/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern money(a,a,a,a,a,a,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UnifyTransformerProof [EQUIVALENT, 0 ms] (2) Prolog (3) UndefinedPredicateHandlerProof [SOUND, 0 ms] (4) Prolog (5) PrologToPiTRSProof [SOUND, 93 ms] (6) PiTRS (7) DependencyPairsProof [EQUIVALENT, 46 ms] (8) PiDP (9) DependencyGraphProof [EQUIVALENT, 2 ms] (10) AND (11) PiDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) PiDP (14) PiDPToQDPProof [SOUND, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [SOUND, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Clauses: demo :- ','(nl, ','(write('Cryptarithmetic puzzle'), ','(nl, ','(nl, ','(write(' SEND'), ','(nl, ','(write(' + MORE'), ','(nl, ','(write(' --------'), ','(nl, ','(write(' MONEY'), ','(nl, ','(nl, ','(nl, ','(time(X2), ','(money(S, E, N, D, M, O, R, Y), ','(time(T), ','(write('The solution is:'), ','(nl, ','(nl, ','(write(' '), ','(write(S), ','(write(E), ','(write(N), ','(write(D), ','(nl, ','(write(' + '), ','(write(M), ','(write(O), ','(write(R), ','(write(E), ','(nl, ','(write(' --------'), ','(nl, ','(write(' '), ','(write(M), ','(write(O), ','(write(N), ','(write(E), ','(write(Y), ','(nl, ','(nl, ','(write('Solved in '), ','(write(T), ','(write(' msec.'), nl))))))))))))))))))))))))))))))))))))))))))))). time(0). solve(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, []))))))))) :- money(S, E, N, D, M, O, R, Y). money(S, E, N, D, M, O, R, Y) :- ','(carry(C1), ','(carry(C2), ','(carry(C3), ','(carry(C4), ','(=(C4, M), ','(=\=(M, 0), ','(domain(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, []))))))))))), ','(=\=(S, 0), ','(calc(C3, S, M, C4, O), ','(calc(C2, E, O, C3, N), ','(calc(C1, N, R, C2, E), calc(0, D, E, C1, Y)))))))))))). calc(C0, D, E, C1, Y) :- ','(sum(C0, D, CD), ','(sum(CD, E, S), ','(carry10(C1, C10), sum(C10, Y, S)))). sum(X, Y, Z) :- is(Z, +(X, Y)). domain([], X3). domain(.(X1, R), L) :- ','(del(X1, L, NL), domain(R, NL)). del(X, .(X, T), T). del(X, .(Y, T), .(Y, NT)) :- del(X, T, NT). carry(1). carry(0). carry10(0, 0). carry10(1, 10). Query: money(a,a,a,a,a,a,a,a) ---------------------------------------- (1) UnifyTransformerProof (EQUIVALENT) Added a fact for the built-in = predicate [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: demo :- ','(nl, ','(write('Cryptarithmetic puzzle'), ','(nl, ','(nl, ','(write(' SEND'), ','(nl, ','(write(' + MORE'), ','(nl, ','(write(' --------'), ','(nl, ','(write(' MONEY'), ','(nl, ','(nl, ','(nl, ','(time(X2), ','(money(S, E, N, D, M, O, R, Y), ','(time(T), ','(write('The solution is:'), ','(nl, ','(nl, ','(write(' '), ','(write(S), ','(write(E), ','(write(N), ','(write(D), ','(nl, ','(write(' + '), ','(write(M), ','(write(O), ','(write(R), ','(write(E), ','(nl, ','(write(' --------'), ','(nl, ','(write(' '), ','(write(M), ','(write(O), ','(write(N), ','(write(E), ','(write(Y), ','(nl, ','(nl, ','(write('Solved in '), ','(write(T), ','(write(' msec.'), nl))))))))))))))))))))))))))))))))))))))))))))). time(0). solve(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, []))))))))) :- money(S, E, N, D, M, O, R, Y). money(S, E, N, D, M, O, R, Y) :- ','(carry(C1), ','(carry(C2), ','(carry(C3), ','(carry(C4), ','(=(C4, M), ','(=\=(M, 0), ','(domain(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, []))))))))))), ','(=\=(S, 0), ','(calc(C3, S, M, C4, O), ','(calc(C2, E, O, C3, N), ','(calc(C1, N, R, C2, E), calc(0, D, E, C1, Y)))))))))))). calc(C0, D, E, C1, Y) :- ','(sum(C0, D, CD), ','(sum(CD, E, S), ','(carry10(C1, C10), sum(C10, Y, S)))). sum(X, Y, Z) :- is(Z, +(X, Y)). domain([], X3). domain(.(X1, R), L) :- ','(del(X1, L, NL), domain(R, NL)). del(X, .(X, T), T). del(X, .(Y, T), .(Y, NT)) :- del(X, T, NT). carry(1). carry(0). carry10(0, 0). carry10(1, 10). =(X, X). Query: money(a,a,a,a,a,a,a,a) ---------------------------------------- (3) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (4) Obligation: Clauses: demo :- ','(nl, ','(write('Cryptarithmetic puzzle'), ','(nl, ','(nl, ','(write(' SEND'), ','(nl, ','(write(' + MORE'), ','(nl, ','(write(' --------'), ','(nl, ','(write(' MONEY'), ','(nl, ','(nl, ','(nl, ','(time(X2), ','(money(S, E, N, D, M, O, R, Y), ','(time(T), ','(write('The solution is:'), ','(nl, ','(nl, ','(write(' '), ','(write(S), ','(write(E), ','(write(N), ','(write(D), ','(nl, ','(write(' + '), ','(write(M), ','(write(O), ','(write(R), ','(write(E), ','(nl, ','(write(' --------'), ','(nl, ','(write(' '), ','(write(M), ','(write(O), ','(write(N), ','(write(E), ','(write(Y), ','(nl, ','(nl, ','(write('Solved in '), ','(write(T), ','(write(' msec.'), nl))))))))))))))))))))))))))))))))))))))))))))). time(0). solve(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, []))))))))) :- money(S, E, N, D, M, O, R, Y). money(S, E, N, D, M, O, R, Y) :- ','(carry(C1), ','(carry(C2), ','(carry(C3), ','(carry(C4), ','(=(C4, M), ','(=\=(M, 0), ','(domain(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, []))))))))))), ','(=\=(S, 0), ','(calc(C3, S, M, C4, O), ','(calc(C2, E, O, C3, N), ','(calc(C1, N, R, C2, E), calc(0, D, E, C1, Y)))))))))))). calc(C0, D, E, C1, Y) :- ','(sum(C0, D, CD), ','(sum(CD, E, S), ','(carry10(C1, C10), sum(C10, Y, S)))). sum(X, Y, Z) :- is(Z, +(X, Y)). domain([], X3). domain(.(X1, R), L) :- ','(del(X1, L, NL), domain(R, NL)). del(X, .(X, T), T). del(X, .(Y, T), .(Y, NT)) :- del(X, T, NT). carry(1). carry(0). carry10(0, 0). carry10(1, 10). =(X, X). nl. write(X0). =\=(X0, X1). is(X0, X1). Query: money(a,a,a,a,a,a,a,a) ---------------------------------------- (5) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: money_in_8: (f,f,f,f,f,f,f,f) domain_in_2: (b,b) del_in_3: (f,b,f) calc_in_5: (b,f,b,b,f) (b,f,f,b,f) sum_in_3: (b,f,f) (f,b,f) (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (6) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) ---------------------------------------- (7) 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: MONEY_IN_AAAAAAAA(S, E, N, D, M, O, R, Y) -> U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_in_a(C1)) MONEY_IN_AAAAAAAA(S, E, N, D, M, O, R, Y) -> CARRY_IN_A(C1) U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> CARRY_IN_A(C2) U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> CARRY_IN_A(C3) U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> CARRY_IN_A(C4) U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> =_IN_GA(C4, M) U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> =\=_IN_GG(M, 0) U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> DOMAIN_IN_GG(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, []))))))))))) DOMAIN_IN_GG(.(X1, R), L) -> U65_GG(X1, R, L, del_in_aga(X1, L, NL)) DOMAIN_IN_GG(.(X1, R), L) -> DEL_IN_AGA(X1, L, NL) DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> U67_AGA(X, Y, T, NT, del_in_aga(X, T, NT)) DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> DEL_IN_AGA(X, T, NT) U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> U66_GG(X1, R, L, domain_in_gg(R, NL)) U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> DOMAIN_IN_GG(R, NL) U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> =\=_IN_AG(S, 0) U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> CALC_IN_GAGGA(C3, S, M, C4, O) CALC_IN_GAGGA(C0, D, E, C1, Y) -> U60_GAGGA(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) CALC_IN_GAGGA(C0, D, E, C1, Y) -> SUM_IN_GAA(C0, D, CD) SUM_IN_GAA(X, Y, Z) -> U64_GAA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_GAA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U60_GAGGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_GAGGA(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) U60_GAGGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> SUM_IN_AGA(CD, E, S) SUM_IN_AGA(X, Y, Z) -> U64_AGA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_AGA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U61_GAGGA(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_GAGGA(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U61_GAGGA(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> CARRY10_IN_GA(C1, C10) U62_GAGGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_GAGGA(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U62_GAGGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> SUM_IN_GAA(C10, Y, S) U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> CALC_IN_GAAGA(C2, E, O, C3, N) CALC_IN_GAAGA(C0, D, E, C1, Y) -> U60_GAAGA(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) CALC_IN_GAAGA(C0, D, E, C1, Y) -> SUM_IN_GAA(C0, D, CD) U60_GAAGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_GAAGA(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) U60_GAAGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> SUM_IN_AAA(CD, E, S) SUM_IN_AAA(X, Y, Z) -> U64_AAA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_AAA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U61_GAAGA(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_GAAGA(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U61_GAAGA(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> CARRY10_IN_GA(C1, C10) U62_GAAGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_GAAGA(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U62_GAAGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> SUM_IN_GAA(C10, Y, S) U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> CALC_IN_GAAGA(C1, N, R, C2, E) U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_AAAAAAAA(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> CALC_IN_GAAGA(0, D, E, C1, Y) The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) MONEY_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = MONEY_IN_AAAAAAAA U48_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_AAAAAAAA(x9) CARRY_IN_A(x1) = CARRY_IN_A U49_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_AAAAAAAA(x9, x10) U50_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_AAAAAAAA(x9, x10, x11) U51_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_AAAAAAAA(x9, x10, x11, x12) U52_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_AAAAAAAA(x9, x10, x11, x12, x13) =_IN_GA(x1, x2) = =_IN_GA(x1) U53_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_AAAAAAAA(x5, x9, x10, x11, x12, x13) =\=_IN_GG(x1, x2) = =\=_IN_GG(x1, x2) U54_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_AAAAAAAA(x5, x9, x10, x11, x12, x13) DOMAIN_IN_GG(x1, x2) = DOMAIN_IN_GG(x1, x2) U65_GG(x1, x2, x3, x4) = U65_GG(x2, x3, x4) DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) U67_AGA(x1, x2, x3, x4, x5) = U67_AGA(x3, x5) U66_GG(x1, x2, x3, x4) = U66_GG(x2, x3, x4) U55_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_AAAAAAAA(x5, x9, x10, x11, x12, x13) =\=_IN_AG(x1, x2) = =\=_IN_AG(x2) U56_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_AAAAAAAA(x5, x9, x10, x11, x12) CALC_IN_GAGGA(x1, x2, x3, x4, x5) = CALC_IN_GAGGA(x1, x3, x4) U60_GAGGA(x1, x2, x3, x4, x5, x6) = U60_GAGGA(x1, x3, x4, x6) SUM_IN_GAA(x1, x2, x3) = SUM_IN_GAA(x1) U64_GAA(x1, x2, x3, x4) = U64_GAA(x1, x4) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U61_GAGGA(x1, x2, x3, x4, x5, x6) = U61_GAGGA(x1, x3, x4, x6) SUM_IN_AGA(x1, x2, x3) = SUM_IN_AGA(x2) U64_AGA(x1, x2, x3, x4) = U64_AGA(x2, x4) U62_GAGGA(x1, x2, x3, x4, x5, x6, x7) = U62_GAGGA(x1, x3, x4, x7) CARRY10_IN_GA(x1, x2) = CARRY10_IN_GA(x1) U63_GAGGA(x1, x2, x3, x4, x5, x6) = U63_GAGGA(x1, x3, x4, x6) U57_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_AAAAAAAA(x5, x9, x10, x11) CALC_IN_GAAGA(x1, x2, x3, x4, x5) = CALC_IN_GAAGA(x1, x4) U60_GAAGA(x1, x2, x3, x4, x5, x6) = U60_GAAGA(x1, x4, x6) U61_GAAGA(x1, x2, x3, x4, x5, x6) = U61_GAAGA(x1, x4, x6) SUM_IN_AAA(x1, x2, x3) = SUM_IN_AAA U64_AAA(x1, x2, x3, x4) = U64_AAA(x4) U62_GAAGA(x1, x2, x3, x4, x5, x6, x7) = U62_GAAGA(x1, x4, x7) U63_GAAGA(x1, x2, x3, x4, x5, x6) = U63_GAAGA(x1, x4, x6) U58_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_AAAAAAAA(x5, x9, x10) U59_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_AAAAAAAA(x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: MONEY_IN_AAAAAAAA(S, E, N, D, M, O, R, Y) -> U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_in_a(C1)) MONEY_IN_AAAAAAAA(S, E, N, D, M, O, R, Y) -> CARRY_IN_A(C1) U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U48_AAAAAAAA(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> CARRY_IN_A(C2) U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U49_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> CARRY_IN_A(C3) U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U50_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> CARRY_IN_A(C4) U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) U51_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> =_IN_GA(C4, M) U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) U52_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> =\=_IN_GG(M, 0) U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) U53_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> DOMAIN_IN_GG(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, []))))))))))) DOMAIN_IN_GG(.(X1, R), L) -> U65_GG(X1, R, L, del_in_aga(X1, L, NL)) DOMAIN_IN_GG(.(X1, R), L) -> DEL_IN_AGA(X1, L, NL) DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> U67_AGA(X, Y, T, NT, del_in_aga(X, T, NT)) DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> DEL_IN_AGA(X, T, NT) U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> U66_GG(X1, R, L, domain_in_gg(R, NL)) U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> DOMAIN_IN_GG(R, NL) U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) U54_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> =\=_IN_AG(S, 0) U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) U55_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> CALC_IN_GAGGA(C3, S, M, C4, O) CALC_IN_GAGGA(C0, D, E, C1, Y) -> U60_GAGGA(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) CALC_IN_GAGGA(C0, D, E, C1, Y) -> SUM_IN_GAA(C0, D, CD) SUM_IN_GAA(X, Y, Z) -> U64_GAA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_GAA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U60_GAGGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_GAGGA(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) U60_GAGGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> SUM_IN_AGA(CD, E, S) SUM_IN_AGA(X, Y, Z) -> U64_AGA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_AGA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U61_GAGGA(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_GAGGA(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U61_GAGGA(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> CARRY10_IN_GA(C1, C10) U62_GAGGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_GAGGA(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U62_GAGGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> SUM_IN_GAA(C10, Y, S) U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) U56_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> CALC_IN_GAAGA(C2, E, O, C3, N) CALC_IN_GAAGA(C0, D, E, C1, Y) -> U60_GAAGA(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) CALC_IN_GAAGA(C0, D, E, C1, Y) -> SUM_IN_GAA(C0, D, CD) U60_GAAGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_GAAGA(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) U60_GAAGA(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> SUM_IN_AAA(CD, E, S) SUM_IN_AAA(X, Y, Z) -> U64_AAA(X, Y, Z, is_in_ag(Z, +(X, Y))) SUM_IN_AAA(X, Y, Z) -> IS_IN_AG(Z, +(X, Y)) U61_GAAGA(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_GAAGA(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U61_GAAGA(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> CARRY10_IN_GA(C1, C10) U62_GAAGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_GAAGA(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U62_GAAGA(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> SUM_IN_GAA(C10, Y, S) U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U57_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> CALC_IN_GAAGA(C1, N, R, C2, E) U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_AAAAAAAA(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U58_AAAAAAAA(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> CALC_IN_GAAGA(0, D, E, C1, Y) The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) MONEY_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = MONEY_IN_AAAAAAAA U48_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_AAAAAAAA(x9) CARRY_IN_A(x1) = CARRY_IN_A U49_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_AAAAAAAA(x9, x10) U50_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_AAAAAAAA(x9, x10, x11) U51_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_AAAAAAAA(x9, x10, x11, x12) U52_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_AAAAAAAA(x9, x10, x11, x12, x13) =_IN_GA(x1, x2) = =_IN_GA(x1) U53_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_AAAAAAAA(x5, x9, x10, x11, x12, x13) =\=_IN_GG(x1, x2) = =\=_IN_GG(x1, x2) U54_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_AAAAAAAA(x5, x9, x10, x11, x12, x13) DOMAIN_IN_GG(x1, x2) = DOMAIN_IN_GG(x1, x2) U65_GG(x1, x2, x3, x4) = U65_GG(x2, x3, x4) DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) U67_AGA(x1, x2, x3, x4, x5) = U67_AGA(x3, x5) U66_GG(x1, x2, x3, x4) = U66_GG(x2, x3, x4) U55_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_AAAAAAAA(x5, x9, x10, x11, x12, x13) =\=_IN_AG(x1, x2) = =\=_IN_AG(x2) U56_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_AAAAAAAA(x5, x9, x10, x11, x12) CALC_IN_GAGGA(x1, x2, x3, x4, x5) = CALC_IN_GAGGA(x1, x3, x4) U60_GAGGA(x1, x2, x3, x4, x5, x6) = U60_GAGGA(x1, x3, x4, x6) SUM_IN_GAA(x1, x2, x3) = SUM_IN_GAA(x1) U64_GAA(x1, x2, x3, x4) = U64_GAA(x1, x4) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U61_GAGGA(x1, x2, x3, x4, x5, x6) = U61_GAGGA(x1, x3, x4, x6) SUM_IN_AGA(x1, x2, x3) = SUM_IN_AGA(x2) U64_AGA(x1, x2, x3, x4) = U64_AGA(x2, x4) U62_GAGGA(x1, x2, x3, x4, x5, x6, x7) = U62_GAGGA(x1, x3, x4, x7) CARRY10_IN_GA(x1, x2) = CARRY10_IN_GA(x1) U63_GAGGA(x1, x2, x3, x4, x5, x6) = U63_GAGGA(x1, x3, x4, x6) U57_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_AAAAAAAA(x5, x9, x10, x11) CALC_IN_GAAGA(x1, x2, x3, x4, x5) = CALC_IN_GAAGA(x1, x4) U60_GAAGA(x1, x2, x3, x4, x5, x6) = U60_GAAGA(x1, x4, x6) U61_GAAGA(x1, x2, x3, x4, x5, x6) = U61_GAAGA(x1, x4, x6) SUM_IN_AAA(x1, x2, x3) = SUM_IN_AAA U64_AAA(x1, x2, x3, x4) = U64_AAA(x4) U62_GAAGA(x1, x2, x3, x4, x5, x6, x7) = U62_GAAGA(x1, x4, x7) U63_GAAGA(x1, x2, x3, x4, x5, x6) = U63_GAAGA(x1, x4, x6) U58_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_AAAAAAAA(x5, x9, x10) U59_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_AAAAAAAA(x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 49 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> DEL_IN_AGA(X, T, NT) The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (13) Obligation: Pi DP problem: The TRS P consists of the following rules: DEL_IN_AGA(X, .(Y, T), .(Y, NT)) -> DEL_IN_AGA(X, T, NT) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: DEL_IN_AGA(.(T)) -> DEL_IN_AGA(T) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (16) 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: *DEL_IN_AGA(.(T)) -> DEL_IN_AGA(T) The graph contains the following edges 1 > 1 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> DOMAIN_IN_GG(R, NL) DOMAIN_IN_GG(.(X1, R), L) -> U65_GG(X1, R, L, del_in_aga(X1, L, NL)) The TRS R consists of the following rules: money_in_aaaaaaaa(S, E, N, D, M, O, R, Y) -> U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_in_a(C1)) carry_in_a(1) -> carry_out_a(1) carry_in_a(0) -> carry_out_a(0) U48_aaaaaaaa(S, E, N, D, M, O, R, Y, carry_out_a(C1)) -> U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_in_a(C2)) U49_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, carry_out_a(C2)) -> U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_in_a(C3)) U50_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, carry_out_a(C3)) -> U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_in_a(C4)) U51_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, carry_out_a(C4)) -> U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_in_ga(C4, M)) =_in_ga(X, X) -> =_out_ga(X, X) U52_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =_out_ga(C4, M)) -> U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_gg(M, 0)) =\=_in_gg(X0, X1) -> =\=_out_gg(X0, X1) U53_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_gg(M, 0)) -> U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_in_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) domain_in_gg([], X3) -> domain_out_gg([], X3) domain_in_gg(.(X1, R), L) -> U65_gg(X1, R, L, del_in_aga(X1, L, NL)) del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) U65_gg(X1, R, L, del_out_aga(X1, L, NL)) -> U66_gg(X1, R, L, domain_in_gg(R, NL)) U66_gg(X1, R, L, domain_out_gg(R, NL)) -> domain_out_gg(.(X1, R), L) U54_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, domain_out_gg(.(S, .(E, .(N, .(D, .(M, .(O, .(R, .(Y, [])))))))), .(0, .(1, .(2, .(3, .(4, .(5, .(6, .(7, .(8, .(9, [])))))))))))) -> U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_in_ag(S, 0)) =\=_in_ag(X0, X1) -> =\=_out_ag(X0, X1) U55_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, C4, =\=_out_ag(S, 0)) -> U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_in_gagga(C3, S, M, C4, O)) calc_in_gagga(C0, D, E, C1, Y) -> U60_gagga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) sum_in_gaa(X, Y, Z) -> U64_gaa(X, Y, Z, is_in_ag(Z, +(X, Y))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U64_gaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_gaa(X, Y, Z) U60_gagga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gagga(C0, D, E, C1, Y, sum_in_aga(CD, E, S)) sum_in_aga(X, Y, Z) -> U64_aga(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aga(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aga(X, Y, Z) U61_gagga(C0, D, E, C1, Y, sum_out_aga(CD, E, S)) -> U62_gagga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) carry10_in_ga(0, 0) -> carry10_out_ga(0, 0) carry10_in_ga(1, 10) -> carry10_out_ga(1, 10) U62_gagga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gagga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gagga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gagga(C0, D, E, C1, Y) U56_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, C3, calc_out_gagga(C3, S, M, C4, O)) -> U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_in_gaaga(C2, E, O, C3, N)) calc_in_gaaga(C0, D, E, C1, Y) -> U60_gaaga(C0, D, E, C1, Y, sum_in_gaa(C0, D, CD)) U60_gaaga(C0, D, E, C1, Y, sum_out_gaa(C0, D, CD)) -> U61_gaaga(C0, D, E, C1, Y, sum_in_aaa(CD, E, S)) sum_in_aaa(X, Y, Z) -> U64_aaa(X, Y, Z, is_in_ag(Z, +(X, Y))) U64_aaa(X, Y, Z, is_out_ag(Z, +(X, Y))) -> sum_out_aaa(X, Y, Z) U61_gaaga(C0, D, E, C1, Y, sum_out_aaa(CD, E, S)) -> U62_gaaga(C0, D, E, C1, Y, S, carry10_in_ga(C1, C10)) U62_gaaga(C0, D, E, C1, Y, S, carry10_out_ga(C1, C10)) -> U63_gaaga(C0, D, E, C1, Y, sum_in_gaa(C10, Y, S)) U63_gaaga(C0, D, E, C1, Y, sum_out_gaa(C10, Y, S)) -> calc_out_gaaga(C0, D, E, C1, Y) U57_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, C2, calc_out_gaaga(C2, E, O, C3, N)) -> U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_in_gaaga(C1, N, R, C2, E)) U58_aaaaaaaa(S, E, N, D, M, O, R, Y, C1, calc_out_gaaga(C1, N, R, C2, E)) -> U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_in_gaaga(0, D, E, C1, Y)) U59_aaaaaaaa(S, E, N, D, M, O, R, Y, calc_out_gaaga(0, D, E, C1, Y)) -> money_out_aaaaaaaa(S, E, N, D, M, O, R, Y) The argument filtering Pi contains the following mapping: money_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_in_aaaaaaaa U48_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_aaaaaaaa(x9) carry_in_a(x1) = carry_in_a carry_out_a(x1) = carry_out_a(x1) U49_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U49_aaaaaaaa(x9, x10) U50_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U50_aaaaaaaa(x9, x10, x11) U51_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U51_aaaaaaaa(x9, x10, x11, x12) U52_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U52_aaaaaaaa(x9, x10, x11, x12, x13) =_in_ga(x1, x2) = =_in_ga(x1) =_out_ga(x1, x2) = =_out_ga(x1, x2) U53_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U53_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_gg(x1, x2) = =\=_in_gg(x1, x2) =\=_out_gg(x1, x2) = =\=_out_gg(x1, x2) 0 = 0 U54_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U54_aaaaaaaa(x5, x9, x10, x11, x12, x13) domain_in_gg(x1, x2) = domain_in_gg(x1, x2) .(x1, x2) = .(x2) [] = [] domain_out_gg(x1, x2) = domain_out_gg(x1, x2) U65_gg(x1, x2, x3, x4) = U65_gg(x2, x3, x4) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) U66_gg(x1, x2, x3, x4) = U66_gg(x2, x3, x4) U55_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U55_aaaaaaaa(x5, x9, x10, x11, x12, x13) =\=_in_ag(x1, x2) = =\=_in_ag(x2) =\=_out_ag(x1, x2) = =\=_out_ag(x2) U56_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U56_aaaaaaaa(x5, x9, x10, x11, x12) calc_in_gagga(x1, x2, x3, x4, x5) = calc_in_gagga(x1, x3, x4) U60_gagga(x1, x2, x3, x4, x5, x6) = U60_gagga(x1, x3, x4, x6) sum_in_gaa(x1, x2, x3) = sum_in_gaa(x1) U64_gaa(x1, x2, x3, x4) = U64_gaa(x1, x4) is_in_ag(x1, x2) = is_in_ag(x2) +(x1, x2) = + is_out_ag(x1, x2) = is_out_ag(x2) sum_out_gaa(x1, x2, x3) = sum_out_gaa(x1) U61_gagga(x1, x2, x3, x4, x5, x6) = U61_gagga(x1, x3, x4, x6) sum_in_aga(x1, x2, x3) = sum_in_aga(x2) U64_aga(x1, x2, x3, x4) = U64_aga(x2, x4) sum_out_aga(x1, x2, x3) = sum_out_aga(x2) U62_gagga(x1, x2, x3, x4, x5, x6, x7) = U62_gagga(x1, x3, x4, x7) carry10_in_ga(x1, x2) = carry10_in_ga(x1) carry10_out_ga(x1, x2) = carry10_out_ga(x1, x2) 1 = 1 U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x3, x4, x6) calc_out_gagga(x1, x2, x3, x4, x5) = calc_out_gagga(x1, x3, x4) U57_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U57_aaaaaaaa(x5, x9, x10, x11) calc_in_gaaga(x1, x2, x3, x4, x5) = calc_in_gaaga(x1, x4) U60_gaaga(x1, x2, x3, x4, x5, x6) = U60_gaaga(x1, x4, x6) U61_gaaga(x1, x2, x3, x4, x5, x6) = U61_gaaga(x1, x4, x6) sum_in_aaa(x1, x2, x3) = sum_in_aaa U64_aaa(x1, x2, x3, x4) = U64_aaa(x4) sum_out_aaa(x1, x2, x3) = sum_out_aaa U62_gaaga(x1, x2, x3, x4, x5, x6, x7) = U62_gaaga(x1, x4, x7) U63_gaaga(x1, x2, x3, x4, x5, x6) = U63_gaaga(x1, x4, x6) calc_out_gaaga(x1, x2, x3, x4, x5) = calc_out_gaaga(x1, x4) U58_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U58_aaaaaaaa(x5, x9, x10) U59_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_aaaaaaaa(x5, x9) money_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = money_out_aaaaaaaa(x5) DOMAIN_IN_GG(x1, x2) = DOMAIN_IN_GG(x1, x2) U65_GG(x1, x2, x3, x4) = U65_GG(x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: U65_GG(X1, R, L, del_out_aga(X1, L, NL)) -> DOMAIN_IN_GG(R, NL) DOMAIN_IN_GG(.(X1, R), L) -> U65_GG(X1, R, L, del_in_aga(X1, L, NL)) The TRS R consists of the following rules: del_in_aga(X, .(X, T), T) -> del_out_aga(X, .(X, T), T) del_in_aga(X, .(Y, T), .(Y, NT)) -> U67_aga(X, Y, T, NT, del_in_aga(X, T, NT)) U67_aga(X, Y, T, NT, del_out_aga(X, T, NT)) -> del_out_aga(X, .(Y, T), .(Y, NT)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) del_in_aga(x1, x2, x3) = del_in_aga(x2) del_out_aga(x1, x2, x3) = del_out_aga(x2, x3) U67_aga(x1, x2, x3, x4, x5) = U67_aga(x3, x5) DOMAIN_IN_GG(x1, x2) = DOMAIN_IN_GG(x1, x2) U65_GG(x1, x2, x3, x4) = U65_GG(x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U65_GG(R, L, del_out_aga(L, NL)) -> DOMAIN_IN_GG(R, NL) DOMAIN_IN_GG(.(R), L) -> U65_GG(R, L, del_in_aga(L)) The TRS R consists of the following rules: del_in_aga(.(T)) -> del_out_aga(.(T), T) del_in_aga(.(T)) -> U67_aga(T, del_in_aga(T)) U67_aga(T, del_out_aga(T, NT)) -> del_out_aga(.(T), .(NT)) The set Q consists of the following terms: del_in_aga(x0) U67_aga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) 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: *DOMAIN_IN_GG(.(R), L) -> U65_GG(R, L, del_in_aga(L)) The graph contains the following edges 1 > 1, 2 >= 2 *U65_GG(R, L, del_out_aga(L, NL)) -> DOMAIN_IN_GG(R, NL) The graph contains the following edges 1 >= 1, 3 > 2 ---------------------------------------- (24) YES