/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern times(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 1 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 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 (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 3 ms] (32) QDP (33) MRRProof [EQUIVALENT, 29 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) TRUE ---------------------------------------- (0) Obligation: Clauses: even(0, true). even(s(0), false). even(s(s(X)), B) :- even(X, B). half(0, 0). half(s(s(X)), s(Y)) :- half(X, Y). plus(0, Y, Y). plus(s(X), Y, s(Z)) :- plus(X, Y, Z). times(0, Y, 0). times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z)). if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z))). if(false, s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)). Query: times(g,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: times_in_3: (b,b,f) even_in_2: (b,f) if_in_4: (b,b,b,f) half_in_2: (b,f) plus_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: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) ---------------------------------------- (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: TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B)) TIMES_IN_GGA(s(X), Y, Z) -> EVEN_IN_GA(s(X), B) EVEN_IN_GA(s(s(X)), B) -> U1_GA(X, B, even_in_ga(X, B)) EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B) U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z) IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1)) IF_IN_GGGA(true, s(X), Y, Z) -> HALF_IN_GA(s(X), X1) HALF_IN_GA(s(s(X)), s(Y)) -> U2_GA(X, Y, half_in_ga(X, Y)) HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1)) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1) U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z)) U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> PLUS_IN_GGA(Y1, Y1, Z) PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) IF_IN_GGGA(false, s(X), Y, Z) -> U9_GGGA(X, Y, Z, times_in_gga(X, Y, U)) IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z)) U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) EVEN_IN_GA(x1, x2) = EVEN_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) U5_GGA(x1, x2, x3, x4) = U5_GGA(x4) IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3) U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x2, x4) HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) U7_GGGA(x1, x2, x3, x4) = U7_GGGA(x4) U8_GGGA(x1, x2, x3, x4) = U8_GGGA(x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) U9_GGGA(x1, x2, x3, x4) = U9_GGGA(x2, x4) U10_GGGA(x1, x2, x3, x4) = U10_GGGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B)) TIMES_IN_GGA(s(X), Y, Z) -> EVEN_IN_GA(s(X), B) EVEN_IN_GA(s(s(X)), B) -> U1_GA(X, B, even_in_ga(X, B)) EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B) U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z) IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1)) IF_IN_GGGA(true, s(X), Y, Z) -> HALF_IN_GA(s(X), X1) HALF_IN_GA(s(s(X)), s(Y)) -> U2_GA(X, Y, half_in_ga(X, Y)) HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1)) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1) U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z)) U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> PLUS_IN_GGA(Y1, Y1, Z) PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) IF_IN_GGGA(false, s(X), Y, Z) -> U9_GGGA(X, Y, Z, times_in_gga(X, Y, U)) IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z)) U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) EVEN_IN_GA(x1, x2) = EVEN_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) U5_GGA(x1, x2, x3, x4) = U5_GGA(x4) IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3) U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x2, x4) HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) U7_GGGA(x1, x2, x3, x4) = U7_GGGA(x4) U8_GGGA(x1, x2, x3, x4) = U8_GGGA(x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) U9_GGGA(x1, x2, x3, x4) = U9_GGGA(x2, x4) U10_GGGA(x1, x2, x3, x4) = U10_GGGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_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: PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) PLUS_IN_GGA(x1, x2, x3) = PLUS_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: PLUS_IN_GGA(s(X), Y) -> PLUS_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: *PLUS_IN_GGA(s(X), Y) -> PLUS_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: HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 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: HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) HALF_IN_GA(x1, x2) = HALF_IN_GA(x1) 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: HALF_IN_GA(s(s(X))) -> HALF_IN_GA(X) 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: *HALF_IN_GA(s(s(X))) -> HALF_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) EVEN_IN_GA(x1, x2) = EVEN_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: EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) EVEN_IN_GA(x1, x2) = EVEN_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: EVEN_IN_GA(s(s(X))) -> EVEN_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: *EVEN_IN_GA(s(s(X))) -> EVEN_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z) IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1)) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1) TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B)) IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) The TRS R consists of the following rules: times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B)) even_in_ga(0, true) -> even_out_ga(0, true) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z)) if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1)) half_in_ga(0, 0) -> half_out_ga(0, 0) half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1)) U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z) if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U)) U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z)) U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z) U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 0 = 0 times_out_gga(x1, x2, x3) = times_out_gga(x3) s(x1) = s(x1) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) U5_gga(x1, x2, x3, x4) = U5_gga(x4) if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3) true = true U6_ggga(x1, x2, x3, x4) = U6_ggga(x2, x4) half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) U7_ggga(x1, x2, x3, x4) = U7_ggga(x4) U8_ggga(x1, x2, x3, x4) = U8_ggga(x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4) false = false U9_ggga(x1, x2, x3, x4) = U9_ggga(x2, x4) U10_ggga(x1, x2, x3, x4) = U10_ggga(x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3) U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z) IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1)) U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1) TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B)) IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) The TRS R consists of the following rules: half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y)) even_in_ga(s(0), false) -> even_out_ga(s(0), false) even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B)) U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y)) U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B) half_in_ga(0, 0) -> half_out_ga(0, 0) even_in_ga(0, true) -> even_out_ga(0, true) The argument filtering Pi contains the following mapping: 0 = 0 s(x1) = s(x1) even_in_ga(x1, x2) = even_in_ga(x1) even_out_ga(x1, x2) = even_out_ga(x2) U1_ga(x1, x2, x3) = U1_ga(x3) true = true half_in_ga(x1, x2) = half_in_ga(x1) half_out_ga(x1, x2) = half_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) false = false TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3) U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGA(X, Y, even_out_ga(B)) -> IF_IN_GGGA(B, s(X), Y) IF_IN_GGGA(true, s(X), Y) -> U6_GGGA(Y, half_in_ga(s(X))) U6_GGGA(Y, half_out_ga(X1)) -> TIMES_IN_GGA(X1, Y) TIMES_IN_GGA(s(X), Y) -> U4_GGA(X, Y, even_in_ga(s(X))) IF_IN_GGGA(false, s(X), Y) -> TIMES_IN_GGA(X, Y) The TRS R consists of the following rules: half_in_ga(s(s(X))) -> U2_ga(half_in_ga(X)) even_in_ga(s(0)) -> even_out_ga(false) even_in_ga(s(s(X))) -> U1_ga(even_in_ga(X)) U2_ga(half_out_ga(Y)) -> half_out_ga(s(Y)) U1_ga(even_out_ga(B)) -> even_out_ga(B) half_in_ga(0) -> half_out_ga(0) even_in_ga(0) -> even_out_ga(true) The set Q consists of the following terms: half_in_ga(x0) even_in_ga(x0) U2_ga(x0) U1_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) MRRProof (EQUIVALENT) 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. Strictly oriented dependency pairs: IF_IN_GGGA(false, s(X), Y) -> TIMES_IN_GGA(X, Y) Strictly oriented rules of the TRS R: half_in_ga(s(s(X))) -> U2_ga(half_in_ga(X)) even_in_ga(s(0)) -> even_out_ga(false) even_in_ga(s(s(X))) -> U1_ga(even_in_ga(X)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(IF_IN_GGGA(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(TIMES_IN_GGA(x_1, x_2)) = 2*x_1 + x_2 POL(U1_ga(x_1)) = x_1 POL(U2_ga(x_1)) = 2 + 2*x_1 POL(U4_GGA(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + x_3 POL(U6_GGGA(x_1, x_2)) = x_1 + x_2 POL(even_in_ga(x_1)) = x_1 POL(even_out_ga(x_1)) = 2*x_1 POL(false) = 0 POL(half_in_ga(x_1)) = x_1 POL(half_out_ga(x_1)) = 2*x_1 POL(s(x_1)) = 1 + 2*x_1 POL(true) = 0 ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGA(X, Y, even_out_ga(B)) -> IF_IN_GGGA(B, s(X), Y) IF_IN_GGGA(true, s(X), Y) -> U6_GGGA(Y, half_in_ga(s(X))) U6_GGGA(Y, half_out_ga(X1)) -> TIMES_IN_GGA(X1, Y) TIMES_IN_GGA(s(X), Y) -> U4_GGA(X, Y, even_in_ga(s(X))) The TRS R consists of the following rules: U2_ga(half_out_ga(Y)) -> half_out_ga(s(Y)) U1_ga(even_out_ga(B)) -> even_out_ga(B) half_in_ga(0) -> half_out_ga(0) even_in_ga(0) -> even_out_ga(true) The set Q consists of the following terms: half_in_ga(x0) even_in_ga(x0) U2_ga(x0) U1_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (36) TRUE