5.12/2.56 YES 5.12/2.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.12/2.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.12/2.60 5.12/2.60 5.12/2.60 Left Termination of the query pattern 5.12/2.60 5.12/2.60 permute(g,a) 5.12/2.60 5.12/2.60 w.r.t. the given Prolog program could successfully be proven: 5.12/2.60 5.12/2.60 (0) Prolog 5.12/2.60 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.12/2.60 (2) PiTRS 5.12/2.60 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.12/2.60 (4) PiDP 5.12/2.60 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.60 (6) AND 5.12/2.60 (7) PiDP 5.12/2.60 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.12/2.60 (9) PiDP 5.12/2.60 (10) PiDPToQDPProof [SOUND, 0 ms] 5.12/2.60 (11) QDP 5.12/2.60 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.12/2.60 (13) YES 5.12/2.60 (14) PiDP 5.12/2.60 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.12/2.60 (16) PiDP 5.12/2.60 (17) PiDPToQDPProof [SOUND, 0 ms] 5.12/2.60 (18) QDP 5.12/2.60 (19) MRRProof [EQUIVALENT, 3 ms] 5.12/2.60 (20) QDP 5.12/2.60 (21) PisEmptyProof [EQUIVALENT, 0 ms] 5.12/2.60 (22) YES 5.12/2.60 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (0) 5.12/2.60 Obligation: 5.12/2.60 Clauses: 5.12/2.60 5.12/2.60 permute([], []). 5.12/2.60 permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)). 5.12/2.60 delete(X, .(X, Y), Y). 5.12/2.60 delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z). 5.12/2.60 5.12/2.60 5.12/2.60 Query: permute(g,a) 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (1) PrologToPiTRSProof (SOUND) 5.12/2.60 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.12/2.60 5.12/2.60 permute_in_2: (b,f) 5.12/2.60 5.12/2.60 delete_in_3: (f,b,f) 5.12/2.60 5.12/2.60 Transforming Prolog into the following Term Rewriting System: 5.12/2.60 5.12/2.60 Pi-finite rewrite system: 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 5.12/2.60 5.12/2.60 5.12/2.60 5.12/2.60 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.12/2.60 5.12/2.60 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (2) 5.12/2.60 Obligation: 5.12/2.60 Pi-finite rewrite system: 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (3) DependencyPairsProof (EQUIVALENT) 5.12/2.60 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 5.12/2.60 5.12/2.60 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 5.12/2.60 5.12/2.60 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 5.12/2.60 5.12/2.60 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 5.12/2.60 5.12/2.60 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (4) 5.12/2.60 Obligation: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 5.12/2.60 5.12/2.60 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 5.12/2.60 5.12/2.60 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 5.12/2.60 5.12/2.60 U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) 5.12/2.60 5.12/2.60 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (5) DependencyGraphProof (EQUIVALENT) 5.12/2.60 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (6) 5.12/2.60 Complex Obligation (AND) 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (7) 5.12/2.60 Obligation: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (8) UsableRulesProof (EQUIVALENT) 5.12/2.60 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (9) 5.12/2.60 Obligation: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) 5.12/2.60 5.12/2.60 R is empty. 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (10) PiDPToQDPProof (SOUND) 5.12/2.60 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (11) 5.12/2.60 Obligation: 5.12/2.60 Q DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y) 5.12/2.60 5.12/2.60 R is empty. 5.12/2.60 Q is empty. 5.12/2.60 We have to consider all (P,Q,R)-chains. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (12) QDPSizeChangeProof (EQUIVALENT) 5.12/2.60 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. 5.12/2.60 5.12/2.60 From the DPs we obtained the following set of size-change graphs: 5.12/2.60 *DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y) 5.12/2.60 The graph contains the following edges 1 > 1 5.12/2.60 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (13) 5.12/2.60 YES 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (14) 5.12/2.60 Obligation: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 permute_in_ga([], []) -> permute_out_ga([], []) 5.12/2.60 permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) 5.12/2.60 U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 permute_in_ga(x1, x2) = permute_in_ga(x1) 5.12/2.60 5.12/2.60 [] = [] 5.12/2.60 5.12/2.60 permute_out_ga(x1, x2) = permute_out_ga(x2) 5.12/2.60 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 5.12/2.60 5.12/2.60 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (15) UsableRulesProof (EQUIVALENT) 5.12/2.60 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (16) 5.12/2.60 Obligation: 5.12/2.60 Pi DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) 5.12/2.60 PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) 5.12/2.60 delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) 5.12/2.60 U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) 5.12/2.60 5.12/2.60 The argument filtering Pi contains the following mapping: 5.12/2.60 .(x1, x2) = .(x1, x2) 5.12/2.60 5.12/2.60 delete_in_aga(x1, x2, x3) = delete_in_aga(x2) 5.12/2.60 5.12/2.60 delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) 5.12/2.60 5.12/2.60 U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) 5.12/2.60 5.12/2.60 PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) 5.12/2.60 5.12/2.60 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 5.12/2.60 5.12/2.60 5.12/2.60 We have to consider all (P,R,Pi)-chains 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (17) PiDPToQDPProof (SOUND) 5.12/2.60 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (18) 5.12/2.60 Obligation: 5.12/2.60 Q DP problem: 5.12/2.60 The TRS P consists of the following rules: 5.12/2.60 5.12/2.60 U1_GA(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) 5.12/2.60 PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) 5.12/2.60 5.12/2.60 The TRS R consists of the following rules: 5.12/2.60 5.12/2.60 delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) 5.12/2.60 delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) 5.12/2.60 U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) 5.12/2.60 5.12/2.60 The set Q consists of the following terms: 5.12/2.60 5.12/2.60 delete_in_aga(x0) 5.12/2.60 U3_aga(x0, x1) 5.12/2.60 5.12/2.60 We have to consider all (P,Q,R)-chains. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (19) MRRProof (EQUIVALENT) 5.12/2.60 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. 5.12/2.60 5.12/2.60 Strictly oriented dependency pairs: 5.12/2.60 5.12/2.60 U1_GA(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) 5.12/2.60 PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) 5.12/2.60 5.12/2.60 Strictly oriented rules of the TRS R: 5.12/2.60 5.12/2.60 delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) 5.12/2.60 delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) 5.12/2.60 U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) 5.12/2.60 5.12/2.60 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 5.12/2.60 5.12/2.60 and weight map: 5.12/2.60 5.12/2.60 delete_in_aga_1=1 5.12/2.60 U1_GA_1=1 5.12/2.60 PERMUTE_IN_GA_1=3 5.12/2.60 ._2=0 5.12/2.60 delete_out_aga_2=1 5.12/2.60 U3_aga_2=0 5.12/2.60 5.12/2.60 The variable weight is 1 5.12/2.60 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (20) 5.12/2.60 Obligation: 5.12/2.60 Q DP problem: 5.12/2.60 P is empty. 5.12/2.60 R is empty. 5.12/2.60 The set Q consists of the following terms: 5.12/2.60 5.12/2.60 delete_in_aga(x0) 5.12/2.60 U3_aga(x0, x1) 5.12/2.60 5.12/2.60 We have to consider all (P,Q,R)-chains. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (21) PisEmptyProof (EQUIVALENT) 5.12/2.60 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.12/2.60 ---------------------------------------- 5.12/2.60 5.12/2.60 (22) 5.12/2.60 YES 5.17/2.63 EOF