DependencyGraphProof (EQUIVALENT) 4.83/2.14 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (6) 4.83/2.14 Complex Obligation (AND) 4.83/2.14 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (7) 4.83/2.14 Obligation: 4.83/2.14 Pi DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 P_IN_GA(s(s(X)), s(s(Y))) -> P_IN_GA(s(X), s(Y)) 4.83/2.14 4.83/2.14 The TRS R consists of the following rules: 4.83/2.14 4.83/2.14 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.14 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.14 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.14 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.14 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.14 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.14 4.83/2.14 The argument filtering Pi contains the following mapping: 4.83/2.14 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.14 4.83/2.14 0 = 0 4.83/2.14 4.83/2.14 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.14 4.83/2.14 s(x1) = s(x1) 4.83/2.14 4.83/2.14 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.14 4.83/2.14 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.14 4.83/2.14 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.14 4.83/2.14 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.14 4.83/2.14 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.14 4.83/2.14 P_IN_GA(x1, x2) = P_IN_GA(x1) 4.83/2.14 4.83/2.14 4.83/2.14 We have to consider all (P,R,Pi)-chains 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (8) UsableRulesProof (EQUIVALENT) 4.83/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (9) 4.83/2.14 Obligation: 4.83/2.14 Pi DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 P_IN_GA(s(s(X)), s(s(Y))) -> P_IN_GA(s(X), s(Y)) 4.83/2.14 4.83/2.14 R is empty. 4.83/2.14 The argument filtering Pi contains the following mapping: 4.83/2.14 s(x1) = s(x1) 4.83/2.14 4.83/2.14 P_IN_GA(x1, x2) = P_IN_GA(x1) 4.83/2.14 4.83/2.14 4.83/2.14 We have to consider all (P,R,Pi)-chains 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (10) PiDPToQDPProof (SOUND) 4.83/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (11) 4.83/2.14 Obligation: 4.83/2.14 Q DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 P_IN_GA(s(s(X))) -> P_IN_GA(s(X)) 4.83/2.14 4.83/2.14 R is empty. 4.83/2.14 Q is empty. 4.83/2.14 We have to consider all (P,Q,R)-chains. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (12) QDPSizeChangeProof (EQUIVALENT) 4.83/2.14 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. 4.83/2.14 4.83/2.14 From the DPs we obtained the following set of size-change graphs: 4.83/2.14 *P_IN_GA(s(s(X))) -> P_IN_GA(s(X)) 4.83/2.14 The graph contains the following edges 1 > 1 4.83/2.14 4.83/2.14 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (13) 4.83/2.14 YES 4.83/2.14 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (14) 4.83/2.14 Obligation: 4.83/2.14 Pi DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> PLUS_IN_GAA(U, Y, Z) 4.83/2.14 PLUS_IN_GAA(s(X), Y, s(Z)) -> U2_GAA(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 4.83/2.14 The TRS R consists of the following rules: 4.83/2.14 4.83/2.14 plus_in_gaa(0, Y, Y) -> plus_out_gaa(0, Y, Y) 4.83/2.14 plus_in_gaa(s(X), Y, s(Z)) -> U2_gaa(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.14 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.14 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.14 U2_gaa(X, Y, Z, p_out_ga(s(X), U)) -> U3_gaa(X, Y, Z, plus_in_gaa(U, Y, Z)) 4.83/2.14 U3_gaa(X, Y, Z, plus_out_gaa(U, Y, Z)) -> plus_out_gaa(s(X), Y, s(Z)) 4.83/2.14 4.83/2.14 The argument filtering Pi contains the following mapping: 4.83/2.14 plus_in_gaa(x1, x2, x3) = plus_in_gaa(x1) 4.83/2.14 4.83/2.14 0 = 0 4.83/2.14 4.83/2.14 plus_out_gaa(x1, x2, x3) = plus_out_gaa 4.83/2.14 4.83/2.14 s(x1) = s(x1) 4.83/2.14 4.83/2.14 U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) 4.83/2.14 4.83/2.14 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.14 4.83/2.14 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.14 4.83/2.14 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.14 4.83/2.14 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 4.83/2.14 4.83/2.14 PLUS_IN_GAA(x1, x2, x3) = PLUS_IN_GAA(x1) 4.83/2.14 4.83/2.14 U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) 4.83/2.14 4.83/2.14 4.83/2.14 We have to consider all (P,R,Pi)-chains 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (15) UsableRulesProof (EQUIVALENT) 4.83/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (16) 4.83/2.14 Obligation: 4.83/2.14 Pi DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 U2_GAA(X, Y, Z, p_out_ga(s(X), U)) -> PLUS_IN_GAA(U, Y, Z) 4.83/2.14 PLUS_IN_GAA(s(X), Y, s(Z)) -> U2_GAA(X, Y, Z, p_in_ga(s(X), U)) 4.83/2.14 4.83/2.14 The TRS R consists of the following rules: 4.83/2.14 4.83/2.14 p_in_ga(s(0), 0) -> p_out_ga(s(0), 0) 4.83/2.14 p_in_ga(s(s(X)), s(s(Y))) -> U1_ga(X, Y, p_in_ga(s(X), s(Y))) 4.83/2.14 U1_ga(X, Y, p_out_ga(s(X), s(Y))) -> p_out_ga(s(s(X)), s(s(Y))) 4.83/2.14 4.83/2.14 The argument filtering Pi contains the following mapping: 4.83/2.14 0 = 0 4.83/2.14 4.83/2.14 s(x1) = s(x1) 4.83/2.14 4.83/2.14 p_in_ga(x1, x2) = p_in_ga(x1) 4.83/2.14 4.83/2.14 p_out_ga(x1, x2) = p_out_ga(x2) 4.83/2.14 4.83/2.14 U1_ga(x1, x2, x3) = U1_ga(x3) 4.83/2.14 4.83/2.14 PLUS_IN_GAA(x1, x2, x3) = PLUS_IN_GAA(x1) 4.83/2.14 4.83/2.14 U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) 4.83/2.14 4.83/2.14 4.83/2.14 We have to consider all (P,R,Pi)-chains 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (17) PiDPToQDPProof (SOUND) 4.83/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (18) 4.83/2.14 Obligation: 4.83/2.14 Q DP problem: 4.83/2.14 The TRS P consists of the following rules: 4.83/2.14 4.83/2.14 U2_GAA(p_out_ga(U)) -> PLUS_IN_GAA(U) 4.83/2.14 PLUS_IN_GAA(s(X)) -> U2_GAA(p_in_ga(s(X))) 4.83/2.14 4.83/2.14 The TRS R consists of the following rules: 4.83/2.14 4.83/2.14 p_in_ga(s(0)) -> p_out_ga(0) 4.83/2.14 p_in_ga(s(s(X))) -> U1_ga(p_in_ga(s(X))) 4.83/2.14 U1_ga(p_out_ga(s(Y))) -> p_out_ga(s(s(Y))) 4.83/2.14 4.83/2.14 The set Q consists of the following terms: 4.83/2.14 4.83/2.14 p_in_ga(x0) 4.83/2.14 U1_ga(x0) 4.83/2.14 4.83/2.14 We have to consider all (P,Q,R)-chains. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (19) MRRProof (EQUIVALENT) 4.83/2.14 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 4.83/2.14 4.83/2.14 Strictly oriented dependency pairs: 4.83/2.14 4.83/2.14 U2_GAA(p_out_ga(U)) -> PLUS_IN_GAA(U) 4.83/2.14 PLUS_IN_GAA(s(X)) -> U2_GAA(p_in_ga(s(X))) 4.83/2.14 4.83/2.14 Strictly oriented rules of the TRS R: 4.83/2.14 4.83/2.14 p_in_ga(s(0)) -> p_out_ga(0) 4.83/2.14 p_in_ga(s(s(X))) -> U1_ga(p_in_ga(s(X))) 4.83/2.14 U1_ga(p_out_ga(s(Y))) -> p_out_ga(s(s(Y))) 4.83/2.14 4.83/2.14 Used ordering: Knuth-Bendix order [KBO] with precedence:0 > s_1 > p_in_ga_1 > U1_ga_1 > U2_GAA_1 > PLUS_IN_GAA_1 > p_out_ga_1 4.83/2.14 4.83/2.14 and weight map: 4.83/2.14 4.83/2.14 0=1 4.83/2.14 p_in_ga_1=1 4.83/2.14 s_1=4 4.83/2.14 p_out_ga_1=3 4.83/2.14 U1_ga_1=4 4.83/2.14 U2_GAA_1=1 4.83/2.14 PLUS_IN_GAA_1=3 4.83/2.14 4.83/2.14 The variable weight is 1 4.83/2.14 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (20) 4.83/2.14 Obligation: 4.83/2.14 Q DP problem: 4.83/2.14 P is empty. 4.83/2.14 R is empty. 4.83/2.14 The set Q consists of the following terms: 4.83/2.14 4.83/2.14 p_in_ga(x0) 4.83/2.14 U1_ga(x0) 4.83/2.14 4.83/2.14 We have to consider all (P,Q,R)-chains. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (21) PisEmptyProof (EQUIVALENT) 4.83/2.14 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.83/2.14 ---------------------------------------- 4.83/2.14 4.83/2.14 (22) 4.83/2.14 YES 4.83/2.17 EOF