4.85/2.11 YES 4.93/2.12 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.93/2.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.93/2.12 4.93/2.12 4.93/2.12 Left Termination of the query pattern 4.93/2.12 4.93/2.12 permute(g,a) 4.93/2.12 4.93/2.12 w.r.t. the given Prolog program could successfully be proven: 4.93/2.12 4.93/2.12 (0) Prolog 4.93/2.12 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.93/2.12 (2) PiTRS 4.93/2.12 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.93/2.12 (4) PiDP 4.93/2.12 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.93/2.12 (6) AND 4.93/2.12 (7) PiDP 4.93/2.12 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.93/2.12 (9) PiDP 4.93/2.12 (10) PiDPToQDPProof [SOUND, 0 ms] 4.93/2.12 (11) QDP 4.93/2.12 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.93/2.12 (13) YES 4.93/2.12 (14) PiDP 4.93/2.12 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.93/2.12 (16) PiDP 4.93/2.12 (17) PiDPToQDPProof [SOUND, 0 ms] 4.93/2.12 (18) QDP 4.93/2.12 (19) MRRProof [EQUIVALENT, 2 ms] 4.93/2.12 (20) QDP 4.93/2.12 (21) PisEmptyProof [EQUIVALENT, 0 ms] 4.93/2.12 (22) YES 4.93/2.12 4.93/2.12 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (0) 4.93/2.12 Obligation: 4.93/2.12 Clauses: 4.93/2.12 4.93/2.12 permute([], []). 4.93/2.12 permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)). 4.93/2.12 delete(X, .(X, Y), Y). 4.93/2.12 delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z). 4.93/2.12 4.93/2.12 4.93/2.12 Query: permute(g,a) 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (1) PrologToPiTRSProof (SOUND) 4.93/2.12 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.93/2.12 4.93/2.12 permute_in_2: (b,f) 4.93/2.12 4.93/2.12 delete_in_3: (f,b,f) 4.93/2.12 4.93/2.12 Transforming Prolog into the following Term Rewriting System: 4.93/2.12 4.93/2.12 Pi-finite rewrite system: 4.93/2.12 The TRS R consists of the following rules: 4.93/2.12 4.93/2.12 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.12 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.12 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.12 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.12 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.12 4.93/2.12 [] = [] 4.93/2.12 4.93/2.12 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.12 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.12 4.93/2.12 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.12 4.93/2.12 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.12 4.93/2.12 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.12 4.93/2.12 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.12 4.93/2.12 4.93/2.12 4.93/2.12 4.93/2.12 4.93/2.12 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.93/2.12 4.93/2.12 4.93/2.12 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (2) 4.93/2.12 Obligation: 4.93/2.12 Pi-finite rewrite system: 4.93/2.12 The TRS R consists of the following rules: 4.93/2.12 4.93/2.12 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.12 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.12 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.12 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.12 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.12 4.93/2.12 [] = [] 4.93/2.12 4.93/2.12 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.12 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.12 4.93/2.12 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.12 4.93/2.12 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.12 4.93/2.12 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.12 4.93/2.12 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.12 4.93/2.12 4.93/2.12 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (3) DependencyPairsProof (EQUIVALENT) 4.93/2.12 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.93/2.12 Pi DP problem: 4.93/2.12 The TRS P consists of the following rules: 4.93/2.12 4.93/2.12 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 4.93/2.12 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 4.93/2.12 4.93/2.12 The TRS R consists of the following rules: 4.93/2.12 4.93/2.12 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.12 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.12 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.12 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.12 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.12 4.93/2.12 [] = [] 4.93/2.12 4.93/2.12 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.12 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.12 4.93/2.12 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.12 4.93/2.12 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.12 4.93/2.12 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.12 4.93/2.12 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.12 4.93/2.12 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 4.93/2.12 4.93/2.12 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 4.93/2.12 4.93/2.12 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 4.93/2.12 4.93/2.12 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 4.93/2.12 4.93/2.12 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) 4.93/2.12 4.93/2.12 4.93/2.12 We have to consider all (P,R,Pi)-chains 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (4) 4.93/2.12 Obligation: 4.93/2.12 Pi DP problem: 4.93/2.12 The TRS P consists of the following rules: 4.93/2.12 4.93/2.12 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 4.93/2.12 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 4.93/2.12 4.93/2.12 The TRS R consists of the following rules: 4.93/2.12 4.93/2.12 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.12 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.12 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.12 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.12 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.12 4.93/2.12 [] = [] 4.93/2.12 4.93/2.12 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.12 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.12 4.93/2.12 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.12 4.93/2.12 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.12 4.93/2.12 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.12 4.93/2.12 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.12 4.93/2.12 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 4.93/2.12 4.93/2.12 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 4.93/2.12 4.93/2.12 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 4.93/2.12 4.93/2.12 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 4.93/2.12 4.93/2.12 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) 4.93/2.12 4.93/2.12 4.93/2.12 We have to consider all (P,R,Pi)-chains 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (5) DependencyGraphProof (EQUIVALENT) 4.93/2.12 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (6) 4.93/2.12 Complex Obligation (AND) 4.93/2.12 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (7) 4.93/2.12 Obligation: 4.93/2.12 Pi DP problem: 4.93/2.12 The TRS P consists of the following rules: 4.93/2.12 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 4.93/2.12 4.93/2.12 The TRS R consists of the following rules: 4.93/2.12 4.93/2.12 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.12 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.12 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.12 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.12 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.12 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.12 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.12 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.12 4.93/2.12 [] = [] 4.93/2.12 4.93/2.12 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.12 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.12 4.93/2.12 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.12 4.93/2.12 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.12 4.93/2.12 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.12 4.93/2.12 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.12 4.93/2.12 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 4.93/2.12 4.93/2.12 4.93/2.12 We have to consider all (P,R,Pi)-chains 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (8) UsableRulesProof (EQUIVALENT) 4.93/2.12 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (9) 4.93/2.12 Obligation: 4.93/2.12 Pi DP problem: 4.93/2.12 The TRS P consists of the following rules: 4.93/2.12 4.93/2.12 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 4.93/2.12 4.93/2.12 R is empty. 4.93/2.12 The argument filtering Pi contains the following mapping: 4.93/2.12 .(x1, x2) = .(x1, x2) 4.93/2.12 4.93/2.12 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 4.93/2.12 4.93/2.12 4.93/2.12 We have to consider all (P,R,Pi)-chains 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (10) PiDPToQDPProof (SOUND) 4.93/2.12 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (11) 4.93/2.12 Obligation: 4.93/2.12 Q DP problem: 4.93/2.12 The TRS P consists of the following rules: 4.93/2.12 4.93/2.12 DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y) 4.93/2.12 4.93/2.12 R is empty. 4.93/2.12 Q is empty. 4.93/2.12 We have to consider all (P,Q,R)-chains. 4.93/2.12 ---------------------------------------- 4.93/2.12 4.93/2.12 (12) QDPSizeChangeProof (EQUIVALENT) 4.93/2.12 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. 4.93/2.13 4.93/2.13 From the DPs we obtained the following set of size-change graphs: 4.93/2.13 *DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y) 4.93/2.13 The graph contains the following edges 1 > 1 4.93/2.13 4.93/2.13 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (13) 4.93/2.13 YES 4.93/2.13 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (14) 4.93/2.13 Obligation: 4.93/2.13 Pi DP problem: 4.93/2.13 The TRS P consists of the following rules: 4.93/2.13 4.93/2.13 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 4.93/2.13 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.13 4.93/2.13 The TRS R consists of the following rules: 4.93/2.13 4.93/2.13 permute_in_ga([], []) -> permute_out_ga([], []) 4.93/2.13 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.13 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.13 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.13 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.13 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 4.93/2.13 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 4.93/2.13 4.93/2.13 The argument filtering Pi contains the following mapping: 4.93/2.13 permute_in_ga(x1, x2) = permute_in_ga(x1) 4.93/2.13 4.93/2.13 [] = [] 4.93/2.13 4.93/2.13 permute_out_ga(x1, x2) = permute_out_ga(x2) 4.93/2.13 4.93/2.13 .(x1, x2) = .(x1, x2) 4.93/2.13 4.93/2.13 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 4.93/2.13 4.93/2.13 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.13 4.93/2.13 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.13 4.93/2.13 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.13 4.93/2.13 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 4.93/2.13 4.93/2.13 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 4.93/2.13 4.93/2.13 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 4.93/2.13 4.93/2.13 4.93/2.13 We have to consider all (P,R,Pi)-chains 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (15) UsableRulesProof (EQUIVALENT) 4.93/2.13 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (16) 4.93/2.13 Obligation: 4.93/2.13 Pi DP problem: 4.93/2.13 The TRS P consists of the following rules: 4.93/2.13 4.93/2.13 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 4.93/2.13 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 4.93/2.13 4.93/2.13 The TRS R consists of the following rules: 4.93/2.13 4.93/2.13 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 4.93/2.13 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 4.93/2.13 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 4.93/2.13 4.93/2.13 The argument filtering Pi contains the following mapping: 4.93/2.13 .(x1, x2) = .(x1, x2) 4.93/2.13 4.93/2.13 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 4.93/2.13 4.93/2.13 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 4.93/2.13 4.93/2.13 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 4.93/2.13 4.93/2.13 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 4.93/2.13 4.93/2.13 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 4.93/2.13 4.93/2.13 4.93/2.13 We have to consider all (P,R,Pi)-chains 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (17) PiDPToQDPProof (SOUND) 4.93/2.13 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (18) 4.93/2.13 Obligation: 4.93/2.13 Q DP problem: 4.93/2.13 The TRS P consists of the following rules: 4.93/2.13 4.93/2.13 U1_GA(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) 4.93/2.13 PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) 4.93/2.13 4.93/2.13 The TRS R consists of the following rules: 4.93/2.13 4.93/2.13 delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) 4.93/2.13 delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) 4.93/2.13 U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) 4.93/2.13 4.93/2.13 The set Q consists of the following terms: 4.93/2.13 4.93/2.13 delete_in_aga(x0) 4.93/2.13 U3_aga(x0, x1) 4.93/2.13 4.93/2.13 We have to consider all (P,Q,R)-chains. 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (19) MRRProof (EQUIVALENT) 4.93/2.13 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 4.93/2.13 4.93/2.13 Strictly oriented dependency pairs: 4.93/2.13 4.93/2.13 U1_GA(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) 4.93/2.13 PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) 4.93/2.13 4.93/2.13 Strictly oriented rules of the TRS R: 4.93/2.13 4.93/2.13 delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) 4.93/2.13 delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) 4.93/2.13 U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) 4.93/2.13 4.93/2.13 Used ordering: Knuth-Bendix order [KBO] with precedence:._2 > delete_in_aga_1 > U3_aga_2 > U1_GA_1 > PERMUTE_IN_GA_1 > delete_out_aga_2 4.93/2.13 4.93/2.13 and weight map: 4.93/2.13 4.93/2.13 delete_in_aga_1=1 4.93/2.13 U1_GA_1=1 4.93/2.13 PERMUTE_IN_GA_1=3 4.93/2.13 ._2=0 4.93/2.13 delete_out_aga_2=1 4.93/2.13 U3_aga_2=0 4.93/2.13 4.93/2.13 The variable weight is 1 4.93/2.13 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (20) 4.93/2.13 Obligation: 4.93/2.13 Q DP problem: 4.93/2.13 P is empty. 4.93/2.13 R is empty. 4.93/2.13 The set Q consists of the following terms: 4.93/2.13 4.93/2.13 delete_in_aga(x0) 4.93/2.13 U3_aga(x0, x1) 4.93/2.13 4.93/2.13 We have to consider all (P,Q,R)-chains. 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (21) PisEmptyProof (EQUIVALENT) 4.93/2.13 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.93/2.13 ---------------------------------------- 4.93/2.13 4.93/2.13 (22) 4.93/2.13 YES 4.93/2.15 EOF