/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern factorial(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 14 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 4 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES ---------------------------------------- (0) Obligation: Clauses: isNat(s(X)) :- isNat(X). isNat(0). notEq(s(X), s(Y)) :- notEq(X, Y). notEq(s(X), 0). notEq(0, s(X)). lt(s(X), s(Y)) :- lt(X, Y). lt(0, s(Y)). gt(s(X), s(Y)) :- gt(X, Y). gt(s(X), 0). le(s(X), s(Y)) :- le(X, Y). le(0, s(Y)). le(0, 0). even(s(X)) :- odd(X). even(0). odd(s(X)) :- even(X). odd(s(0)). add(s(X), Y, s(Z)) :- add(X, Y, Z). add(0, X, X). mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)). mult(0, Y, 0). factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)). factorial(0, s(0)). Query: factorial(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: factorial_in_2: (b,f) mult_in_3: (b,b,f) add_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: FACTORIAL_IN_GA(s(X), R) -> U11_GA(X, R, factorial_in_ga(X, Y)) FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) U11_GA(X, R, factorial_out_ga(X, Y)) -> U12_GA(X, R, mult_in_gga(s(X), Y, R)) U11_GA(X, R, factorial_out_ga(X, Y)) -> MULT_IN_GGA(s(X), Y, R) MULT_IN_GGA(s(X), Y, R) -> U9_GGA(X, Y, R, mult_in_gga(X, Y, Z)) MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_GGA(X, Y, R, add_in_gga(Y, Z, R)) U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R) ADD_IN_GGA(s(X), Y, s(Z)) -> U8_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) U11_GA(x1, x2, x3) = U11_GA(x1, x3) U12_GA(x1, x2, x3) = U12_GA(x3) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4) U10_GGA(x1, x2, x3, x4) = U10_GGA(x4) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U8_GGA(x1, x2, x3, x4) = U8_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: FACTORIAL_IN_GA(s(X), R) -> U11_GA(X, R, factorial_in_ga(X, Y)) FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) U11_GA(X, R, factorial_out_ga(X, Y)) -> U12_GA(X, R, mult_in_gga(s(X), Y, R)) U11_GA(X, R, factorial_out_ga(X, Y)) -> MULT_IN_GGA(s(X), Y, R) MULT_IN_GGA(s(X), Y, R) -> U9_GGA(X, Y, R, mult_in_gga(X, Y, Z)) MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_GGA(X, Y, R, add_in_gga(Y, Z, R)) U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R) ADD_IN_GGA(s(X), Y, s(Z)) -> U8_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) U11_GA(x1, x2, x3) = U11_GA(x1, x3) U12_GA(x1, x2, x3) = U12_GA(x3) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4) U10_GGA(x1, x2, x3, x4) = U10_GGA(x4) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U8_GGA(x1, x2, x3, x4) = U8_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) The TRS R consists of the following rules: factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) The argument filtering Pi contains the following mapping: factorial_in_ga(x1, x2) = factorial_in_ga(x1) s(x1) = s(x1) U11_ga(x1, x2, x3) = U11_ga(x1, x3) 0 = 0 factorial_out_ga(x1, x2) = factorial_out_ga(x2) U12_ga(x1, x2, x3) = U12_ga(x3) mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) mult_out_gga(x1, x2, x3) = mult_out_gga(x3) U10_gga(x1, x2, x3, x4) = U10_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) U8_gga(x1, x2, x3, x4) = U8_gga(x4) add_out_gga(x1, x2, x3) = add_out_gga(x3) FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: FACTORIAL_IN_GA(s(X)) -> FACTORIAL_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *FACTORIAL_IN_GA(s(X)) -> FACTORIAL_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES