7.94/2.87 YES 7.94/2.89 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 7.94/2.89 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.94/2.89 7.94/2.89 7.94/2.89 Termination of the given RelTRS could be proven: 7.94/2.89 7.94/2.89 (0) RelTRS 7.94/2.89 (1) RelTRSRRRProof [EQUIVALENT, 43 ms] 7.94/2.89 (2) RelTRS 7.94/2.89 (3) RelTRSRRRProof [EQUIVALENT, 0 ms] 7.94/2.89 (4) RelTRS 7.94/2.89 (5) RelTRSRRRProof [EQUIVALENT, 13 ms] 7.94/2.89 (6) RelTRS 7.94/2.89 (7) RelTRSRRRProof [EQUIVALENT, 10 ms] 7.94/2.89 (8) RelTRS 7.94/2.89 (9) RelTRSRRRProof [EQUIVALENT, 9 ms] 7.94/2.89 (10) RelTRS 7.94/2.89 (11) RIsEmptyProof [EQUIVALENT, 0 ms] 7.94/2.89 (12) YES 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (0) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 The relative TRS consists of the following R rules: 7.94/2.89 7.94/2.89 not(true) -> false 7.94/2.89 not(false) -> true 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 evenodd(0, s(0)) -> false 7.94/2.89 evenodd(s(x), s(0)) -> evenodd(x, 0) 7.94/2.89 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> x 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (1) RelTRSRRRProof (EQUIVALENT) 7.94/2.89 We used the following monotonic ordering for rule removal: 7.94/2.89 Polynomial interpretation [POLO]: 7.94/2.89 7.94/2.89 POL(0) = 0 7.94/2.89 POL(evenodd(x_1, x_2)) = 1 + x_1 + x_2 7.94/2.89 POL(false) = 1 7.94/2.89 POL(not(x_1)) = x_1 7.94/2.89 POL(rand(x_1)) = 1 + x_1 7.94/2.89 POL(s(x_1)) = x_1 7.94/2.89 POL(true) = 1 7.94/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.94/2.89 Rules from R: 7.94/2.89 none 7.94/2.89 Rules from S: 7.94/2.89 7.94/2.89 rand(x) -> x 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (2) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 The relative TRS consists of the following R rules: 7.94/2.89 7.94/2.89 not(true) -> false 7.94/2.89 not(false) -> true 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 evenodd(0, s(0)) -> false 7.94/2.89 evenodd(s(x), s(0)) -> evenodd(x, 0) 7.94/2.89 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (3) RelTRSRRRProof (EQUIVALENT) 7.94/2.89 We used the following monotonic ordering for rule removal: 7.94/2.89 Polynomial interpretation [POLO]: 7.94/2.89 7.94/2.89 POL(0) = 1 7.94/2.89 POL(evenodd(x_1, x_2)) = 1 + x_1 + x_2 7.94/2.89 POL(false) = 1 7.94/2.89 POL(not(x_1)) = x_1 7.94/2.89 POL(rand(x_1)) = x_1 7.94/2.89 POL(s(x_1)) = x_1 7.94/2.89 POL(true) = 1 7.94/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.94/2.89 Rules from R: 7.94/2.89 7.94/2.89 evenodd(0, s(0)) -> false 7.94/2.89 Rules from S: 7.94/2.89 none 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (4) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 The relative TRS consists of the following R rules: 7.94/2.89 7.94/2.89 not(true) -> false 7.94/2.89 not(false) -> true 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 evenodd(s(x), s(0)) -> evenodd(x, 0) 7.94/2.89 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (5) RelTRSRRRProof (EQUIVALENT) 7.94/2.89 We used the following monotonic ordering for rule removal: 7.94/2.89 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(not(x_1)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(true) = [[0], [1]] 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(false) = [[0], [1]] 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(evenodd(x_1, x_2)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(0) = [[0], [0]] 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(s(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(rand(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.94/2.89 Rules from R: 7.94/2.89 7.94/2.89 not(true) -> false 7.94/2.89 not(false) -> true 7.94/2.89 Rules from S: 7.94/2.89 none 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (6) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 The relative TRS consists of the following R rules: 7.94/2.89 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 evenodd(s(x), s(0)) -> evenodd(x, 0) 7.94/2.89 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (7) RelTRSRRRProof (EQUIVALENT) 7.94/2.89 We used the following monotonic ordering for rule removal: 7.94/2.89 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(evenodd(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(0) = [[0], [0]] 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(not(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(s(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(rand(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.94/2.89 Rules from R: 7.94/2.89 7.94/2.89 evenodd(s(x), s(0)) -> evenodd(x, 0) 7.94/2.89 Rules from S: 7.94/2.89 none 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (8) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 The relative TRS consists of the following R rules: 7.94/2.89 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (9) RelTRSRRRProof (EQUIVALENT) 7.94/2.89 We used the following monotonic ordering for rule removal: 7.94/2.89 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(evenodd(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 0]] * x_2 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(0) = [[0], [1]] 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(not(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(s(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 <<< 7.94/2.89 POL(rand(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 7.94/2.89 >>> 7.94/2.89 7.94/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.94/2.89 Rules from R: 7.94/2.89 7.94/2.89 evenodd(x, 0) -> not(evenodd(x, s(0))) 7.94/2.89 Rules from S: 7.94/2.89 none 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (10) 7.94/2.89 Obligation: 7.94/2.89 Relative term rewrite system: 7.94/2.89 R is empty. 7.94/2.89 The relative TRS consists of the following S rules: 7.94/2.89 7.94/2.89 rand(x) -> rand(s(x)) 7.94/2.89 7.94/2.89 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (11) RIsEmptyProof (EQUIVALENT) 7.94/2.89 The TRS R is empty. Hence, termination is trivially proven. 7.94/2.89 ---------------------------------------- 7.94/2.89 7.94/2.89 (12) 7.94/2.89 YES 8.30/2.96 EOF