4.58/1.95 YES 4.58/1.96 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.58/1.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.58/1.96 4.58/1.96 4.58/1.96 Left Termination of the query pattern 4.58/1.96 4.58/1.96 in_order(g,a) 4.58/1.96 4.58/1.96 w.r.t. the given Prolog program could successfully be proven: 4.58/1.96 4.58/1.96 (0) Prolog 4.58/1.96 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.58/1.96 (2) PiTRS 4.58/1.96 (3) DependencyPairsProof [EQUIVALENT, 1 ms] 4.58/1.96 (4) PiDP 4.58/1.96 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.58/1.96 (6) AND 4.58/1.96 (7) PiDP 4.58/1.96 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.58/1.96 (9) PiDP 4.58/1.96 (10) PiDPToQDPProof [SOUND, 3 ms] 4.58/1.96 (11) QDP 4.58/1.96 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.58/1.96 (13) YES 4.58/1.96 (14) PiDP 4.58/1.96 (15) PiDPToQDPProof [SOUND, 0 ms] 4.58/1.96 (16) QDP 4.58/1.96 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.58/1.96 (18) YES 4.58/1.96 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (0) 4.58/1.96 Obligation: 4.58/1.96 Clauses: 4.58/1.96 4.58/1.96 in_order(void, []). 4.58/1.96 in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs))). 4.58/1.96 app([], X, X). 4.58/1.96 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 4.58/1.96 4.58/1.96 4.58/1.96 Query: in_order(g,a) 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (1) PrologToPiTRSProof (SOUND) 4.58/1.96 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.58/1.96 4.58/1.96 in_order_in_2: (b,f) 4.58/1.96 4.58/1.96 app_in_3: (b,b,f) 4.58/1.96 4.58/1.96 Transforming Prolog into the following Term Rewriting System: 4.58/1.96 4.58/1.96 Pi-finite rewrite system: 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 4.58/1.96 4.58/1.96 4.58/1.96 4.58/1.96 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.58/1.96 4.58/1.96 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (2) 4.58/1.96 Obligation: 4.58/1.96 Pi-finite rewrite system: 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (3) DependencyPairsProof (EQUIVALENT) 4.58/1.96 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.58/1.96 Pi DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) 4.58/1.96 U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_GA(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) 4.58/1.96 U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_GA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> APP_IN_GGA(Ls, .(X, Rs), Xs) 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.58/1.96 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) 4.58/1.96 4.58/1.96 U1_GA(x1, x2, x3, x4, x5) = U1_GA'(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.58/1.96 4.58/1.96 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 4.58/1.96 We have to consider all (P,R,Pi)-chains 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (4) 4.58/1.96 Obligation: 4.58/1.96 Pi DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) 4.58/1.96 U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_GA(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) 4.58/1.96 U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_GA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> APP_IN_GGA(Ls, .(X, Rs), Xs) 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.58/1.96 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) 4.58/1.96 4.58/1.96 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.58/1.96 4.58/1.96 U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 4.58/1.96 We have to consider all (P,R,Pi)-chains 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (5) DependencyGraphProof (EQUIVALENT) 4.58/1.96 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (6) 4.58/1.96 Complex Obligation (AND) 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (7) 4.58/1.96 Obligation: 4.58/1.96 Pi DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.58/1.96 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.58/1.96 4.58/1.96 4.58/1.96 We have to consider all (P,R,Pi)-chains 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (8) UsableRulesProof (EQUIVALENT) 4.58/1.96 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (9) 4.58/1.96 Obligation: 4.58/1.96 Pi DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) 4.58/1.96 4.58/1.96 R is empty. 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) 4.58/1.96 4.58/1.96 4.58/1.96 We have to consider all (P,R,Pi)-chains 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (10) PiDPToQDPProof (SOUND) 4.58/1.96 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (11) 4.58/1.96 Obligation: 4.58/1.96 Q DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 4.58/1.96 4.58/1.96 R is empty. 4.58/1.96 Q is empty. 4.58/1.96 We have to consider all (P,Q,R)-chains. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (12) QDPSizeChangeProof (EQUIVALENT) 4.58/1.96 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. 4.58/1.96 4.58/1.96 From the DPs we obtained the following set of size-change graphs: 4.58/1.96 *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) 4.58/1.96 The graph contains the following edges 1 > 1, 2 >= 2 4.58/1.96 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (13) 4.58/1.96 YES 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (14) 4.58/1.96 Obligation: 4.58/1.96 Pi DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) 4.58/1.96 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void, []) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) 4.58/1.96 U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) 4.58/1.96 U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) 4.58/1.96 app_in_gga([], X, X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) 4.58/1.96 U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The argument filtering Pi contains the following mapping: 4.58/1.96 in_order_in_ga(x1, x2) = in_order_in_ga(x1) 4.58/1.96 4.58/1.96 void = void 4.58/1.96 4.58/1.96 in_order_out_ga(x1, x2) = in_order_out_ga(x1, x2) 4.58/1.96 4.58/1.96 tree(x1, x2, x3) = tree(x1, x2, x3) 4.58/1.96 4.58/1.96 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) 4.58/1.96 4.58/1.96 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) 4.58/1.96 4.58/1.96 [] = [] 4.58/1.96 4.58/1.96 app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3) 4.58/1.96 4.58/1.96 .(x1, x2) = .(x1, x2) 4.58/1.96 4.58/1.96 U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) 4.58/1.96 4.58/1.96 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 4.58/1.96 4.58/1.96 4.58/1.96 We have to consider all (P,R,Pi)-chains 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (15) PiDPToQDPProof (SOUND) 4.58/1.96 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (16) 4.58/1.96 Obligation: 4.58/1.96 Q DP problem: 4.58/1.96 The TRS P consists of the following rules: 4.58/1.96 4.58/1.96 U1_GA(X, Left, Right, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right)) -> U1_GA(X, Left, Right, in_order_in_ga(Left)) 4.58/1.96 IN_ORDER_IN_GA(tree(X, Left, Right)) -> IN_ORDER_IN_GA(Left) 4.58/1.96 4.58/1.96 The TRS R consists of the following rules: 4.58/1.96 4.58/1.96 in_order_in_ga(void) -> in_order_out_ga(void, []) 4.58/1.96 in_order_in_ga(tree(X, Left, Right)) -> U1_ga(X, Left, Right, in_order_in_ga(Left)) 4.58/1.96 U1_ga(X, Left, Right, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Ls, in_order_in_ga(Right)) 4.58/1.96 U2_ga(X, Left, Right, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, app_in_gga(Ls, .(X, Rs))) 4.58/1.96 app_in_gga([], X) -> app_out_gga([], X, X) 4.58/1.96 app_in_gga(.(X, Xs), Ys) -> U4_gga(X, Xs, Ys, app_in_gga(Xs, Ys)) 4.58/1.96 U4_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) 4.58/1.96 U3_ga(X, Left, Right, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) 4.58/1.96 4.58/1.96 The set Q consists of the following terms: 4.58/1.96 4.58/1.96 in_order_in_ga(x0) 4.58/1.96 U1_ga(x0, x1, x2, x3) 4.58/1.96 U2_ga(x0, x1, x2, x3, x4) 4.58/1.96 app_in_gga(x0, x1) 4.58/1.96 U4_gga(x0, x1, x2, x3) 4.58/1.96 U3_ga(x0, x1, x2, x3) 4.58/1.96 4.58/1.96 We have to consider all (P,Q,R)-chains. 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (17) QDPSizeChangeProof (EQUIVALENT) 4.58/1.96 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. 4.58/1.96 4.58/1.96 From the DPs we obtained the following set of size-change graphs: 4.58/1.96 *IN_ORDER_IN_GA(tree(X, Left, Right)) -> U1_GA(X, Left, Right, in_order_in_ga(Left)) 4.58/1.96 The graph contains the following edges 1 > 1, 1 > 2, 1 > 3 4.58/1.96 4.58/1.96 4.58/1.96 *IN_ORDER_IN_GA(tree(X, Left, Right)) -> IN_ORDER_IN_GA(Left) 4.58/1.96 The graph contains the following edges 1 > 1 4.58/1.96 4.58/1.96 4.58/1.96 *U1_GA(X, Left, Right, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right) 4.58/1.96 The graph contains the following edges 3 >= 1 4.58/1.96 4.58/1.96 4.58/1.96 ---------------------------------------- 4.58/1.96 4.58/1.96 (18) 4.58/1.96 YES 4.58/1.98 EOF