4.46/1.92 YES 4.64/2.00 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.64/2.00 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.64/2.00 4.64/2.00 4.64/2.00 Left Termination of the query pattern 4.64/2.00 4.64/2.00 factorial(g,a) 4.64/2.00 4.64/2.00 w.r.t. the given Prolog program could successfully be proven: 4.64/2.00 4.64/2.00 (0) Prolog 4.64/2.00 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.64/2.00 (2) PiTRS 4.64/2.00 (3) DependencyPairsProof [EQUIVALENT, 7 ms] 4.64/2.00 (4) PiDP 4.64/2.00 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.64/2.00 (6) AND 4.64/2.00 (7) PiDP 4.64/2.00 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.64/2.00 (9) PiDP 4.64/2.00 (10) PiDPToQDPProof [SOUND, 1 ms] 4.64/2.00 (11) QDP 4.64/2.00 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.64/2.00 (13) YES 4.64/2.00 (14) PiDP 4.64/2.00 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.64/2.00 (16) PiDP 4.64/2.00 (17) PiDPToQDPProof [SOUND, 0 ms] 4.64/2.00 (18) QDP 4.64/2.00 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.64/2.00 (20) YES 4.64/2.00 (21) PiDP 4.64/2.00 (22) UsableRulesProof [EQUIVALENT, 0 ms] 4.64/2.00 (23) PiDP 4.64/2.00 (24) PiDPToQDPProof [SOUND, 0 ms] 4.64/2.00 (25) QDP 4.64/2.00 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.64/2.00 (27) YES 4.64/2.00 4.64/2.00 4.64/2.00 ---------------------------------------- 4.64/2.00 4.64/2.00 (0) 4.64/2.00 Obligation: 4.64/2.00 Clauses: 4.64/2.00 4.64/2.00 isNat(s(X)) :- isNat(X). 4.64/2.00 isNat(0). 4.64/2.00 notEq(s(X), s(Y)) :- notEq(X, Y). 4.64/2.00 notEq(s(X), 0). 4.64/2.00 notEq(0, s(X)). 4.64/2.00 lt(s(X), s(Y)) :- lt(X, Y). 4.64/2.00 lt(0, s(Y)). 4.64/2.00 gt(s(X), s(Y)) :- gt(X, Y). 4.64/2.00 gt(s(X), 0). 4.64/2.00 le(s(X), s(Y)) :- le(X, Y). 4.64/2.00 le(0, s(Y)). 4.64/2.00 le(0, 0). 4.64/2.00 even(s(X)) :- odd(X). 4.64/2.00 even(0). 4.64/2.00 odd(s(X)) :- even(X). 4.64/2.00 odd(s(0)). 4.64/2.00 add(s(X), Y, s(Z)) :- add(X, Y, Z). 4.64/2.00 add(0, X, X). 4.64/2.00 mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)). 4.64/2.00 mult(0, Y, 0). 4.64/2.00 factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)). 4.64/2.00 factorial(0, s(0)). 4.64/2.00 4.64/2.00 4.64/2.00 Query: factorial(g,a) 4.64/2.00 ---------------------------------------- 4.64/2.00 4.64/2.00 (1) PrologToPiTRSProof (SOUND) 4.64/2.00 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.64/2.00 4.64/2.00 factorial_in_2: (b,f) 4.64/2.00 4.64/2.00 mult_in_3: (b,b,f) 4.64/2.00 4.64/2.00 add_in_3: (b,b,f) 4.64/2.00 4.64/2.00 Transforming Prolog into the following Term Rewriting System: 4.64/2.00 4.64/2.00 Pi-finite rewrite system: 4.64/2.00 The TRS R consists of the following rules: 4.64/2.00 4.64/2.00 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.00 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.00 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.00 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.00 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.00 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.00 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.00 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.00 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.00 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.00 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.00 4.64/2.00 The argument filtering Pi contains the following mapping: 4.64/2.00 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.00 4.64/2.00 s(x1) = s(x1) 4.64/2.00 4.64/2.00 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.00 4.64/2.00 0 = 0 4.64/2.00 4.64/2.00 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.00 4.64/2.00 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.00 4.64/2.00 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 4.64/2.01 4.64/2.01 4.64/2.01 4.64/2.01 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.64/2.01 4.64/2.01 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (2) 4.64/2.01 Obligation: 4.64/2.01 Pi-finite rewrite system: 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (3) DependencyPairsProof (EQUIVALENT) 4.64/2.01 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> U11_GA(X, R, factorial_in_ga(X, Y)) 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) 4.64/2.01 U11_GA(X, R, factorial_out_ga(X, Y)) -> U12_GA(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 U11_GA(X, R, factorial_out_ga(X, Y)) -> MULT_IN_GGA(s(X), Y, R) 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> U9_GGA(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) 4.64/2.01 U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_GGA(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R) 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> U8_GGA(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) 4.64/2.01 4.64/2.01 U11_GA(x1, x2, x3) = U11_GA(x1, x3) 4.64/2.01 4.64/2.01 U12_GA(x1, x2, x3) = U12_GA(x3) 4.64/2.01 4.64/2.01 MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4) 4.64/2.01 4.64/2.01 U10_GGA(x1, x2, x3, x4) = U10_GGA(x4) 4.64/2.01 4.64/2.01 ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 U8_GGA(x1, x2, x3, x4) = U8_GGA(x4) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (4) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> U11_GA(X, R, factorial_in_ga(X, Y)) 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) 4.64/2.01 U11_GA(X, R, factorial_out_ga(X, Y)) -> U12_GA(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 U11_GA(X, R, factorial_out_ga(X, Y)) -> MULT_IN_GGA(s(X), Y, R) 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> U9_GGA(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) 4.64/2.01 U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_GGA(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R) 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> U8_GGA(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) 4.64/2.01 4.64/2.01 U11_GA(x1, x2, x3) = U11_GA(x1, x3) 4.64/2.01 4.64/2.01 U12_GA(x1, x2, x3) = U12_GA(x3) 4.64/2.01 4.64/2.01 MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4) 4.64/2.01 4.64/2.01 U10_GGA(x1, x2, x3, x4) = U10_GGA(x4) 4.64/2.01 4.64/2.01 ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 U8_GGA(x1, x2, x3, x4) = U8_GGA(x4) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (5) DependencyGraphProof (EQUIVALENT) 4.64/2.01 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (6) 4.64/2.01 Complex Obligation (AND) 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (7) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (8) UsableRulesProof (EQUIVALENT) 4.64/2.01 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (9) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (10) PiDPToQDPProof (SOUND) 4.64/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (11) 4.64/2.01 Obligation: 4.64/2.01 Q DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 Q is empty. 4.64/2.01 We have to consider all (P,Q,R)-chains. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (12) QDPSizeChangeProof (EQUIVALENT) 4.64/2.01 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. 4.64/2.01 4.64/2.01 From the DPs we obtained the following set of size-change graphs: 4.64/2.01 *ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) 4.64/2.01 The graph contains the following edges 1 > 1, 2 >= 2 4.64/2.01 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (13) 4.64/2.01 YES 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (14) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (15) UsableRulesProof (EQUIVALENT) 4.64/2.01 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (16) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (17) PiDPToQDPProof (SOUND) 4.64/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (18) 4.64/2.01 Obligation: 4.64/2.01 Q DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 Q is empty. 4.64/2.01 We have to consider all (P,Q,R)-chains. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (19) QDPSizeChangeProof (EQUIVALENT) 4.64/2.01 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. 4.64/2.01 4.64/2.01 From the DPs we obtained the following set of size-change graphs: 4.64/2.01 *MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y) 4.64/2.01 The graph contains the following edges 1 > 1, 2 >= 2 4.64/2.01 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (20) 4.64/2.01 YES 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (21) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) 4.64/2.01 4.64/2.01 The TRS R consists of the following rules: 4.64/2.01 4.64/2.01 factorial_in_ga(s(X), R) -> U11_ga(X, R, factorial_in_ga(X, Y)) 4.64/2.01 factorial_in_ga(0, s(0)) -> factorial_out_ga(0, s(0)) 4.64/2.01 U11_ga(X, R, factorial_out_ga(X, Y)) -> U12_ga(X, R, mult_in_gga(s(X), Y, R)) 4.64/2.01 mult_in_gga(s(X), Y, R) -> U9_gga(X, Y, R, mult_in_gga(X, Y, Z)) 4.64/2.01 mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0) 4.64/2.01 U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U10_gga(X, Y, R, add_in_gga(Y, Z, R)) 4.64/2.01 add_in_gga(s(X), Y, s(Z)) -> U8_gga(X, Y, Z, add_in_gga(X, Y, Z)) 4.64/2.01 add_in_gga(0, X, X) -> add_out_gga(0, X, X) 4.64/2.01 U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) 4.64/2.01 U10_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R) 4.64/2.01 U12_ga(X, R, mult_out_gga(s(X), Y, R)) -> factorial_out_ga(s(X), R) 4.64/2.01 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 factorial_in_ga(x1, x2) = factorial_in_ga(x1) 4.64/2.01 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 U11_ga(x1, x2, x3) = U11_ga(x1, x3) 4.64/2.01 4.64/2.01 0 = 0 4.64/2.01 4.64/2.01 factorial_out_ga(x1, x2) = factorial_out_ga(x2) 4.64/2.01 4.64/2.01 U12_ga(x1, x2, x3) = U12_ga(x3) 4.64/2.01 4.64/2.01 mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4) 4.64/2.01 4.64/2.01 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) 4.64/2.01 4.64/2.01 U10_gga(x1, x2, x3, x4) = U10_gga(x4) 4.64/2.01 4.64/2.01 add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) 4.64/2.01 4.64/2.01 U8_gga(x1, x2, x3, x4) = U8_gga(x4) 4.64/2.01 4.64/2.01 add_out_gga(x1, x2, x3) = add_out_gga(x3) 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (22) UsableRulesProof (EQUIVALENT) 4.64/2.01 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (23) 4.64/2.01 Obligation: 4.64/2.01 Pi DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(s(X), R) -> FACTORIAL_IN_GA(X, Y) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 The argument filtering Pi contains the following mapping: 4.64/2.01 s(x1) = s(x1) 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1) 4.64/2.01 4.64/2.01 4.64/2.01 We have to consider all (P,R,Pi)-chains 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (24) PiDPToQDPProof (SOUND) 4.64/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (25) 4.64/2.01 Obligation: 4.64/2.01 Q DP problem: 4.64/2.01 The TRS P consists of the following rules: 4.64/2.01 4.64/2.01 FACTORIAL_IN_GA(s(X)) -> FACTORIAL_IN_GA(X) 4.64/2.01 4.64/2.01 R is empty. 4.64/2.01 Q is empty. 4.64/2.01 We have to consider all (P,Q,R)-chains. 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (26) QDPSizeChangeProof (EQUIVALENT) 4.64/2.01 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. 4.64/2.01 4.64/2.01 From the DPs we obtained the following set of size-change graphs: 4.64/2.01 *FACTORIAL_IN_GA(s(X)) -> FACTORIAL_IN_GA(X) 4.64/2.01 The graph contains the following edges 1 > 1 4.64/2.01 4.64/2.01 4.64/2.01 ---------------------------------------- 4.64/2.01 4.64/2.01 (27) 4.64/2.01 YES 4.79/2.04 EOF