/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRSRRRProof [EQUIVALENT, 131 ms] (2) RelTRS (3) RelTRSRRRProof [EQUIVALENT, 30 ms] (4) RelTRS (5) RelTRSRRRProof [EQUIVALENT, 20 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 18 ms] (8) RelTRS (9) RelTRSRRRProof [EQUIVALENT, 188 ms] (10) RelTRS (11) RelTRSRRRProof [EQUIVALENT, 14 ms] (12) RelTRS (13) RelTRSRRRProof [EQUIVALENT, 12 ms] (14) RelTRS (15) RelTRSRRRProof [EQUIVALENT, 1 ms] (16) RelTRS (17) RIsEmptyProof [EQUIVALENT, 4 ms] (18) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAo(R) -> R RAn(R) -> R WAo(W) -> W WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_r(read(RAo(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_w(read(RAo(r), x), write(W, Ww)))) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(R, Rw), write(W, WIo(y))))) -> top(check(sys_r(read(R, Rw), write(WAo(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIo(y))))) -> top(check(sys_w(read(R, Rw), write(WAo(W), y)))) top(ok(sys_r(read(r, RIo(x)), write(W, y)))) -> top(check(sys_w(read(RAo(r), x), write(W, y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIo(y))))) -> top(check(sys_r(read(R, x), write(WAo(W), y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RIo(x)) -> ok(RIo(x)) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIo(x)) -> RIo(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (1) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = x_1 POL(RIn(x_1)) = x_1 POL(RIo(x_1)) = x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = 1 + x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(read(x_1, x_2)) = x_1 + x_2 POL(sys_r(x_1, x_2)) = x_1 + x_2 POL(sys_w(x_1, x_2)) = x_1 + x_2 POL(top(x_1)) = x_1 POL(write(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: none Rules from S: top(ok(sys_r(read(R, Rw), write(W, WIo(y))))) -> top(check(sys_r(read(R, Rw), write(WAo(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIo(y))))) -> top(check(sys_w(read(R, Rw), write(WAo(W), y)))) top(ok(sys_w(read(R, x), write(W, WIo(y))))) -> top(check(sys_r(read(R, x), write(WAo(W), y)))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAo(R) -> R RAn(R) -> R WAo(W) -> W WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_r(read(RAo(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_w(read(RAo(r), x), write(W, Ww)))) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIo(x)), write(W, y)))) -> top(check(sys_w(read(RAo(r), x), write(W, y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RIo(x)) -> ok(RIo(x)) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIo(x)) -> RIo(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (3) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = 1 + x_1 POL(RIn(x_1)) = x_1 POL(RIo(x_1)) = 1 + x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(read(x_1, x_2)) = x_1 + x_2 POL(sys_r(x_1, x_2)) = 1 + x_1 + x_2 POL(sys_w(x_1, x_2)) = 1 + x_1 + x_2 POL(top(x_1)) = x_1 POL(write(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: RAo(R) -> R Rules from S: none ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAn(R) -> R WAo(W) -> W WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_r(read(RAo(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_w(read(RAo(r), x), write(W, Ww)))) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIo(x)), write(W, y)))) -> top(check(sys_w(read(RAo(r), x), write(W, y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RIo(x)) -> ok(RIo(x)) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIo(x)) -> RIo(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (5) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = 1 + x_1 POL(RIn(x_1)) = x_1 POL(RIo(x_1)) = 1 + x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = 1 + x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(read(x_1, x_2)) = x_1 + x_2 POL(sys_r(x_1, x_2)) = 1 + x_1 + x_2 POL(sys_w(x_1, x_2)) = 1 + x_1 + x_2 POL(top(x_1)) = x_1 POL(write(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: WAo(W) -> W Rules from S: none ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAn(R) -> R WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_r(read(RAo(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_w(read(RAo(r), x), write(W, Ww)))) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIo(x)), write(W, y)))) -> top(check(sys_w(read(RAo(r), x), write(W, y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RIo(x)) -> ok(RIo(x)) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIo(x)) -> RIo(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = x_1 POL(RIn(x_1)) = x_1 POL(RIo(x_1)) = 1 + x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(read(x_1, x_2)) = x_1 + x_2 POL(sys_r(x_1, x_2)) = x_1 + x_2 POL(sys_w(x_1, x_2)) = x_1 + x_2 POL(top(x_1)) = x_1 POL(write(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: none Rules from S: top(ok(sys_r(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_r(read(RAo(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIo(x)), write(W, Ww)))) -> top(check(sys_w(read(RAo(r), x), write(W, Ww)))) top(ok(sys_r(read(r, RIo(x)), write(W, y)))) -> top(check(sys_w(read(RAo(r), x), write(W, y)))) ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAn(R) -> R WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RIo(x)) -> ok(RIo(x)) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIo(x)) -> RIo(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (9) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(RAn(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(R) = [[0], [0]] >>> <<< POL(WAn(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(W) = [[0], [0]] >>> <<< POL(Rw) = [[0], [0]] >>> <<< POL(RIn(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(Ww) = [[0], [0]] >>> <<< POL(WIn(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(top(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(ok(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(sys_r(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(read(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(write(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(check(x_1)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 >>> <<< POL(sys_w(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(RIo(x_1)) = [[1], [1]] + [[1, 1], [0, 1]] * x_1 >>> <<< POL(RAo(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(WAo(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(WIo(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: none Rules from S: check(RIo(x)) -> ok(RIo(x)) check(RIo(x)) -> RIo(check(x)) ---------------------------------------- (10) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAn(R) -> R WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (11) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = x_1 POL(RIn(x_1)) = x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = 1 + x_1 POL(read(x_1, x_2)) = x_1 + x_2 POL(sys_r(x_1, x_2)) = x_1 + x_2 POL(sys_w(x_1, x_2)) = x_1 + x_2 POL(top(x_1)) = x_1 POL(write(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: none Rules from S: top(ok(sys_r(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_r(read(RAn(r), x), write(W, Ww)))) top(ok(sys_w(read(r, RIn(x)), write(W, Ww)))) -> top(check(sys_w(read(RAn(r), x), write(W, Ww)))) top(ok(sys_r(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_r(read(R, Rw), write(WAn(W), y)))) top(ok(sys_w(read(R, Rw), write(W, WIn(y))))) -> top(check(sys_w(read(R, Rw), write(WAn(W), y)))) top(ok(sys_r(read(r, RIn(x)), write(W, y)))) -> top(check(sys_w(read(RAn(r), x), write(W, y)))) top(ok(sys_w(read(R, x), write(W, WIn(y))))) -> top(check(sys_r(read(R, x), write(WAn(W), y)))) ---------------------------------------- (12) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: RAn(R) -> R WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (13) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(R) = 0 POL(RAn(x_1)) = 1 + x_1 POL(RAo(x_1)) = x_1 POL(RIn(x_1)) = x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(sys_r(x_1, x_2)) = x_1 + x_2 POL(sys_w(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: RAn(R) -> R Rules from S: none ---------------------------------------- (14) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: WAn(W) -> W The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (15) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(RAn(x_1)) = x_1 POL(RAo(x_1)) = x_1 POL(RIn(x_1)) = x_1 POL(Rw) = 0 POL(W) = 0 POL(WAn(x_1)) = 1 + x_1 POL(WAo(x_1)) = x_1 POL(WIn(x_1)) = x_1 POL(WIo(x_1)) = x_1 POL(Ww) = 0 POL(check(x_1)) = x_1 POL(ok(x_1)) = x_1 POL(sys_r(x_1, x_2)) = x_1 + x_2 POL(sys_w(x_1, x_2)) = x_1 + x_2 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: WAn(W) -> W Rules from S: none ---------------------------------------- (16) Obligation: Relative term rewrite system: R is empty. The relative TRS consists of the following S rules: Rw -> RIn(Rw) Ww -> WIn(Ww) check(RAo(x)) -> RAo(check(x)) check(RAn(x)) -> RAn(check(x)) check(WAo(x)) -> WAo(check(x)) check(WAn(x)) -> WAn(check(x)) check(RIn(x)) -> RIn(check(x)) check(WIo(x)) -> WIo(check(x)) check(WIn(x)) -> WIn(check(x)) check(sys_r(x, y)) -> sys_r(check(x), y) check(sys_r(x, y)) -> sys_r(x, check(y)) check(sys_w(x, y)) -> sys_w(check(x), y) check(sys_w(x, y)) -> sys_w(x, check(y)) RAo(ok(x)) -> ok(RAo(x)) RAn(ok(x)) -> ok(RAn(x)) WAo(ok(x)) -> ok(WAo(x)) WAn(ok(x)) -> ok(WAn(x)) RIn(ok(x)) -> ok(RIn(x)) WIo(ok(x)) -> ok(WIo(x)) WIn(ok(x)) -> ok(WIn(x)) sys_r(ok(x), y) -> ok(sys_r(x, y)) sys_r(x, ok(y)) -> ok(sys_r(x, y)) sys_w(ok(x), y) -> ok(sys_w(x, y)) sys_w(x, ok(y)) -> ok(sys_w(x, y)) ---------------------------------------- (17) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (18) YES