7.62/2.75 YES 7.62/2.76 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.62/2.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.62/2.76 7.62/2.76 7.62/2.76 Termination of the given RelTRS could be proven: 7.62/2.76 7.62/2.76 (0) RelTRS 7.62/2.76 (1) RelTRSRRRProof [EQUIVALENT, 12 ms] 7.62/2.76 (2) RelTRS 7.62/2.76 (3) RelTRSRRRProof [EQUIVALENT, 8 ms] 7.62/2.76 (4) RelTRS 7.62/2.76 (5) RelTRSRRRProof [EQUIVALENT, 18 ms] 7.62/2.76 (6) RelTRS 7.62/2.76 (7) RelTRSRRRProof [EQUIVALENT, 19 ms] 7.62/2.76 (8) RelTRS 7.62/2.76 (9) RelTRSRRRProof [EQUIVALENT, 1 ms] 7.62/2.76 (10) RelTRS 7.62/2.76 (11) RIsEmptyProof [EQUIVALENT, 3 ms] 7.62/2.76 (12) YES 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (0) 7.62/2.76 Obligation: 7.62/2.76 Relative term rewrite system: 7.62/2.76 The relative TRS consists of the following R rules: 7.62/2.76 7.62/2.76 g(c(x, s(y))) -> g(c(s(x), y)) 7.62/2.76 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.76 f(f(x)) -> f(d(f(x))) 7.62/2.76 f(x) -> x 7.62/2.76 7.62/2.76 The relative TRS consists of the following S rules: 7.62/2.76 7.62/2.76 rand(x) -> x 7.62/2.76 rand(x) -> rand(s(x)) 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (1) RelTRSRRRProof (EQUIVALENT) 7.62/2.76 We used the following monotonic ordering for rule removal: 7.62/2.76 Polynomial interpretation [POLO]: 7.62/2.76 7.62/2.76 POL(c(x_1, x_2)) = x_1 + x_2 7.62/2.76 POL(d(x_1)) = x_1 7.62/2.76 POL(f(x_1)) = 1 + x_1 7.62/2.76 POL(g(x_1)) = x_1 7.62/2.76 POL(rand(x_1)) = x_1 7.62/2.76 POL(s(x_1)) = x_1 7.62/2.76 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.62/2.76 Rules from R: 7.62/2.76 7.62/2.76 f(x) -> x 7.62/2.76 Rules from S: 7.62/2.76 none 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (2) 7.62/2.76 Obligation: 7.62/2.76 Relative term rewrite system: 7.62/2.76 The relative TRS consists of the following R rules: 7.62/2.76 7.62/2.76 g(c(x, s(y))) -> g(c(s(x), y)) 7.62/2.76 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.76 f(f(x)) -> f(d(f(x))) 7.62/2.76 7.62/2.76 The relative TRS consists of the following S rules: 7.62/2.76 7.62/2.76 rand(x) -> x 7.62/2.76 rand(x) -> rand(s(x)) 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (3) RelTRSRRRProof (EQUIVALENT) 7.62/2.76 We used the following monotonic ordering for rule removal: 7.62/2.76 Polynomial interpretation [POLO]: 7.62/2.76 7.62/2.76 POL(c(x_1, x_2)) = x_1 + x_2 7.62/2.76 POL(d(x_1)) = x_1 7.62/2.76 POL(f(x_1)) = x_1 7.62/2.76 POL(g(x_1)) = x_1 7.62/2.76 POL(rand(x_1)) = 1 + x_1 7.62/2.76 POL(s(x_1)) = x_1 7.62/2.76 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.62/2.76 Rules from R: 7.62/2.76 none 7.62/2.76 Rules from S: 7.62/2.76 7.62/2.76 rand(x) -> x 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (4) 7.62/2.76 Obligation: 7.62/2.76 Relative term rewrite system: 7.62/2.76 The relative TRS consists of the following R rules: 7.62/2.76 7.62/2.76 g(c(x, s(y))) -> g(c(s(x), y)) 7.62/2.76 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.76 f(f(x)) -> f(d(f(x))) 7.62/2.76 7.62/2.76 The relative TRS consists of the following S rules: 7.62/2.76 7.62/2.76 rand(x) -> rand(s(x)) 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (5) RelTRSRRRProof (EQUIVALENT) 7.62/2.76 We used the following monotonic ordering for rule removal: 7.62/2.76 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(g(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(c(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(s(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(f(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(d(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(rand(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.62/2.76 Rules from R: 7.62/2.76 7.62/2.76 f(f(x)) -> f(d(f(x))) 7.62/2.76 Rules from S: 7.62/2.76 none 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (6) 7.62/2.76 Obligation: 7.62/2.76 Relative term rewrite system: 7.62/2.76 The relative TRS consists of the following R rules: 7.62/2.76 7.62/2.76 g(c(x, s(y))) -> g(c(s(x), y)) 7.62/2.76 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.76 7.62/2.76 The relative TRS consists of the following S rules: 7.62/2.76 7.62/2.76 rand(x) -> rand(s(x)) 7.62/2.76 7.62/2.76 7.62/2.76 ---------------------------------------- 7.62/2.76 7.62/2.76 (7) RelTRSRRRProof (EQUIVALENT) 7.62/2.76 We used the following monotonic ordering for rule removal: 7.62/2.76 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(g(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(c(x_1, x_2)) = [[1], [1]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(s(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(f(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.62/2.76 >>> 7.62/2.76 7.62/2.76 <<< 7.62/2.76 POL(rand(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 7.62/2.77 >>> 7.62/2.77 7.62/2.77 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.62/2.77 Rules from R: 7.62/2.77 7.62/2.77 g(c(x, s(y))) -> g(c(s(x), y)) 7.62/2.77 Rules from S: 7.62/2.77 none 7.62/2.77 7.62/2.77 7.62/2.77 7.62/2.77 7.62/2.77 ---------------------------------------- 7.62/2.77 7.62/2.77 (8) 7.62/2.77 Obligation: 7.62/2.77 Relative term rewrite system: 7.62/2.77 The relative TRS consists of the following R rules: 7.62/2.77 7.62/2.77 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.77 7.62/2.77 The relative TRS consists of the following S rules: 7.62/2.77 7.62/2.77 rand(x) -> rand(s(x)) 7.62/2.77 7.62/2.77 7.62/2.77 ---------------------------------------- 7.62/2.77 7.62/2.77 (9) RelTRSRRRProof (EQUIVALENT) 7.62/2.77 We used the following monotonic ordering for rule removal: 7.62/2.77 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.62/2.77 7.62/2.77 <<< 7.62/2.77 POL(f(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 7.62/2.77 >>> 7.62/2.77 7.62/2.77 <<< 7.62/2.77 POL(c(x_1, x_2)) = [[1], [1]] + [[1, 1], [0, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 7.62/2.77 >>> 7.62/2.77 7.62/2.77 <<< 7.62/2.77 POL(s(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 7.62/2.77 >>> 7.62/2.77 7.62/2.77 <<< 7.62/2.77 POL(rand(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 7.62/2.77 >>> 7.62/2.77 7.62/2.77 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.62/2.77 Rules from R: 7.62/2.77 7.62/2.77 f(c(s(x), y)) -> f(c(x, s(y))) 7.62/2.77 Rules from S: 7.62/2.77 none 7.62/2.77 7.62/2.77 7.62/2.77 7.62/2.77 7.62/2.77 ---------------------------------------- 7.62/2.77 7.62/2.77 (10) 7.62/2.77 Obligation: 7.62/2.77 Relative term rewrite system: 7.62/2.77 R is empty. 7.62/2.77 The relative TRS consists of the following S rules: 7.62/2.77 7.62/2.77 rand(x) -> rand(s(x)) 7.62/2.77 7.62/2.77 7.62/2.77 ---------------------------------------- 7.62/2.77 7.62/2.77 (11) RIsEmptyProof (EQUIVALENT) 7.62/2.77 The TRS R is empty. Hence, termination is trivially proven. 7.62/2.77 ---------------------------------------- 7.62/2.77 7.62/2.77 (12) 7.62/2.77 YES 7.98/2.83 EOF