23.55/7.59 YES 23.55/7.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.55/7.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.55/7.60 23.55/7.60 23.55/7.60 Termination of the given RelTRS could be proven: 23.55/7.60 23.55/7.60 (0) RelTRS 23.55/7.60 (1) FlatCCProof [EQUIVALENT, 0 ms] 23.55/7.60 (2) RelTRS 23.55/7.60 (3) RootLabelingProof [EQUIVALENT, 45 ms] 23.55/7.60 (4) RelTRS 23.55/7.60 (5) RelTRSRRRProof [EQUIVALENT, 1005 ms] 23.55/7.60 (6) RelTRS 23.55/7.60 (7) RelTRSRRRProof [EQUIVALENT, 209 ms] 23.55/7.60 (8) RelTRS 23.55/7.60 (9) RelTRSRRRProof [EQUIVALENT, 221 ms] 23.55/7.60 (10) RelTRS 23.55/7.60 (11) RelTRSRRRProof [EQUIVALENT, 194 ms] 23.55/7.60 (12) RelTRS 23.55/7.60 (13) RelTRSRRRProof [EQUIVALENT, 52 ms] 23.55/7.60 (14) RelTRS 23.55/7.60 (15) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (16) RelTRS 23.55/7.60 (17) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (18) RelTRS 23.55/7.60 (19) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (20) RelTRS 23.55/7.60 (21) RelTRSRRRProof [EQUIVALENT, 12 ms] 23.55/7.60 (22) RelTRS 23.55/7.60 (23) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (24) RelTRS 23.55/7.60 (25) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (26) RelTRS 23.55/7.60 (27) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (28) RelTRS 23.55/7.60 (29) RelTRSRRRProof [EQUIVALENT, 1 ms] 23.55/7.60 (30) RelTRS 23.55/7.60 (31) RelTRSRRRProof [EQUIVALENT, 0 ms] 23.55/7.60 (32) RelTRS 23.55/7.60 (33) RelTRSRRRProof [EQUIVALENT, 5 ms] 23.55/7.60 (34) RelTRS 23.55/7.60 (35) RIsEmptyProof [EQUIVALENT, 0 ms] 23.55/7.60 (36) YES 23.55/7.60 23.55/7.60 23.55/7.60 ---------------------------------------- 23.55/7.60 23.55/7.60 (0) 23.55/7.60 Obligation: 23.55/7.60 Relative term rewrite system: 23.55/7.60 The relative TRS consists of the following R rules: 23.55/7.60 23.55/7.60 f(a, g(y), z) -> f(b, g(y), g(y)) 23.55/7.60 f(b, g(y), z) -> f(a, y, z) 23.55/7.60 a -> b 23.55/7.60 23.55/7.60 The relative TRS consists of the following S rules: 23.55/7.60 23.55/7.60 f(x, y, z) -> f(x, y, g(z)) 23.55/7.60 23.55/7.60 23.55/7.60 ---------------------------------------- 23.55/7.60 23.55/7.60 (1) FlatCCProof (EQUIVALENT) 23.55/7.60 We used flat context closure [ROOTLAB] 23.55/7.60 23.55/7.60 ---------------------------------------- 23.55/7.60 23.55/7.60 (2) 23.55/7.60 Obligation: 23.55/7.60 Relative term rewrite system: 23.55/7.60 The relative TRS consists of the following R rules: 23.55/7.60 23.55/7.60 f(a, g(y), z) -> f(b, g(y), g(y)) 23.55/7.60 f(b, g(y), z) -> f(a, y, z) 23.55/7.60 f(a, z1, z2) -> f(b, z1, z2) 23.55/7.60 f(z0, a, z2) -> f(z0, b, z2) 23.55/7.60 f(z0, z1, a) -> f(z0, z1, b) 23.55/7.60 g(a) -> g(b) 23.55/7.60 23.55/7.60 The relative TRS consists of the following S rules: 23.55/7.60 23.55/7.60 f(x, y, z) -> f(x, y, g(z)) 23.55/7.60 23.55/7.60 23.55/7.60 ---------------------------------------- 23.55/7.60 23.55/7.60 (3) RootLabelingProof (EQUIVALENT) 23.55/7.60 We used plain root labeling [ROOTLAB] with the following heuristic: 23.55/7.60 LabelAll: All function symbols get labeled 23.55/7.60 23.55/7.60 23.55/7.60 ---------------------------------------- 23.55/7.60 23.55/7.60 (4) 23.55/7.60 Obligation: 23.55/7.60 Relative term rewrite system: 23.55/7.60 The relative TRS consists of the following R rules: 23.55/7.60 23.55/7.60 f_{a,g_1,f_3}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.60 f_{a,g_1,a}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.60 f_{a,g_1,g_1}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.60 f_{a,g_1,b}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.60 f_{a,g_1,f_3}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.60 f_{a,g_1,a}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.60 f_{a,g_1,g_1}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.60 f_{a,g_1,b}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.60 f_{a,g_1,f_3}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.60 f_{a,g_1,a}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.60 f_{a,g_1,g_1}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.60 f_{a,g_1,b}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.60 f_{a,g_1,f_3}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.60 f_{a,g_1,a}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.60 f_{a,g_1,g_1}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.60 f_{a,g_1,b}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.60 f_{b,g_1,f_3}(b, g_{f_3}(y), z) -> f_{a,f_3,f_3}(a, y, z) 23.55/7.60 f_{b,g_1,a}(b, g_{f_3}(y), z) -> f_{a,f_3,a}(a, y, z) 23.55/7.60 f_{b,g_1,g_1}(b, g_{f_3}(y), z) -> f_{a,f_3,g_1}(a, y, z) 23.55/7.60 f_{b,g_1,b}(b, g_{f_3}(y), z) -> f_{a,f_3,b}(a, y, z) 23.55/7.60 f_{b,g_1,f_3}(b, g_{a}(y), z) -> f_{a,a,f_3}(a, y, z) 23.55/7.60 f_{b,g_1,a}(b, g_{a}(y), z) -> f_{a,a,a}(a, y, z) 23.55/7.60 f_{b,g_1,g_1}(b, g_{a}(y), z) -> f_{a,a,g_1}(a, y, z) 23.55/7.60 f_{b,g_1,b}(b, g_{a}(y), z) -> f_{a,a,b}(a, y, z) 23.55/7.60 f_{b,g_1,f_3}(b, g_{g_1}(y), z) -> f_{a,g_1,f_3}(a, y, z) 23.55/7.60 f_{b,g_1,a}(b, g_{g_1}(y), z) -> f_{a,g_1,a}(a, y, z) 23.55/7.60 f_{b,g_1,g_1}(b, g_{g_1}(y), z) -> f_{a,g_1,g_1}(a, y, z) 23.55/7.60 f_{b,g_1,b}(b, g_{g_1}(y), z) -> f_{a,g_1,b}(a, y, z) 23.55/7.60 f_{b,g_1,f_3}(b, g_{b}(y), z) -> f_{a,b,f_3}(a, y, z) 23.55/7.60 f_{b,g_1,a}(b, g_{b}(y), z) -> f_{a,b,a}(a, y, z) 23.55/7.60 f_{b,g_1,g_1}(b, g_{b}(y), z) -> f_{a,b,g_1}(a, y, z) 23.55/7.60 f_{b,g_1,b}(b, g_{b}(y), z) -> f_{a,b,b}(a, y, z) 23.55/7.60 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.55/7.60 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.55/7.60 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.55/7.60 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.55/7.60 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.55/7.60 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.55/7.60 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.55/7.60 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.55/7.60 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.55/7.60 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.55/7.60 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.55/7.60 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.55/7.60 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.55/7.60 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.55/7.60 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.55/7.60 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.55/7.60 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.55/7.60 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.55/7.60 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.55/7.60 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.55/7.60 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.55/7.60 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.55/7.60 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.55/7.60 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.55/7.60 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.55/7.60 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.55/7.60 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.55/7.60 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.55/7.60 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.55/7.60 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.55/7.60 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.55/7.60 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.55/7.60 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.55/7.60 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.55/7.60 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.55/7.60 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.55/7.60 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.55/7.60 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.55/7.60 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.55/7.60 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.55/7.60 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.55/7.60 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.55/7.60 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.55/7.60 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.55/7.60 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.55/7.60 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.55/7.60 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.55/7.60 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.55/7.60 g_{a}(a) -> g_{b}(b) 23.55/7.60 23.55/7.60 The relative TRS consists of the following S rules: 23.55/7.60 23.55/7.60 f_{f_3,f_3,f_3}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{f_3,g_1,f_3}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{g_1,f_3,f_3}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.60 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.60 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.60 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.60 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,g_1,f_3}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 23.55/7.61 23.55/7.61 ---------------------------------------- 23.55/7.61 23.55/7.61 (5) RelTRSRRRProof (EQUIVALENT) 23.55/7.61 We used the following monotonic ordering for rule removal: 23.55/7.61 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,f_3}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 1], [2, 1]] * x_1 + [[2, 1], [0, 2]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(a) = [[0], [0]] 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{f_3}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(b) = [[0], [0]] 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,a}(x_1, x_2, x_3)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 + [[3, 1], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[3, 1], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{a}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{g_1}(x_1)) = [[0], [3]] + [[1, 0], [1, 1]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{b}(x_1)) = [[0], [0]] + [[1, 0], [0, 2]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,f_3}(x_1, x_2, x_3)) = [[0], [2]] + [[1, 2], [2, 2]] * x_1 + [[2, 1], [0, 2]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 2], [2, 2]] * x_1 + [[3, 0], [2, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,a}(x_1, x_2, x_3)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 + [[3, 1], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[3, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 1], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[3, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,f_3}(x_1, x_2, x_3)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,a}(x_1, x_2, x_3)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,b,a}(x_1, x_2, x_3)) = [[0], [2]] + [[1, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 2], [2, 2]] * x_1 + [[3, 0], [1, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[3, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[3, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,a,a}(x_1, x_2, x_3)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,b,a}(x_1, x_2, x_3)) = [[0], [1]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,a,f_3}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[3, 1], [2, 2]] * x_2 + [[3, 3], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 1], [3, 3]] * x_1 + [[2, 1], [2, 2]] * x_2 + [[3, 3], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,a,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,b,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 2], [2, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 2], [2, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 2], [2, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[2, 1], [2, 2]] * x_2 + [[3, 3], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[3, 1], [2, 2]] * x_2 + [[3, 0], [3, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,a,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,b,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 2], [1, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 2], [2, 3]] * x_1 + [[2, 0], [0, 0]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 2], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 1], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 2], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 0], [1, 2]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 0], [2, 2]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 1], [1, 2]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = [[0], [2]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 0], [1, 2]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[2, 0], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,f_3,f_3}(x_1, x_2, x_3)) = [[3], [1]] + [[3, 3], [3, 2]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[3, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 2]] * x_1 + [[3, 3], [3, 2]] * x_2 + [[2, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,g_1,f_3}(x_1, x_2, x_3)) = [[1], [1]] + [[3, 3], [3, 2]] * x_1 + [[3, 2], [3, 3]] * x_2 + [[1, 1], [1, 3]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 3], [3, 2]] * x_1 + [[3, 2], [3, 2]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,f_3,f_3}(x_1, x_2, x_3)) = [[3], [1]] + [[3, 2], [3, 3]] * x_1 + [[3, 3], [2, 3]] * x_2 + [[1, 3], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[2, 2], [3, 3]] * x_1 + [[2, 3], [2, 3]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,g_1,f_3}(x_1, x_2, x_3)) = [[3], [1]] + [[3, 3], [3, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[3, 2], [2, 3]] * x_1 + [[3, 3], [3, 3]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.55/7.61 Rules from R: 23.55/7.61 23.55/7.61 f_{b,g_1,f_3}(b, g_{g_1}(y), z) -> f_{a,g_1,f_3}(a, y, z) 23.55/7.61 f_{b,g_1,a}(b, g_{g_1}(y), z) -> f_{a,g_1,a}(a, y, z) 23.55/7.61 f_{b,g_1,g_1}(b, g_{g_1}(y), z) -> f_{a,g_1,g_1}(a, y, z) 23.55/7.61 f_{b,g_1,b}(b, g_{g_1}(y), z) -> f_{a,g_1,b}(a, y, z) 23.55/7.61 Rules from S: 23.55/7.61 23.55/7.61 f_{f_3,f_3,f_3}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{f_3,g_1,f_3}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,f_3,f_3}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,g_1,f_3}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 23.55/7.61 23.55/7.61 23.55/7.61 23.55/7.61 ---------------------------------------- 23.55/7.61 23.55/7.61 (6) 23.55/7.61 Obligation: 23.55/7.61 Relative term rewrite system: 23.55/7.61 The relative TRS consists of the following R rules: 23.55/7.61 23.55/7.61 f_{a,g_1,f_3}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.61 f_{a,g_1,a}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.61 f_{a,g_1,g_1}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.61 f_{a,g_1,b}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.55/7.61 f_{a,g_1,f_3}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.61 f_{a,g_1,a}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.61 f_{a,g_1,g_1}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.61 f_{a,g_1,b}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.55/7.61 f_{a,g_1,f_3}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.61 f_{a,g_1,a}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.61 f_{a,g_1,g_1}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.61 f_{a,g_1,b}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.55/7.61 f_{a,g_1,f_3}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.61 f_{a,g_1,a}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.61 f_{a,g_1,g_1}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.61 f_{a,g_1,b}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.55/7.61 f_{b,g_1,f_3}(b, g_{f_3}(y), z) -> f_{a,f_3,f_3}(a, y, z) 23.55/7.61 f_{b,g_1,a}(b, g_{f_3}(y), z) -> f_{a,f_3,a}(a, y, z) 23.55/7.61 f_{b,g_1,g_1}(b, g_{f_3}(y), z) -> f_{a,f_3,g_1}(a, y, z) 23.55/7.61 f_{b,g_1,b}(b, g_{f_3}(y), z) -> f_{a,f_3,b}(a, y, z) 23.55/7.61 f_{b,g_1,f_3}(b, g_{a}(y), z) -> f_{a,a,f_3}(a, y, z) 23.55/7.61 f_{b,g_1,a}(b, g_{a}(y), z) -> f_{a,a,a}(a, y, z) 23.55/7.61 f_{b,g_1,g_1}(b, g_{a}(y), z) -> f_{a,a,g_1}(a, y, z) 23.55/7.61 f_{b,g_1,b}(b, g_{a}(y), z) -> f_{a,a,b}(a, y, z) 23.55/7.61 f_{b,g_1,f_3}(b, g_{b}(y), z) -> f_{a,b,f_3}(a, y, z) 23.55/7.61 f_{b,g_1,a}(b, g_{b}(y), z) -> f_{a,b,a}(a, y, z) 23.55/7.61 f_{b,g_1,g_1}(b, g_{b}(y), z) -> f_{a,b,g_1}(a, y, z) 23.55/7.61 f_{b,g_1,b}(b, g_{b}(y), z) -> f_{a,b,b}(a, y, z) 23.55/7.61 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.55/7.61 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.55/7.61 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.55/7.61 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.55/7.61 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.55/7.61 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.55/7.61 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.55/7.61 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.55/7.61 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.55/7.61 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.55/7.61 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.55/7.61 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.55/7.61 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.55/7.61 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.55/7.61 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.55/7.61 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.55/7.61 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.55/7.61 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.55/7.61 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.55/7.61 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.55/7.61 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.55/7.61 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.55/7.61 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.55/7.61 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.55/7.61 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.55/7.61 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.55/7.61 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.55/7.61 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.55/7.61 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.55/7.61 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.55/7.61 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.55/7.61 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.55/7.61 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.55/7.61 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.55/7.61 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.55/7.61 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.55/7.61 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.55/7.61 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.55/7.61 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.55/7.61 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.55/7.61 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.55/7.61 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.55/7.61 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.55/7.61 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.55/7.61 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.55/7.61 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.55/7.61 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.55/7.61 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.55/7.61 g_{a}(a) -> g_{b}(b) 23.55/7.61 23.55/7.61 The relative TRS consists of the following S rules: 23.55/7.61 23.55/7.61 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.55/7.61 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.55/7.61 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.55/7.61 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.55/7.61 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.55/7.61 23.55/7.61 23.55/7.61 ---------------------------------------- 23.55/7.61 23.55/7.61 (7) RelTRSRRRProof (EQUIVALENT) 23.55/7.61 We used the following monotonic ordering for rule removal: 23.55/7.61 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(a) = [[0], [1]] 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{f_3}(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(b) = [[0], [0]] 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{a}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{g_1}(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(g_{b}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{b,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.55/7.61 >>> 23.55/7.61 23.55/7.61 <<< 23.55/7.61 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{a,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{a,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{a,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{a,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{b,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{f_3,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{f_3,b,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.61 POL(f_{f_3,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.61 >>> 23.72/7.61 23.72/7.61 <<< 23.72/7.62 POL(f_{f_3,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 23.72/7.62 f_{a,g_1,f_3}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{f_3}(y), z) -> f_{b,g_1,g_1}(b, g_{f_3}(y), g_{f_3}(y)) 23.72/7.62 Rules from S: 23.72/7.62 none 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (8) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,g_1,f_3}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{b,g_1,f_3}(b, g_{f_3}(y), z) -> f_{a,f_3,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{f_3}(y), z) -> f_{a,f_3,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{f_3}(y), z) -> f_{a,f_3,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{f_3}(y), z) -> f_{a,f_3,b}(a, y, z) 23.72/7.62 f_{b,g_1,f_3}(b, g_{a}(y), z) -> f_{a,a,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{a}(y), z) -> f_{a,a,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{a}(y), z) -> f_{a,a,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{a}(y), z) -> f_{a,a,b}(a, y, z) 23.72/7.62 f_{b,g_1,f_3}(b, g_{b}(y), z) -> f_{a,b,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{b}(y), z) -> f_{a,b,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{b}(y), z) -> f_{a,b,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{b}(y), z) -> f_{a,b,b}(a, y, z) 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.72/7.62 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.72/7.62 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.72/7.62 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.72/7.62 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.72/7.62 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.72/7.62 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.72/7.62 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.72/7.62 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.72/7.62 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.72/7.62 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.72/7.62 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.72/7.62 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.72/7.62 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.72/7.62 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.72/7.62 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.72/7.62 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.72/7.62 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.72/7.62 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.72/7.62 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.72/7.62 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.72/7.62 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.72/7.62 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.72/7.62 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.72/7.62 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.72/7.62 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.72/7.62 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.72/7.62 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.72/7.62 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.72/7.62 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.72/7.62 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.72/7.62 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.72/7.62 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.72/7.62 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.72/7.62 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.72/7.62 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.72/7.62 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.72/7.62 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.72/7.62 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.72/7.62 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.72/7.62 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.72/7.62 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.72/7.62 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.72/7.62 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.72/7.62 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.72/7.62 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.72/7.62 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.72/7.62 g_{a}(a) -> g_{b}(b) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (9) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(a) = [[0], [1]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{a}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(b) = [[0], [1]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{g_1}(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{b}(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{f_3}(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [0, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 23.72/7.62 f_{b,g_1,f_3}(b, g_{f_3}(y), z) -> f_{a,f_3,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{f_3}(y), z) -> f_{a,f_3,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{f_3}(y), z) -> f_{a,f_3,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{f_3}(y), z) -> f_{a,f_3,b}(a, y, z) 23.72/7.62 f_{b,g_1,f_3}(b, g_{a}(y), z) -> f_{a,a,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{a}(y), z) -> f_{a,a,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{a}(y), z) -> f_{a,a,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{a}(y), z) -> f_{a,a,b}(a, y, z) 23.72/7.62 f_{b,g_1,f_3}(b, g_{b}(y), z) -> f_{a,b,f_3}(a, y, z) 23.72/7.62 f_{b,g_1,a}(b, g_{b}(y), z) -> f_{a,b,a}(a, y, z) 23.72/7.62 f_{b,g_1,g_1}(b, g_{b}(y), z) -> f_{a,b,g_1}(a, y, z) 23.72/7.62 f_{b,g_1,b}(b, g_{b}(y), z) -> f_{a,b,b}(a, y, z) 23.72/7.62 Rules from S: 23.72/7.62 none 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (10) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,g_1,f_3}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.72/7.62 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.72/7.62 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.72/7.62 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.72/7.62 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.72/7.62 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.72/7.62 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.72/7.62 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.72/7.62 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.72/7.62 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.72/7.62 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.72/7.62 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.72/7.62 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.72/7.62 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.72/7.62 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.72/7.62 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.72/7.62 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.72/7.62 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.72/7.62 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.72/7.62 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.72/7.62 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.72/7.62 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.72/7.62 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.72/7.62 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.72/7.62 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.72/7.62 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.72/7.62 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.72/7.62 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.72/7.62 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.72/7.62 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.72/7.62 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.72/7.62 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.72/7.62 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.72/7.62 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.72/7.62 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.72/7.62 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.72/7.62 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.72/7.62 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.72/7.62 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.72/7.62 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.72/7.62 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.72/7.62 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.72/7.62 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.72/7.62 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.72/7.62 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.72/7.62 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.72/7.62 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.72/7.62 g_{a}(a) -> g_{b}(b) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (11) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(a) = [[0], [1]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{a}(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [0, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(b) = [[0], [0]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{g_1}(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{b}(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,f_3}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,f_3}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,a}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,b}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{f_3}(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [0, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 23.72/7.62 f_{a,g_1,f_3}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{a}(y), z) -> f_{b,g_1,g_1}(b, g_{a}(y), g_{a}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{g_1}(y), z) -> f_{b,g_1,g_1}(b, g_{g_1}(y), g_{g_1}(y)) 23.72/7.62 f_{a,g_1,f_3}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,a}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,g_1}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 f_{a,g_1,b}(a, g_{b}(y), z) -> f_{b,g_1,g_1}(b, g_{b}(y), g_{b}(y)) 23.72/7.62 Rules from S: 23.72/7.62 none 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (12) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.72/7.62 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.72/7.62 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.72/7.62 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.72/7.62 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.72/7.62 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.72/7.62 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.72/7.62 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.72/7.62 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.72/7.62 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.72/7.62 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.72/7.62 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.72/7.62 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.72/7.62 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.72/7.62 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.72/7.62 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.72/7.62 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.72/7.62 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.72/7.62 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.72/7.62 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.72/7.62 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.72/7.62 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.72/7.62 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.72/7.62 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.72/7.62 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.72/7.62 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.72/7.62 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.72/7.62 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.72/7.62 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.72/7.62 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.72/7.62 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.72/7.62 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.72/7.62 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.72/7.62 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.72/7.62 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.72/7.62 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.72/7.62 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.72/7.62 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.72/7.62 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.72/7.62 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.72/7.62 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.72/7.62 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.72/7.62 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.72/7.62 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.72/7.62 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.72/7.62 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.72/7.62 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.72/7.62 g_{a}(a) -> g_{b}(b) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (13) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Polynomial interpretation [POLO]: 23.72/7.62 23.72/7.62 POL(a) = 1 23.72/7.62 POL(b) = 0 23.72/7.62 POL(f_{a,a,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,a,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,a,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(g_{a}(x_1)) = x_1 23.72/7.62 POL(g_{b}(x_1)) = x_1 23.72/7.62 POL(g_{f_3}(x_1)) = x_1 23.72/7.62 POL(g_{g_1}(x_1)) = x_1 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 23.72/7.62 f_{a,f_3,a}(a, z1, z2) -> f_{b,f_3,a}(b, z1, z2) 23.72/7.62 f_{a,f_3,g_1}(a, z1, z2) -> f_{b,f_3,g_1}(b, z1, z2) 23.72/7.62 f_{a,f_3,b}(a, z1, z2) -> f_{b,f_3,b}(b, z1, z2) 23.72/7.62 f_{a,a,f_3}(a, z1, z2) -> f_{b,a,f_3}(b, z1, z2) 23.72/7.62 f_{a,a,a}(a, z1, z2) -> f_{b,a,a}(b, z1, z2) 23.72/7.62 f_{a,a,g_1}(a, z1, z2) -> f_{b,a,g_1}(b, z1, z2) 23.72/7.62 f_{a,a,b}(a, z1, z2) -> f_{b,a,b}(b, z1, z2) 23.72/7.62 f_{a,g_1,f_3}(a, z1, z2) -> f_{b,g_1,f_3}(b, z1, z2) 23.72/7.62 f_{a,g_1,a}(a, z1, z2) -> f_{b,g_1,a}(b, z1, z2) 23.72/7.62 f_{a,g_1,g_1}(a, z1, z2) -> f_{b,g_1,g_1}(b, z1, z2) 23.72/7.62 f_{a,g_1,b}(a, z1, z2) -> f_{b,g_1,b}(b, z1, z2) 23.72/7.62 f_{a,b,f_3}(a, z1, z2) -> f_{b,b,f_3}(b, z1, z2) 23.72/7.62 f_{a,b,a}(a, z1, z2) -> f_{b,b,a}(b, z1, z2) 23.72/7.62 f_{a,b,g_1}(a, z1, z2) -> f_{b,b,g_1}(b, z1, z2) 23.72/7.62 f_{a,b,b}(a, z1, z2) -> f_{b,b,b}(b, z1, z2) 23.72/7.62 f_{f_3,a,f_3}(z0, a, z2) -> f_{f_3,b,f_3}(z0, b, z2) 23.72/7.62 f_{f_3,a,a}(z0, a, z2) -> f_{f_3,b,a}(z0, b, z2) 23.72/7.62 f_{f_3,a,g_1}(z0, a, z2) -> f_{f_3,b,g_1}(z0, b, z2) 23.72/7.62 f_{f_3,a,b}(z0, a, z2) -> f_{f_3,b,b}(z0, b, z2) 23.72/7.62 f_{a,a,f_3}(z0, a, z2) -> f_{a,b,f_3}(z0, b, z2) 23.72/7.62 f_{a,a,a}(z0, a, z2) -> f_{a,b,a}(z0, b, z2) 23.72/7.62 f_{a,a,g_1}(z0, a, z2) -> f_{a,b,g_1}(z0, b, z2) 23.72/7.62 f_{a,a,b}(z0, a, z2) -> f_{a,b,b}(z0, b, z2) 23.72/7.62 f_{g_1,a,f_3}(z0, a, z2) -> f_{g_1,b,f_3}(z0, b, z2) 23.72/7.62 f_{g_1,a,a}(z0, a, z2) -> f_{g_1,b,a}(z0, b, z2) 23.72/7.62 f_{g_1,a,g_1}(z0, a, z2) -> f_{g_1,b,g_1}(z0, b, z2) 23.72/7.62 f_{g_1,a,b}(z0, a, z2) -> f_{g_1,b,b}(z0, b, z2) 23.72/7.62 f_{b,a,f_3}(z0, a, z2) -> f_{b,b,f_3}(z0, b, z2) 23.72/7.62 f_{b,a,a}(z0, a, z2) -> f_{b,b,a}(z0, b, z2) 23.72/7.62 f_{b,a,g_1}(z0, a, z2) -> f_{b,b,g_1}(z0, b, z2) 23.72/7.62 f_{b,a,b}(z0, a, z2) -> f_{b,b,b}(z0, b, z2) 23.72/7.62 f_{f_3,f_3,a}(z0, z1, a) -> f_{f_3,f_3,b}(z0, z1, b) 23.72/7.62 f_{f_3,a,a}(z0, z1, a) -> f_{f_3,a,b}(z0, z1, b) 23.72/7.62 f_{f_3,g_1,a}(z0, z1, a) -> f_{f_3,g_1,b}(z0, z1, b) 23.72/7.62 f_{f_3,b,a}(z0, z1, a) -> f_{f_3,b,b}(z0, z1, b) 23.72/7.62 f_{a,f_3,a}(z0, z1, a) -> f_{a,f_3,b}(z0, z1, b) 23.72/7.62 f_{a,a,a}(z0, z1, a) -> f_{a,a,b}(z0, z1, b) 23.72/7.62 f_{a,g_1,a}(z0, z1, a) -> f_{a,g_1,b}(z0, z1, b) 23.72/7.62 f_{a,b,a}(z0, z1, a) -> f_{a,b,b}(z0, z1, b) 23.72/7.62 f_{g_1,f_3,a}(z0, z1, a) -> f_{g_1,f_3,b}(z0, z1, b) 23.72/7.62 f_{g_1,a,a}(z0, z1, a) -> f_{g_1,a,b}(z0, z1, b) 23.72/7.62 f_{g_1,g_1,a}(z0, z1, a) -> f_{g_1,g_1,b}(z0, z1, b) 23.72/7.62 f_{g_1,b,a}(z0, z1, a) -> f_{g_1,b,b}(z0, z1, b) 23.72/7.62 f_{b,f_3,a}(z0, z1, a) -> f_{b,f_3,b}(z0, z1, b) 23.72/7.62 f_{b,a,a}(z0, z1, a) -> f_{b,a,b}(z0, z1, b) 23.72/7.62 f_{b,g_1,a}(z0, z1, a) -> f_{b,g_1,b}(z0, z1, b) 23.72/7.62 f_{b,b,a}(z0, z1, a) -> f_{b,b,b}(z0, z1, b) 23.72/7.62 g_{a}(a) -> g_{b}(b) 23.72/7.62 Rules from S: 23.72/7.62 23.72/7.62 f_{f_3,a,f_3}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,a,a}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,a,b}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,f_3}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{f_3,b,a}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,b,b}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,f_3,a}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,f_3,b}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,a,f_3}(x, y, z) -> f_{a,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,a,a}(x, y, z) -> f_{a,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,a,b}(x, y, z) -> f_{a,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,g_1,f_3}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,g_1,a}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,g_1,b}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{a,b,f_3}(x, y, z) -> f_{a,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,b,a}(x, y, z) -> f_{a,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{a,b,b}(x, y, z) -> f_{a,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,f_3}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,a,a}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,a,b}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,f_3}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{g_1,b,a}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,b,b}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,f_3,f_3}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,f_3,a}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,f_3,b}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,a,f_3}(x, y, z) -> f_{b,a,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,a,a}(x, y, z) -> f_{b,a,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,a,b}(x, y, z) -> f_{b,a,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,g_1,f_3}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,g_1,a}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,g_1,b}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{b,b,f_3}(x, y, z) -> f_{b,b,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{b,b,a}(x, y, z) -> f_{b,b,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{b,b,b}(x, y, z) -> f_{b,b,g_1}(x, y, g_{b}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (14) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (15) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(a) = [[1], [1]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(b) = [[1], [1]] 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{a}(x_1)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{g_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{b}(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(g_{f_3}(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 <<< 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 1], [1, 1]] * x_3 23.72/7.62 >>> 23.72/7.62 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 none 23.72/7.62 Rules from S: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{f_3}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (16) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (17) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Polynomial interpretation [POLO]: 23.72/7.62 23.72/7.62 POL(a) = 1 23.72/7.62 POL(b) = 1 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(g_{a}(x_1)) = x_1 23.72/7.62 POL(g_{b}(x_1)) = x_1 23.72/7.62 POL(g_{g_1}(x_1)) = x_1 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 none 23.72/7.62 Rules from S: 23.72/7.62 23.72/7.62 f_{g_1,f_3,a}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (18) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (19) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Polynomial interpretation [POLO]: 23.72/7.62 23.72/7.62 POL(a) = 1 23.72/7.62 POL(b) = 1 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(g_{a}(x_1)) = x_1 23.72/7.62 POL(g_{b}(x_1)) = x_1 23.72/7.62 POL(g_{g_1}(x_1)) = x_1 23.72/7.62 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.62 Rules from R: 23.72/7.62 none 23.72/7.62 Rules from S: 23.72/7.62 23.72/7.62 f_{f_3,f_3,b}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (20) 23.72/7.62 Obligation: 23.72/7.62 Relative term rewrite system: 23.72/7.62 The relative TRS consists of the following R rules: 23.72/7.62 23.72/7.62 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.62 23.72/7.62 The relative TRS consists of the following S rules: 23.72/7.62 23.72/7.62 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.62 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.62 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.62 23.72/7.62 23.72/7.62 ---------------------------------------- 23.72/7.62 23.72/7.62 (21) RelTRSRRRProof (EQUIVALENT) 23.72/7.62 We used the following monotonic ordering for rule removal: 23.72/7.62 Polynomial interpretation [POLO]: 23.72/7.62 23.72/7.62 POL(a) = 1 23.72/7.62 POL(b) = 1 23.72/7.62 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.62 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.62 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = 1 + x_1 23.72/7.63 POL(g_{b}(x_1)) = x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{f_3,g_1,b}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (22) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (23) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = 1 + x_1 23.72/7.63 POL(g_{b}(x_1)) = x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{g_1,g_1,b}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{b}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (24) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (25) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = x_1 23.72/7.63 POL(g_{b}(x_1)) = 1 + x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{g_1,g_1,a}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (26) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (27) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,b}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = 1 + x_1 23.72/7.63 POL(g_{b}(x_1)) = x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{g_1,f_3,b}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{b}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (28) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (29) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{f_3,g_1,a}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{a}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (30) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (31) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,a}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{a}(x_1)) = x_1 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 none 23.72/7.63 Rules from S: 23.72/7.63 23.72/7.63 f_{f_3,f_3,a}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{a}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (32) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 The relative TRS consists of the following R rules: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (33) RelTRSRRRProof (EQUIVALENT) 23.72/7.63 We used the following monotonic ordering for rule removal: 23.72/7.63 Polynomial interpretation [POLO]: 23.72/7.63 23.72/7.63 POL(a) = 1 23.72/7.63 POL(b) = 1 23.72/7.63 POL(f_{a,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,f_3}(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{a,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,f_3}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{b,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{f_3,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,a,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,b,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,f_3,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(f_{g_1,g_1,g_1}(x_1, x_2, x_3)) = x_1 + x_2 + x_3 23.72/7.63 POL(g_{g_1}(x_1)) = x_1 23.72/7.63 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 23.72/7.63 Rules from R: 23.72/7.63 23.72/7.63 f_{a,f_3,f_3}(a, z1, z2) -> f_{b,f_3,f_3}(b, z1, z2) 23.72/7.63 Rules from S: 23.72/7.63 none 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (34) 23.72/7.63 Obligation: 23.72/7.63 Relative term rewrite system: 23.72/7.63 R is empty. 23.72/7.63 The relative TRS consists of the following S rules: 23.72/7.63 23.72/7.63 f_{f_3,f_3,g_1}(x, y, z) -> f_{f_3,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,a,g_1}(x, y, z) -> f_{f_3,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,g_1,g_1}(x, y, z) -> f_{f_3,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{f_3,b,g_1}(x, y, z) -> f_{f_3,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,f_3,g_1}(x, y, z) -> f_{a,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,a,g_1}(x, y, z) -> f_{a,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,g_1,g_1}(x, y, z) -> f_{a,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{a,b,g_1}(x, y, z) -> f_{a,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,f_3,g_1}(x, y, z) -> f_{g_1,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,a,g_1}(x, y, z) -> f_{g_1,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,g_1,g_1}(x, y, z) -> f_{g_1,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{g_1,b,g_1}(x, y, z) -> f_{g_1,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,f_3,g_1}(x, y, z) -> f_{b,f_3,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,a,g_1}(x, y, z) -> f_{b,a,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,g_1,g_1}(x, y, z) -> f_{b,g_1,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 f_{b,b,g_1}(x, y, z) -> f_{b,b,g_1}(x, y, g_{g_1}(z)) 23.72/7.63 23.72/7.63 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (35) RIsEmptyProof (EQUIVALENT) 23.72/7.63 The TRS R is empty. Hence, termination is trivially proven. 23.72/7.63 ---------------------------------------- 23.72/7.63 23.72/7.63 (36) 23.72/7.63 YES 23.77/7.67 EOF