4.01/1.81 YES 4.01/1.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.01/1.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.01/1.82 4.01/1.82 4.01/1.82 Left Termination of the query pattern 4.01/1.82 4.01/1.82 subset(g,g) 4.01/1.82 4.01/1.82 w.r.t. the given Prolog program could successfully be proven: 4.01/1.82 4.01/1.82 (0) Prolog 4.01/1.82 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.01/1.82 (2) PiTRS 4.01/1.82 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.01/1.82 (4) PiDP 4.01/1.82 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.01/1.82 (6) AND 4.01/1.82 (7) PiDP 4.01/1.82 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.01/1.82 (9) PiDP 4.01/1.82 (10) PiDPToQDPProof [EQUIVALENT, 1 ms] 4.01/1.82 (11) QDP 4.01/1.82 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.01/1.82 (13) YES 4.01/1.82 (14) PiDP 4.01/1.82 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.01/1.82 (16) PiDP 4.01/1.82 (17) PiDPToQDPProof [SOUND, 0 ms] 4.01/1.82 (18) QDP 4.01/1.82 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.01/1.82 (20) YES 4.01/1.82 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (0) 4.01/1.82 Obligation: 4.01/1.82 Clauses: 4.01/1.82 4.01/1.82 member(X, .(Y, Xs)) :- member(X, Xs). 4.01/1.82 member(X, .(X, Xs)). 4.01/1.82 subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)). 4.01/1.82 subset([], Ys). 4.01/1.82 member1(X, .(Y, Xs)) :- member1(X, Xs). 4.01/1.82 member1(X, .(X, Xs)). 4.01/1.82 subset1(.(X, Xs), Ys) :- ','(member1(X, Ys), subset1(Xs, Ys)). 4.01/1.82 subset1([], Ys). 4.01/1.82 4.01/1.82 4.01/1.82 Query: subset(g,g) 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (1) PrologToPiTRSProof (SOUND) 4.01/1.82 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.01/1.82 4.01/1.82 subset_in_2: (b,b) 4.01/1.82 4.01/1.82 member_in_2: (b,b) 4.01/1.82 4.01/1.82 Transforming Prolog into the following Term Rewriting System: 4.01/1.82 4.01/1.82 Pi-finite rewrite system: 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 4.01/1.82 4.01/1.82 4.01/1.82 4.01/1.82 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.01/1.82 4.01/1.82 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (2) 4.01/1.82 Obligation: 4.01/1.82 Pi-finite rewrite system: 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (3) DependencyPairsProof (EQUIVALENT) 4.01/1.82 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> MEMBER_IN_GG(X, Ys) 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> U1_GG(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_GG(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 SUBSET_IN_GG(x1, x2) = SUBSET_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U2_GG(x1, x2, x3, x4) = U2_GG(x2, x3, x4) 4.01/1.82 4.01/1.82 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U1_GG(x1, x2, x3, x4) = U1_GG(x4) 4.01/1.82 4.01/1.82 U3_GG(x1, x2, x3, x4) = U3_GG(x4) 4.01/1.82 4.01/1.82 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (4) 4.01/1.82 Obligation: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> MEMBER_IN_GG(X, Ys) 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> U1_GG(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_GG(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 SUBSET_IN_GG(x1, x2) = SUBSET_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U2_GG(x1, x2, x3, x4) = U2_GG(x2, x3, x4) 4.01/1.82 4.01/1.82 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U1_GG(x1, x2, x3, x4) = U1_GG(x4) 4.01/1.82 4.01/1.82 U3_GG(x1, x2, x3, x4) = U3_GG(x4) 4.01/1.82 4.01/1.82 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (5) DependencyGraphProof (EQUIVALENT) 4.01/1.82 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (6) 4.01/1.82 Complex Obligation (AND) 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (7) 4.01/1.82 Obligation: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (8) UsableRulesProof (EQUIVALENT) 4.01/1.82 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (9) 4.01/1.82 Obligation: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 4.01/1.82 R is empty. 4.01/1.82 Pi is empty. 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (10) PiDPToQDPProof (EQUIVALENT) 4.01/1.82 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (11) 4.01/1.82 Obligation: 4.01/1.82 Q DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 4.01/1.82 R is empty. 4.01/1.82 Q is empty. 4.01/1.82 We have to consider all (P,Q,R)-chains. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (12) QDPSizeChangeProof (EQUIVALENT) 4.01/1.82 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. 4.01/1.82 4.01/1.82 From the DPs we obtained the following set of size-change graphs: 4.01/1.82 *MEMBER_IN_GG(X, .(Y, Xs)) -> MEMBER_IN_GG(X, Xs) 4.01/1.82 The graph contains the following edges 1 >= 1, 2 > 2 4.01/1.82 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (13) 4.01/1.82 YES 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (14) 4.01/1.82 Obligation: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 subset_in_gg(.(X, Xs), Ys) -> U2_gg(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 U2_gg(X, Xs, Ys, member_out_gg(X, Ys)) -> U3_gg(X, Xs, Ys, subset_in_gg(Xs, Ys)) 4.01/1.82 subset_in_gg([], Ys) -> subset_out_gg([], Ys) 4.01/1.82 U3_gg(X, Xs, Ys, subset_out_gg(Xs, Ys)) -> subset_out_gg(.(X, Xs), Ys) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 subset_in_gg(x1, x2) = subset_in_gg(x1, x2) 4.01/1.82 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 U2_gg(x1, x2, x3, x4) = U2_gg(x2, x3, x4) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 U3_gg(x1, x2, x3, x4) = U3_gg(x4) 4.01/1.82 4.01/1.82 [] = [] 4.01/1.82 4.01/1.82 subset_out_gg(x1, x2) = subset_out_gg 4.01/1.82 4.01/1.82 SUBSET_IN_GG(x1, x2) = SUBSET_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U2_GG(x1, x2, x3, x4) = U2_GG(x2, x3, x4) 4.01/1.82 4.01/1.82 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (15) UsableRulesProof (EQUIVALENT) 4.01/1.82 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (16) 4.01/1.82 Obligation: 4.01/1.82 Pi DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 U2_GG(X, Xs, Ys, member_out_gg(X, Ys)) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(X, Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(X, Y, Xs, member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg(X, .(X, Xs)) 4.01/1.82 U1_gg(X, Y, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(Y, Xs)) 4.01/1.82 4.01/1.82 The argument filtering Pi contains the following mapping: 4.01/1.82 .(x1, x2) = .(x1, x2) 4.01/1.82 4.01/1.82 member_in_gg(x1, x2) = member_in_gg(x1, x2) 4.01/1.82 4.01/1.82 U1_gg(x1, x2, x3, x4) = U1_gg(x4) 4.01/1.82 4.01/1.82 member_out_gg(x1, x2) = member_out_gg 4.01/1.82 4.01/1.82 SUBSET_IN_GG(x1, x2) = SUBSET_IN_GG(x1, x2) 4.01/1.82 4.01/1.82 U2_GG(x1, x2, x3, x4) = U2_GG(x2, x3, x4) 4.01/1.82 4.01/1.82 4.01/1.82 We have to consider all (P,R,Pi)-chains 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (17) PiDPToQDPProof (SOUND) 4.01/1.82 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (18) 4.01/1.82 Obligation: 4.01/1.82 Q DP problem: 4.01/1.82 The TRS P consists of the following rules: 4.01/1.82 4.01/1.82 U2_GG(Xs, Ys, member_out_gg) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 4.01/1.82 The TRS R consists of the following rules: 4.01/1.82 4.01/1.82 member_in_gg(X, .(Y, Xs)) -> U1_gg(member_in_gg(X, Xs)) 4.01/1.82 member_in_gg(X, .(X, Xs)) -> member_out_gg 4.01/1.82 U1_gg(member_out_gg) -> member_out_gg 4.01/1.82 4.01/1.82 The set Q consists of the following terms: 4.01/1.82 4.01/1.82 member_in_gg(x0, x1) 4.01/1.82 U1_gg(x0) 4.01/1.82 4.01/1.82 We have to consider all (P,Q,R)-chains. 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (19) QDPSizeChangeProof (EQUIVALENT) 4.01/1.82 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. 4.01/1.82 4.01/1.82 From the DPs we obtained the following set of size-change graphs: 4.01/1.82 *SUBSET_IN_GG(.(X, Xs), Ys) -> U2_GG(Xs, Ys, member_in_gg(X, Ys)) 4.01/1.82 The graph contains the following edges 1 > 1, 2 >= 2 4.01/1.82 4.01/1.82 4.01/1.82 *U2_GG(Xs, Ys, member_out_gg) -> SUBSET_IN_GG(Xs, Ys) 4.01/1.82 The graph contains the following edges 1 >= 1, 2 >= 2 4.01/1.82 4.01/1.82 4.01/1.82 ---------------------------------------- 4.01/1.82 4.01/1.82 (20) 4.01/1.82 YES 4.01/1.85 EOF