/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRSRRRProof [EQUIVALENT, 147 ms] (2) RelTRS (3) RelTRSRRRProof [EQUIVALENT, 157 ms] (4) RelTRS (5) RelTRSRRRProof [EQUIVALENT, 113 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 130 ms] (8) RelTRS (9) RelTRSRRRProof [EQUIVALENT, 77 ms] (10) RelTRS (11) RelTRSRRRProof [EQUIVALENT, 67 ms] (12) RelTRS (13) RelTRSRRRProof [EQUIVALENT, 63 ms] (14) RelTRS (15) RelTRSRRRProof [EQUIVALENT, 37 ms] (16) RelTRS (17) RelTRSRRRProof [EQUIVALENT, 0 ms] (18) RelTRS (19) RelTRSRRRProof [EQUIVALENT, 0 ms] (20) RelTRS (21) RelTRSRRRProof [EQUIVALENT, 6 ms] (22) RelTRS (23) RIsEmptyProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(old(n), e, s, w)) -> top(east(n, e, s, w)) top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(north(new(n), e, s, old(w))) -> top(east(n, e, s, old(w))) top(east(n, old(e), s, w)) -> top(south(n, e, s, w)) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(n, e, old(s), w)) -> top(west(n, e, s, w)) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, old(e), new(s), w)) -> top(west(n, old(e), s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(n, e, s, old(w))) -> top(north(n, e, s, w)) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(west(n, e, old(s), new(w))) -> top(north(n, e, old(s), w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) top(north(bot, e, s, old(w))) -> top(east(bot, e, s, old(w))) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, old(e), bot, w)) -> top(west(n, old(e), bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) top(west(n, e, old(s), bot)) -> top(north(n, e, old(s), bot)) The relative TRS consists of the following S rules: top(north(old(n), e, s, w)) -> top(north(n, e, s, w)) top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, old(e), s, w)) -> top(east(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, old(s), w)) -> top(south(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, old(w))) -> top(west(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (1) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(bot) = 0 POL(east(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(new(x_1)) = x_1 POL(north(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(old(x_1)) = 1 + x_1 POL(south(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(top(x_1)) = x_1 POL(west(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(north(old(n), e, s, w)) -> top(east(n, e, s, w)) top(east(n, old(e), s, w)) -> top(south(n, e, s, w)) top(south(n, e, old(s), w)) -> top(west(n, e, s, w)) top(west(n, e, s, old(w))) -> top(north(n, e, s, w)) Rules from S: top(north(old(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, old(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, old(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, old(w))) -> top(west(n, e, s, w)) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(north(new(n), e, s, old(w))) -> top(east(n, e, s, old(w))) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, old(e), new(s), w)) -> top(west(n, old(e), s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(west(n, e, old(s), new(w))) -> top(north(n, e, old(s), w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) top(north(bot, e, s, old(w))) -> top(east(bot, e, s, old(w))) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, old(e), bot, w)) -> top(west(n, old(e), bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) top(west(n, e, old(s), bot)) -> top(north(n, e, old(s), bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (3) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(west(n, e, old(s), new(w))) -> top(north(n, e, old(s), w)) top(west(n, e, old(s), bot)) -> top(north(n, e, old(s), bot)) Rules from S: none ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(north(new(n), e, s, old(w))) -> top(east(n, e, s, old(w))) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, old(e), new(s), w)) -> top(west(n, old(e), s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) top(north(bot, e, s, old(w))) -> top(east(bot, e, s, old(w))) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, old(e), bot, w)) -> top(west(n, old(e), bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (5) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(south(n, old(e), new(s), w)) -> top(west(n, old(e), s, w)) top(south(n, old(e), bot, w)) -> top(west(n, old(e), bot, w)) Rules from S: none ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(north(new(n), e, s, old(w))) -> top(east(n, e, s, old(w))) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) top(north(bot, e, s, old(w))) -> top(east(bot, e, s, old(w))) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 1]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 0], [0, 1]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 0], [0, 1]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(north(new(n), e, s, old(w))) -> top(east(n, e, s, old(w))) top(north(bot, e, s, old(w))) -> top(east(bot, e, s, old(w))) Rules from S: none ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (9) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 0], [0, 1]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 0], [0, 1]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 0], [0, 1]] * x_3 + [[1, 0], [0, 1]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(north(new(n), e, old(s), w)) -> top(east(n, e, old(s), w)) top(north(bot, e, old(s), w)) -> top(east(bot, e, old(s), w)) Rules from S: none ---------------------------------------- (10) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (11) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(east(old(n), new(e), s, w)) -> top(south(old(n), e, s, w)) top(east(old(n), bot, s, w)) -> top(south(old(n), bot, s, w)) Rules from S: none ---------------------------------------- (12) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (13) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(south(old(n), e, new(s), w)) -> top(west(old(n), e, s, w)) top(south(old(n), e, bot, w)) -> top(west(old(n), e, bot, w)) Rules from S: none ---------------------------------------- (14) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (15) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(top(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(north(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 0], [1, 0]] * x_4 >>> <<< POL(new(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(old(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(east(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 0], [1, 0]] * x_4 >>> <<< POL(south(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[1, 1], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 0], [1, 0]] * x_4 >>> <<< POL(west(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 + [[1, 0], [1, 0]] * x_4 >>> <<< POL(bot) = [[0], [0]] >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(west(old(n), e, s, new(w))) -> top(north(old(n), e, s, w)) top(west(n, old(e), s, new(w))) -> top(north(n, old(e), s, w)) top(west(old(n), e, s, bot)) -> top(north(old(n), e, s, bot)) top(west(n, old(e), s, bot)) -> top(north(n, old(e), s, bot)) Rules from S: none ---------------------------------------- (16) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (17) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(bot) = 0 POL(east(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(new(x_1)) = x_1 POL(north(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(old(x_1)) = x_1 POL(south(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(top(x_1)) = x_1 POL(west(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(south(n, e, new(s), old(w))) -> top(west(n, e, s, old(w))) top(south(n, e, bot, old(w))) -> top(west(n, e, bot, old(w))) Rules from S: none ---------------------------------------- (18) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (19) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(bot) = 0 POL(east(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(new(x_1)) = x_1 POL(north(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(old(x_1)) = x_1 POL(south(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(top(x_1)) = x_1 POL(west(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(east(n, new(e), old(s), w)) -> top(south(n, e, old(s), w)) top(east(n, new(e), s, old(w))) -> top(south(n, e, s, old(w))) top(east(n, bot, old(s), w)) -> top(south(n, bot, old(s), w)) top(east(n, bot, s, old(w))) -> top(south(n, bot, s, old(w))) Rules from S: none ---------------------------------------- (20) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (21) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(bot) = 0 POL(east(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(new(x_1)) = x_1 POL(north(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(old(x_1)) = x_1 POL(south(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(top(x_1)) = x_1 POL(west(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: top(north(new(n), old(e), s, w)) -> top(east(n, old(e), s, w)) top(north(bot, old(e), s, w)) -> top(east(bot, old(e), s, w)) Rules from S: none ---------------------------------------- (22) Obligation: Relative term rewrite system: R is empty. The relative TRS consists of the following S rules: top(north(new(n), e, s, w)) -> top(north(n, e, s, w)) top(east(n, new(e), s, w)) -> top(east(n, e, s, w)) top(south(n, e, new(s), w)) -> top(south(n, e, s, w)) top(west(n, e, s, new(w))) -> top(west(n, e, s, w)) bot -> new(bot) ---------------------------------------- (23) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (24) YES