4.70/2.06 YES 4.70/2.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.70/2.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.70/2.08 4.70/2.08 4.70/2.08 Left Termination of the query pattern 4.70/2.08 4.70/2.08 front(g,a) 4.70/2.08 4.70/2.08 w.r.t. the given Prolog program could successfully be proven: 4.70/2.08 4.70/2.08 (0) Prolog 4.70/2.08 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.70/2.08 (2) PiTRS 4.70/2.08 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.70/2.08 (4) PiDP 4.70/2.08 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.70/2.08 (6) AND 4.70/2.08 (7) PiDP 4.70/2.08 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.70/2.08 (9) PiDP 4.70/2.08 (10) PiDPToQDPProof [SOUND, 0 ms] 4.70/2.08 (11) QDP 4.70/2.08 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.70/2.08 (13) YES 4.70/2.08 (14) PiDP 4.70/2.08 (15) PiDPToQDPProof [SOUND, 0 ms] 4.70/2.08 (16) QDP 4.70/2.08 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.70/2.08 (18) YES 4.70/2.08 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (0) 4.70/2.08 Obligation: 4.70/2.08 Clauses: 4.70/2.08 4.70/2.08 front(void, []). 4.70/2.08 front(tree(X, void, void), .(X, [])). 4.70/2.08 front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs))). 4.70/2.08 app([], X, X). 4.70/2.08 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 4.70/2.08 4.70/2.08 4.70/2.08 Query: front(g,a) 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (1) PrologToPiTRSProof (SOUND) 4.70/2.08 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.70/2.08 4.70/2.08 front_in_2: (b,f) 4.70/2.08 4.70/2.08 app_in_3: (b,b,f) 4.70/2.08 4.70/2.08 Transforming Prolog into the following Term Rewriting System: 4.70/2.08 4.70/2.08 Pi-finite rewrite system: 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 4.70/2.08 4.70/2.08 4.70/2.08 4.70/2.08 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.70/2.08 4.70/2.08 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (2) 4.70/2.08 Obligation: 4.70/2.08 Pi-finite rewrite system: 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (3) DependencyPairsProof (EQUIVALENT) 4.70/2.08 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.70/2.08 Pi DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> U1_GA(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> FRONT_IN_GA(L, Ls) 4.70/2.08 U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) -> FRONT_IN_GA(R, Rs) 4.70/2.08 U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.70/2.08 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 FRONT_IN_GA(x1, x2) = FRONT_IN_GA(x1) 4.70/2.08 4.70/2.08 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5) 4.70/2.08 4.70/2.08 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6) 4.70/2.08 4.70/2.08 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) 4.70/2.08 4.70/2.08 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.70/2.08 4.70/2.08 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) 4.70/2.08 4.70/2.08 4.70/2.08 We have to consider all (P,R,Pi)-chains 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (4) 4.70/2.08 Obligation: 4.70/2.08 Pi DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> U1_GA(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> FRONT_IN_GA(L, Ls) 4.70/2.08 U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) -> FRONT_IN_GA(R, Rs) 4.70/2.08 U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> APP_IN_GGA(Ls, Rs, Xs) 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.70/2.08 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 FRONT_IN_GA(x1, x2) = FRONT_IN_GA(x1) 4.70/2.08 4.70/2.08 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5) 4.70/2.08 4.70/2.08 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6) 4.70/2.08 4.70/2.08 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) 4.70/2.08 4.70/2.08 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.70/2.08 4.70/2.08 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) 4.70/2.08 4.70/2.08 4.70/2.08 We have to consider all (P,R,Pi)-chains 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (5) DependencyGraphProof (EQUIVALENT) 4.70/2.08 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (6) 4.70/2.08 Complex Obligation (AND) 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (7) 4.70/2.08 Obligation: 4.70/2.08 Pi DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.70/2.08 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.70/2.08 4.70/2.08 4.70/2.08 We have to consider all (P,R,Pi)-chains 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (8) UsableRulesProof (EQUIVALENT) 4.70/2.08 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (9) 4.70/2.08 Obligation: 4.70/2.08 Pi DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.70/2.08 4.70/2.08 R is empty. 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.70/2.08 4.70/2.08 4.70/2.08 We have to consider all (P,R,Pi)-chains 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (10) PiDPToQDPProof (SOUND) 4.70/2.08 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (11) 4.70/2.08 Obligation: 4.70/2.08 Q DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 4.70/2.08 4.70/2.08 R is empty. 4.70/2.08 Q is empty. 4.70/2.08 We have to consider all (P,Q,R)-chains. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (12) QDPSizeChangeProof (EQUIVALENT) 4.70/2.08 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 4.70/2.08 4.70/2.08 From the DPs we obtained the following set of size-change graphs: 4.70/2.08 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 4.70/2.08 The graph contains the following edges 1 > 1, 2 >= 2 4.70/2.08 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (13) 4.70/2.08 YES 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (14) 4.70/2.08 Obligation: 4.70/2.08 Pi DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) -> FRONT_IN_GA(R, Rs) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> U1_GA(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R), Xs) -> FRONT_IN_GA(L, Ls) 4.70/2.08 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void, []) -> front_out_ga(void, []) 4.70/2.08 front_in_ga(tree(X, void, void), .(X, [])) -> front_out_ga(tree(X, void, void), .(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R), Xs) -> U1_ga(X1, L, R, Xs, front_in_ga(L, Ls)) 4.70/2.08 U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) -> U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs)) 4.70/2.08 U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) -> U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs)) 4.70/2.08 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.70/2.08 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.70/2.08 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.70/2.08 U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) -> front_out_ga(tree(X1, L, R), Xs) 4.70/2.08 4.70/2.08 The argument filtering Pi contains the following mapping: 4.70/2.08 front_in_ga(x1, x2) = front_in_ga(x1) 4.70/2.08 4.70/2.08 void = void 4.70/2.08 4.70/2.08 front_out_ga(x1, x2) = front_out_ga(x2) 4.70/2.08 4.70/2.08 tree(x1, x2, x3) = tree(x1, x2, x3) 4.70/2.08 4.70/2.08 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5) 4.70/2.08 4.70/2.08 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6) 4.70/2.08 4.70/2.08 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) 4.70/2.08 4.70/2.08 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.70/2.08 4.70/2.08 [] = [] 4.70/2.08 4.70/2.08 app_out_gga(x1, x2, x3) = app_out_gga(x3) 4.70/2.08 4.70/2.08 .(x1, x2) = .(x1, x2) 4.70/2.08 4.70/2.08 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) 4.70/2.08 4.70/2.08 FRONT_IN_GA(x1, x2) = FRONT_IN_GA(x1) 4.70/2.08 4.70/2.08 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5) 4.70/2.08 4.70/2.08 4.70/2.08 We have to consider all (P,R,Pi)-chains 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (15) PiDPToQDPProof (SOUND) 4.70/2.08 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (16) 4.70/2.08 Obligation: 4.70/2.08 Q DP problem: 4.70/2.08 The TRS P consists of the following rules: 4.70/2.08 4.70/2.08 U1_GA(R, front_out_ga(Ls)) -> FRONT_IN_GA(R) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R)) -> U1_GA(R, front_in_ga(L)) 4.70/2.08 FRONT_IN_GA(tree(X1, L, R)) -> FRONT_IN_GA(L) 4.70/2.08 4.70/2.08 The TRS R consists of the following rules: 4.70/2.08 4.70/2.08 front_in_ga(void) -> front_out_ga([]) 4.70/2.08 front_in_ga(tree(X, void, void)) -> front_out_ga(.(X, [])) 4.70/2.08 front_in_ga(tree(X1, L, R)) -> U1_ga(R, front_in_ga(L)) 4.70/2.08 U1_ga(R, front_out_ga(Ls)) -> U2_ga(Ls, front_in_ga(R)) 4.70/2.08 U2_ga(Ls, front_out_ga(Rs)) -> U3_ga(app_in_gga(Ls, Rs)) 4.70/2.08 app_in_gga([], X) -> app_out_gga(X) 4.70/2.08 app_in_gga(.(X, Xs), Ys) -> U4_gga(X, app_in_gga(Xs, Ys)) 4.70/2.08 U4_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) 4.70/2.08 U3_ga(app_out_gga(Xs)) -> front_out_ga(Xs) 4.70/2.08 4.70/2.08 The set Q consists of the following terms: 4.70/2.08 4.70/2.08 front_in_ga(x0) 4.70/2.08 U1_ga(x0, x1) 4.70/2.08 U2_ga(x0, x1) 4.70/2.08 app_in_gga(x0, x1) 4.70/2.08 U4_gga(x0, x1) 4.70/2.08 U3_ga(x0) 4.70/2.08 4.70/2.08 We have to consider all (P,Q,R)-chains. 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (17) QDPSizeChangeProof (EQUIVALENT) 4.70/2.08 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 4.70/2.08 4.70/2.08 From the DPs we obtained the following set of size-change graphs: 4.70/2.08 *FRONT_IN_GA(tree(X1, L, R)) -> U1_GA(R, front_in_ga(L)) 4.70/2.08 The graph contains the following edges 1 > 1 4.70/2.08 4.70/2.08 4.70/2.08 *FRONT_IN_GA(tree(X1, L, R)) -> FRONT_IN_GA(L) 4.70/2.08 The graph contains the following edges 1 > 1 4.70/2.08 4.70/2.08 4.70/2.08 *U1_GA(R, front_out_ga(Ls)) -> FRONT_IN_GA(R) 4.70/2.08 The graph contains the following edges 1 >= 1 4.70/2.08 4.70/2.08 4.70/2.08 ---------------------------------------- 4.70/2.08 4.70/2.08 (18) 4.70/2.08 YES 4.88/2.13 EOF